ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 2 Issue 4 - 2015
Second Order Optimality of Sequential Designs with Application in Software Reliability Estimation
Kamel Rekab* and Xing Song
Department of Mathematics and Statistics, University of Missouri-Kansas City, USA
Received: April 13, 2015 | Published: April 27, 2015
*Corresponding author: Kamel Rekab, Department of Mathematics and Statistics, University of Missouri-Kansas City, PO Box 32464 Kansas City, MO 64171, USA, Tel: 816-269-4432; Email:
Citation: Rekab K, Song X (2014) Second Order Optimality of Sequential Designs with Application in Software Reliability Estimation. Biom Biostat Int J 2(4): 00037. DOI: 10.15406/bbij.2015.02.00037

Abstract

We propose three efficient sequential designs in the software reliability estimation. The fully sequential design the multistage sequential design and the accelerated sequential design. These designs make allocation decisions dynamically throughout the testing process. We then refine these estimated reliabilities in an iterative manner as we sample. Monte Carlo simulation seems to indicate that these sequential designs are second order optimal.

Keywords: Software reliability, Partition testing, Fully sequential design, Multistage sequential design, Accelerated sequential design.

Introduction

Reliability of a system is an important aspect of any system design since any user of the system would expect some type of guarantee that the system will function to some level of confidence. Failing to meet such guarantee will result in disastrous consequences. On the other hand, overly exceeding such guarantee level may incur additional and unnecessary expense to the developers. Moreover, for any non-trivial software system, an exhaustive testing among the entire input domain can be very expensive. By adopting the partition testing strategy, we attempt to break up the testable input domain of possible test cases into partitions, which must be non-overlapping, such that if test case i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyAaa aa@3768@ belongs to partition j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOAaa aa@3769@ , then no partition other than j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOAaa aa@3769@ will contain i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyAaa aa@3768@ . Sayre and Poore[11] have given several possible mechanics to partition the domain into finitely many subdomains, X ij ={ 1,if testjtaken from partitioniis processed correctly 0,otherwise MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGybqcfa4damaaBaaajeaibaqcLbmapeGaamyAaiaadQga aSWdaeqaaKqzGeWdbiabg2da9Kqbaoaaceaak8aabaqcLbsafaqabe Gabaaakeaajugib8qacaaIXaGaaiilaiaabMgacaqGMbGaaeiiaiaa bshacaqGLbGaae4CaiaabshacaWGQbGaaeiDaiaabggacaqGRbGaae yzaiaab6gacaqGGaGaaeOzaiaabkhacaqGVbGaaeyBaiaabccacaqG WbGaaeyyaiaabkhacaqG0bGaaeyAaiaabshacaqGPbGaae4Baiaab6 gacaWGPbGaaeyAaiaabohacaqGGaGaaeiCaiaabkhacaqGVbGaae4y aiaabwgacaqGZbGaae4CaiaabwgacaqGKbGaaeiiaiaabogacaqGVb GaaeOCaiaabkhacaqGLbGaae4yaiaabshacaqGSbGaaeyEaaGcpaqa aKqzGeWdbiaaicdacaGGSaGaae4BaiaabshacaqGObGaaeyzaiaabk hacaqG3bGaaeyAaiaabohacaqGLbaaaaGccaGL7baaaaa@7A87@ , such that :
D= i=1 k D i ; D i D j =,ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGebGaeyypa0tcfa4damaawahakeqajeaibaqcLbmacaWG PbGaeyypa0JaaGymaaqcbasaaKqzadGaam4AaaqdbaGaeSOkIufaaK qzGeWdbiaadseajuaGpaWaaSbaaKqaGeaajugWa8qacaWGPbaal8aa beaajugib8qacaGG7aGaamiraKqba+aadaWgaaqcbasaaKqzadWdbi aadMgaaSWdaeqaaKqzGeGaeyykIC8dbiaadseajuaGpaWaaSbaaKqa GeaajugWa8qacaWGQbaal8aabeaajugib8qacqGH9aqpcqGHfiIXca GGSaGaamyAaiabgcMi5kaadQgaaaa@58BE@ which allows us to define the system reliability by a weighted sum of reliabilities of these  subdomains, i.e.
R= i=1 k p i R i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGsbGaeyypa0tcfa4aaybCaOqabKqaG8aabaqcLbmacaWG PbGaeyypa0JaaGymaaqcbasaaKqzadWdbiaadUgaaKWaG=aabaqcLb EacqGHris5aaqcLbsapeGaamiCaKqba+aadaWgaaqcbasaaKqzadWd biaadMgaaSWdaeqaaKqzGeWdbiaadkfajuaGpaWaaSbaaKqaGeaaju gWa8qacaWGPbaal8aabeaaaaa@4D60@ Where R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaa aa@3752@ denotes the system reliability R i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaK qbaoaaBaaajuaibaqcLbmacaWGPbaajuaGbeaaaaa@3AD9@ and is the reliability of each subdomain D i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiraK qbaoaaBaaajuaibaqcLbmacaWGPbaajuaGbeaaaaa@3ACB@ ; and p i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiCaK qbaoaaBaaajuaibaqcLbmacaWGPbaajuaGbeaaaaa@3AF7@ , parameters of the operational profile is the likelihood of this test case belongs to partition D i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiraK qbaoaaBaaajuaibaqcLbmacaWGPbaajuaGbeaaaaa@3ACB@ , which are assumed to be known [12].  As mentioned above, a complete testing of any software system of non-trivial size is practically impossible, R i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaK qbaoaaBaaajuaibaqcLbmacaWGPbaajuaGbeaaaaa@3AD9@  are usually unknown parameters to us. So as to gain knowledge about R i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaK qbaoaaBaaajuaibaqcLbmacaWGPbaajuaGbeaaaaa@3AD9@ , we must distribute the k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4Aaa aa@376A@ test cases among these k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4Aaa aa@376A@ partitions, and generate reasonable estimates for each. Specifically, we denote n 1 , n 2 ,, n k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGUbqcfa4damaaBaaajeaibaqcLbmapeGaaGymaaWcpaqa baqcLbsapeGaaiilaiaad6gajuaGpaWaaSbaaKqaGeaajugWa8qaca aIYaaal8aabeaajugib8qacaGGSaGaeyOjGWRaaiilaiaad6gajuaG paWaaSbaaKqaGeaajugWa8qacaWGRbaal8aabeaaaaa@4777@ as sizes of the samples which are taken from sub domain D 1 , D 2 ,,' D k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGebqcfa4damaaBaaajeaibaqcLbmapeGaaGymaaWcpaqa baqcLbsapeGaaiilaiaadseajuaGpaWaaSbaaKqaGeaajugWa8qaca aIYaaal8aabeaajugib8qacaGGSaGaeyOjGWRaaiilaiaacEcacaWG ebqcfa4damaaBaaajeaibaqcLbmapeGaam4AaaWcpaqabaaaaa@47A4@

, respectively, where i=1 k n i =N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyyeIu Ecfa4aa0baaKqaGeaajugWaiaadMgacqGH9aqpcaaIXaaajeaibaqc LbmacaWGRbaaaKqzGeGaamOBaKqbaoaaBaaajeaibaqcLbmacaWGPb aaleqaaKqzGeGaeyypa0JaamOtaaaa@4607@ .

We model the outcome of the j th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOAaK qbaoaaCaaaleqajeaibaqcLbmacaWG0bGaamiAaaaaaaa@3B62@ taken from the i th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyAaK qbaoaaCaaaleqajeaibaqcLbmacaWG0bGaamiAaaaaaaa@3B61@ partition as a Bernoulli random variable X ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiwaK qbaoaaBaaajeaibaqcLbmacaWGPbGaamOAaaWcbeaaaaa@3B46@ such that:

X ij ={ 1,if testjtaken from partitioniis processed correctly 0,otherwise MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGybqcfa4damaaBaaajeaibaqcLbmapeGaamyAaiaadQga aSWdaeqaaKqzGeWdbiabg2da9Kqbaoaaceaak8aabaqcLbsafaqabe Gabaaakeaajugib8qacaaIXaGaaiilaiaabMgacaqGMbGaaeiiaiaa bshacaqGLbGaae4CaiaabshacaWGQbGaaeiDaiaabggacaqGRbGaae yzaiaab6gacaqGGaGaaeOzaiaabkhacaqGVbGaaeyBaiaabccacaqG WbGaaeyyaiaabkhacaqG0bGaaeyAaiaabshacaqGPbGaae4Baiaab6 gacaWGPbGaaeyAaiaabohacaqGGaGaaeiCaiaabkhacaqGVbGaae4y aiaabwgacaqGZbGaae4CaiaabwgacaqGKbGaaeiiaiaabogacaqGVb GaaeOCaiaabkhacaqGLbGaae4yaiaabshacaqGSbGaaeyEaaGcpaqa aKqzGeWdbiaaicdacaGGSaGaae4BaiaabshacaqGObGaaeyzaiaabk hacaqG3bGaaeyAaiaabohacaqGLbaaaaGccaGL7baaaaa@7A87@

and each X ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiwaK qbaoaaBaaajeaibaqcLbmacaWGPbGaamOAaaWcbeaaaaa@3B46@ follows a Bernoulli distribution with parameter R i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaK qbaoaaBaaajuaibaqcLbmacaWGPbaajuaGbeaaaaa@3AD9@ . Then, the estimate of the overall system reliability, R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaa aa@3752@ denoted by R ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOuay aajaaaaa@3761@ can thus be defined as:

R ^ = i=1 k p i R ^ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiqadkfagaqcaiabg2da9KqbaoaaxadakeaajugibiabgQYi XdqcbasaaKqzadGaamyAaiabg2da9iaaigdaaKqaGeaajugWaiaadU gaaaqcLbsacaWGWbqcfa4damaaBaaaleaajugib8qacaWGPbaal8aa beaajugibiqadkfagaqcaKqbaoaaBaaaleaajugib8qacaWGPbaal8 aabeaaaaa@49EA@ where R ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGsb GbaKaalmaaBaaajeaibaqcLbmacaWGPbaajeaibeaaaaa@3A09@  is the estimate of R i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb WcdaWgaaqcbasaaKqzadGaamyAaaqcbasabaaaaa@39F9@  after n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb WcdaWgaaqcbasaaKqzadGaamyAaaqcbasabaaaaa@3A15@  test cases have been allocated to partition such that:
R ^ i = j=1 n i X ij n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGsb GbaKaalmaaBaaajqwaa+FaaKqzadaeaaaaaaaaa8qacaWGPbaajqwa a+=daeqaaKqzGeWdbiabg2da9KqbaoaalaaabaWaaabmaeaajugibi aadIfalmaaBaaajuaibaqcLbmacaWGPbGaamOAaaqcfayabaaajuai baqcLbmacaWGQbGaeyypa0JaaGymaaqcfayaaKqzadGaamOBaKqbao aaBaaajuaibaqcLbmacaWGPbaajuaGbeaaaKqzGeGaeyyeIuoaaKqb agaajugibiaad6galmaaBaaajuaibaqcLbmacaWGPbaajuaibeaaaa aaaa@56CF@ and Var( R ^ )= i=1 k p i 2 R i ( 1 R i ) n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadAfacaWGHbGaamOCaKqbaoaabmaak8aabaqcLbsaceWG sbGbaKaaaOWdbiaawIcacaGLPaaajugibiabg2da9Kqba+aadaGfWb GcbeqcKfaG=haajugWaiaadMgacqGH9aqpcaaIXaaajqwaa+FaaKqz adGaam4AaaqcKnaOaeaajugWbiabggHiLdaajuaGpeWaaSaaaOWdae aajugib8qacaWGWbqcfa4damaaDaaajeaibaqcLbmapeGaamyAaaqc baYdaeaajugWa8qacaaIYaaaaKqzGeGaamOuaSWdamaaBaaajeaiba qcLbmapeGaamyAaaqcbaYdaeqaaKqba+qadaqadaGcpaqaaKqzGeWd biaaigdacqGHsislcaWGsbWcpaWaaSbaaKqaGeaajugWa8qacaWGPb aajeaipaqabaaak8qacaGLOaGaayzkaaaapaqaaKqzGeWdbiaad6ga l8aadaWgaaqcbasaaKqzadWdbiaadMgaaKqaG8aabeaaaaaaaa@661F@                                           (1.1)

Optimal Sampling Scheme

Ideally, we would like to execute all possible test paths through the software and determine the true overall reliability of the system. In practice though, resources are often limited, sample test cases must be chosen and allocated strategically to attain the best reliability estimate possible given all kinds of constraints. One of the criteria of distributing test cases among the  partitions, which proceeds from rewriting (1.1) as follows:

Var( R ^ )= [ i=1 k p i R i ( 1 R i ) ] 2 N + 1 N i=1 k1 j=i+1 k [ n i p j R j ( 1 R j ) n j p i R i ( 1 R i ) ] 2 n i n j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaamyyaiaadkhajuaGdaqadaGcbaqcLbsaceWGsbGbaKaaaOGaayjk aiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqbaoaadmaakeaaju aGdaqfWaGcbeqcbasaaKqzadGaamyAaiabg2da9iaaigdaaKqaGeaa jugWaiaadUgaaKazdamabaqcLbEacqGHris5aaqcLbsacaWGWbWcda WgaaqcbasaaKqzadGaamyAaaqcbasabaqcfa4aaOaaaOqaaKqzGeGa amOuaSWaaSbaaKqaGeaajugWaiaadMgaaKqaGeqaaKqbaoaabmaake aajugibiaaigdacqGHsislcaWGsbWcdaWgaaqcbasaaKqzadGaamyA aaqcbasabaaakiaawIcacaGLPaaaaSqabaaakiaawUfacaGLDbaaju aGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaOqaaKqzGeGaamOtaaaa cqGHRaWkjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaad6eaaa qcfa4aaybCaOqabKqaGeaajugWaiaadMgacqGH9aqpcaaIXaaajeai baqcLbmacaWGRbGaeyOeI0IaaGymaaqcKnaWaeaajug4biabggHiLd aajuaGdaGfWbGcbeqcbasaaKqzadGaamOAaiabg2da9iaadMgacqGH RaWkcaaIXaaajeaibaqcLbmacaWGRbaajq2aadqaaKqzGhGaeyyeIu oaaKqbaoaalaaakeaajuaGdaWadaGcbaqcLbsacaWGUbqcfa4aaSba aKqaGeaajugWaiaadMgaaSqabaqcLbsacaWGWbqcfa4aaSbaaKqaGe aajugWaiaadQgaaSqabaqcfa4aaOaaaOqaaKqzGeGaamOuaKqbaoaa BaaajeaibaqcLbmacaWGQbaaleqaaKqbaoaabmaakeaajugibiaaig dacqGHsislcaWGsbqcfa4aaSbaaKqaGeaajugWaiaadQgaaSqabaaa kiaawIcacaGLPaaaaSqabaqcLbsacqGHsislcaWGUbqcfa4aaSbaaK qaGeaajugWaiaadQgaaSqabaqcLbsacaWGWbqcfa4aaSbaaKqaGeaa jugWaiaadMgaaSqabaqcfa4aaOaaaOqaaKqzGeGaamOuaKqbaoaaBa aajeaibaqcLbmacaWGPbaaleqaaKqbaoaabmaakeaajugibiaaigda cqGHsislcaWGsbqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaaki aawIcacaGLPaaaaSqabaaakiaawUfacaGLDbaajuaGdaahaaWcbeqc basaaKqzadGaaGOmaaaaaOqaaKqzGeGaamOBaKqbaoaaBaaajeaiba qcLbmacaWGPbaaleqaaKqzGeGaamOBaKqbaoaaBaaajeaibaqcLbma caWGQbaaleqaaaaaaaa@BD90@

which is bounded below by the first term:

Var( R ^ ) [ i=1 k p i R i ( 1 R i ) ] 2 N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaamyyaiaadkhajuaGdaqadaGcbaqcLbsaceWGsbGbaKaaaOGaayjk aiaawMcaaKqzGeGaeyyzImBcfa4aaSaaaOqaaKqbaoaadmaakeaaju aGdaqfWaGcbeqcbasaaKqzadGaamyAaiabg2da9iaaigdaaKqaGeaa jugWaiaadUgaaKazdakabaqcLbCacqGHris5aaqcLbsacaWGWbqcfa 4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcfa4aaOaaaOqaaKqzGeGa amOuaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqbaoaabmaake aajugibiaaigdacqGHsislcaWGsbqcfa4aaSbaaKqaGeaajugWaiaa dMgaaSqabaaakiaawIcacaGLPaaaaSqabaaakiaawUfacaGLDbaaju aGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaOqaaKqzGeGaamOtaaaa aaa@6397@

with equality of (2.1) to hold is and only if:

n i n j = p i R i ( 1 R i ) p j R j ( 1 R i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaGcpaqaaKqzGeWdbiaad6gajuaGpaWaaSbaaKqaGeaa jugWa8qacaWGPbaal8aabeaaaOqaaKqzGeWdbiaad6gajuaGpaWaaS baaKqaGeaajugWa8qacaWGQbaal8aabeaaaaqcLbsapeGaeyypa0tc fa4aaSaaaOWdaeaajugib8qacaWGWbqcfa4damaaBaaajeaibaqcLb mapeGaamyAaaWcpaqabaqcfa4dbmaakaaak8aabaqcLbsapeGaamOu aKqba+aadaWgaaqcbasaaKqzadWdbiaadMgaaSWdaeqaaKqba+qada qadaGcpaqaaKqzGeWdbiaaigdacqGHsislcaWGsbqcfa4damaaBaaa jeaibaqcLbmapeGaamyAaaWcpaqabaaak8qacaGLOaGaayzkaaaale qaaaGcpaqaaKqzGeWdbiaadchajuaGpaWaaSbaaKqaGeaajugWa8qa caWGQbaal8aabeaajuaGpeWaaOaaaOWdaeaajugib8qacaWGsbqcfa 4damaaBaaajeaibaqcLbmapeGaamOAaaWcpaqabaqcfa4dbmaabmaa k8aabaqcLbsapeGaaGymaiabgkHiTiaadkfajuaGpaWaaSbaaKqaGe aajugWa8qacaWGPbaal8aabeaaaOWdbiaawIcacaGLPaaaaSqabaaa aaaa@688C@

for all . Hence, the optimal allocation is determined by:

n i N = p i R i ( 1 R i ) j=1 k p j R j ( 1 R i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamOBaSWaaSbaaKqaGeaajugWaiaadMgaaKqaGeqaaaGc baqcLbsacaWGobaaaiabg2da9Kqbaoaalaaakeaajugibiaadchaju aGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajuaGdaGcaaGcbaqcLbsa caWGsbqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcfa4aaeWaaO qaaKqzGeGaaGymaiabgkHiTiaadkfajuaGdaWgaaqcbasaaKqzadGa amyAaaWcbeaaaOGaayjkaiaawMcaaaWcbeaaaOqaaKqbaoaavadake qajeaibaqcLbmacaWGQbGaeyypa0JaaGymaaqcbasaaKqzadGaam4A aaqcdaEaaKqzagGaeyyeIuoaaKqzGeGaamiCaKqbaoaaBaaajeaiba qcLbmacaWGQbaaleqaaKqbaoaakaaakeaajugibiaadkfajuaGdaWg aaqcbasaaKqzadGaamOAaaWcbeaajuaGdaqadaGcbaqcLbsacaaIXa GaeyOeI0IaamOuaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaaGc caGLOaGaayzkaaaaleqaaaaaaaa@6C43@

for  i=1,2,,k1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaacckacaWGPbGaeyypa0JaaGymaiaacYcacaaIYaGaaiil aiabgAci8kaacYcacaWGRbGaeyOeI0IaaGymaaaa@416A@ , and

n k =N i=1 k1 n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6gajuaGpaWaaSbaaKqaGeaajugWa8qacaWGRbaal8aa beaajugib8qacqGH9aqpcaWGobGaeyOeI0scfa4aaybCaOqabKqaG8 aabaqcLbmacaWGPbGaeyypa0JaaGymaaqcbasaaKqzadWdbiaadUga cqGHsislcaaIXaaajq2aaeWdaeaajugiciabgQYiXdaajugib8qaca WGUbqcfa4damaaBaaajeaibaqcLbmapeGaamyAaaWcpaqabaaaaa@4F04@

and the variance incurred by this optimal allocation is:

Var( O )= [ j=1 k p j R j ( 1 R i ) ] 2 N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadAfacaWGHbGaamOCaKqbaoaabmaak8aabaqcLbsapeGa am4taaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcpaqaaK qba+qadaWadaGcpaqaaKqba+qadaqfWaGcbeqcbaYdaeaajugWaiaa dQgacqGH9aqpcaaIXaaajeaibaqcLbmapeGaam4Aaaqcda+daeaaju gGb8qacqGHris5aaqcLbsacaWGWbqcfa4damaaBaaajeaibaqcLbma peGaamOAaaWcpaqabaqcfa4dbmaakaaak8aabaqcLbsapeGaamOuaK qba+aadaWgaaqcbasaaKqzadWdbiaadQgaaSWdaeqaaKqba+qadaqa daGcpaqaaKqzGeWdbiaaigdacqGHsislcaWGsbqcfa4damaaBaaaje aibaqcLbmapeGaamyAaaWcpaqabaaak8qacaGLOaGaayzkaaaaleqa aaGccaGLBbGaayzxaaqcfa4damaaCaaaleqajeaibaqcLbmapeGaaG OmaaaaaOWdaeaajugib8qacaWGobaaaaaa@6455@

Note that the optimal allocation depends on the actual reliabilities R 1 , R 2 ,, R k , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadkfajuaGpaWaaSbaaKqaGeaajugWa8qacaaIXaaal8aa beaajugib8qacaGGSaGaamOuaKqba+aadaWgaaqcbasaaKqzadWdbi aaikdaaSWdaeqaaKqzGeWdbiaacYcacqGHMacVcaGGSaGaamOuaKqb a+aadaWgaaqcbasaaKqzadWdbiaadUgaaSWdaeqaaKqzGeWdbiaacY caaaa@487C@ which are unknown. Therefore the optimal sampling scheme is not practical.  It is this shortcoming that motivates us to adopt such dynamic allocation approaches that will be discussed in the following three sections.

Fully Sequential Sampling Scheme

By adopting a fully Bayesian approach with Beta priors, Rekab, Thompson and Wei [5] proposed a fully sequential design shown to be first order optimal.  The fully sequential design is defined as follows;

We first test one case from each of the  partitions and estimate the reliability for each of these partitions.

Stage 1:

After  cases have been allocated, lk, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadYgacqGHLjYScaWGRbGaaiilaaaa@3AFC@ the next test will be taken from partition i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMgaaaa@3793@ if for all j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadQgaaaa@3794@

n l,i n l,j < p i R ^ l,i ( 1 R ^ l,i ) p j R ^ l,j ( 1 R ^ l,j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaGcpaqaaKqzGeWdbiaad6gajuaGpaWaaSbaaKqaGeaa jugWa8qacaWGSbGaaiilaiaadMgaaSWdaeqaaaGcbaqcLbsapeGaam OBaKqba+aadaWgaaqcbasaaKqzadWdbiaadYgacaGGSaGaamOAaaWc paqabaaaaKqzGeWdbiabgYda8Kqbaoaalaaak8aabaqcLbsapeGaam iCaKqba+aadaWgaaqcbasaaKqzadWdbiaadMgaaSWdaeqaaKqba+qa daGcaaGcpaqaaKqzGeGabmOuayaajaqcfa4aaSbaaKqaGeaajugWa8 qacaWGSbGaaiilaiaadMgaaSWdaeqaaKqba+qadaqadaGcpaqaaKqz GeWdbiaaigdacqGHsislceWGsbGbaKaajuaGpaWaaSbaaKqaGeaaju gWa8qacaWGSbGaaiilaiaadMgaaSWdaeqaaaGcpeGaayjkaiaawMca aaWcbeaaaOWdaeaajugib8qacaWGWbqcfa4damaaBaaajeaibaqcLb mapeGaamOAaaWcpaqabaqcfa4dbmaakaaak8aabaqcLbsaceWGsbGb aKaajuaGdaWgaaqcbasaaKqzadWdbiaadYgacaGGSaGaamOAaaWcpa qabaqcfa4dbmaabmaak8aabaqcLbsapeGaaGymaiabgkHiTiqadkfa gaqcaKqba+aadaWgaaqcbasaaKqzadWdbiaadYgacaGGSaGaamOAaa Wcpaqabaaak8qacaGLOaGaayzkaaaaleqaaaaaaaa@7253@

where n l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6gajuaGpaWaaSbaaKazba4=baqcLbmapeGaamiBaiaa cYcacaWGPbaal8aabeaaaaa@3E2B@ is the cumulative test cases allocated to partition i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMgaaaa@3794@ after l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadYgaaaa@3797@ tests have been allocated and the current  estimated reliability for partition i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMgaaaa@3794@ is determined by:

R ^ l,i = m=1 n l,i X im n l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGsb GbaKaajuaGdaWgaaqcbasaaKqzadaeaaaaaaaaa8qacaWGSbGaaiil aiaadMgaaSWdaeqaaKqzGeWdbiabg2da9Kqbaoaalaaak8aabaqcfa 4dbmaavadakeqajeaipaqaaKqzadGaamyBaiabg2da9iaaigdaaKqa GeaajugWa8qacaWGUbWcdaWgaaqccasaaKqzadGaamiBaiaacYcaca WGPbaajiaibeaaaKWaG=aabaqcLbyapeGaeyyeIuoaaKqzGeGaamiw aKqba+aadaWgaaqcbasaaKqzadWdbiaadMgacaWGTbaal8aabeaaaO qaaKqzGeWdbiaad6gajuaGpaWaaSbaaKqaGeaajugWa8qacaWGSbGa aiilaiaadMgaaSWdaeqaaaaaaaa@59D7@

Stage 2:

Repeat step 2 sequentially until all the  test cases are allocated, and the final estimate of reliability for partition  is:

R ^ i = m=1 n N,i X im n N,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGsb GbaKaajuaGdaWgaaqcbasaaKqzadaeaaaaaaaaa8qacaWGPbaal8aa beaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqba+qadaqfWaGcbe qcbaYdaeaajugWaiaad2gacqGH9aqpcaaIXaaajeaibaqcLbmapeGa amOBaSWaaSbaaKGaGeaajugWaiaad6eacaGGSaGaamyAaaqccasaba aajq2aacWdaeaajugOb8qacqGHris5aaqcLbsacaWGybqcfa4damaa BaaajeaibaqcLbmapeGaamyAaiaad2gaaSWdaeqaaaGcbaqcLbsape GaamOBaKqba+aadaWgaaqcbasaaKqzadWdbiaad6eacaGGSaGaamyA aaWcpaqabaaaaaaa@581E@

And thus, the estimate of the overall reliability of the system is:

R ^ = i=1 k p i R ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiqadkfagaqcaiabg2da9KqbaoaawahakeqajeaipaqaaKqz adGaamyAaiabg2da9iaaigdaaKqaGeaajugWa8qacaWGRbaajq2aGc WdaeaajugWb8qacqGHQms8aaqcLbsacaWGWbqcfa4damaaBaaajeai baqcLbmapeGaamyAaaWcpaqabaqcLbsaceWGsbGbaKaajuaGdaWgaa qcbasaaKqzadWdbiaadMgaaSWdaeqaaaaa@4D96@

Multistage Sequential Sampling

By adopting a fully Bayesian approach with Beta priors, Rekab, Thompson and Wei [6] proposed a multistage sequential design shown to be first order optimalInstead of making an allocation decision for each test at a time, the multistage sequential sampling allocates  test cases among the partitions in  stages in batches, where  and   are both pre-specified. The multistage sequential design is defined as follows:

Stage 1:

We start with an initial sample of  test cases, which are allocated among the  partitions with a balanced allocation scheme, such that:

S 1,i = S 1 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadofajuaGpaWaaSbaaKqaGeaajugWa8qacaaIXaGaaiil aiaadMgaaSWdaeqaaKqzGeWdbiabg2da9Kqbaoaalaaak8aabaqcLb sapeGaam4uaKqba+aadaWgaaqcbasaaKqzadWdbiaaigdaaSWdaeqa aaGcbaqcLbsapeGaam4Aaaaaaaa@447E@

and estimate the reliability for partition  by:

R ^ 1,i = m=1 s 1,i X im S 1,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGsb GbaKaajuaGdaWgaaqcbasaaKqzadaeaaaaaaaaa8qacaaIXaGaaiil aiaadMgaaSWdaeqaaKqzGeWdbiabg2da9Kqbaoaalaaak8aabaqcfa 4dbmaavadakeqajeaipaqaaKqzadGaamyBaiabg2da9iaaigdaaKqa GeaajugWa8qacaWGZbWcdaWgaaqccasaaKqzadGaaGymaiaacYcaca WGPbaajiaibeaaaKWaG=aabaqcLbyapeGaeyyeIuoaaKqzGeGaamiw aKqba+aadaWgaaqcbasaaKqzadWdbiaadMgacaWGTbaal8aabeaaaO qaaKqzGeWdbiaadofajuaGpaWaaSbaaKqaGeaajugWa8qacaaIXaGa aiilaiaadMgaaSWdaeqaaaaaaaa@591F@

Therefore,

C ^ i ( S 1 )= p i R ^ 1,i ( 1 R ^ 1,i ) j=1 k p j R ^ 1,j ( 1 R ^ 1,j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGdb GbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajuaGdaqadaGc baqcLbsacaWGtbqcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaaaki aawIcacaGLPaaajugibiabg2da9Kqbaoaalaaakeaajugibiaadcha juaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajuaGdaGcaaGcbaqcLb saceWGsbGbaKaajuaGdaWgaaqcbasaaKqzadGaaGymaiaacYcacaWG PbaaleqaaKqbaoaabmaakeaajugibiaaigdacqGHsislceWGsbGbaK aajuaGdaWgaaqcbasaaKqzadGaaGymaiaacYcacaWGPbaaleqaaaGc caGLOaGaayzkaaaaleqaaaGcbaqcfa4aaubmaOqabKqaGeaajugWai aadQgacqGH9aqpcaaIXaaajeaibaqcLbmacaWGRbaaneaajugibiab ggHiLdaacaWGWbqcfa4aaSbaaKqaGeaajugWaiaadQgaaSqabaqcfa 4aaOaaaOqaaKqzGeGabmOuayaajaqcfa4aaSbaaKqaGeaajugWaiaa igdacaGGSaGaamOAaaWcbeaajuaGdaqadaGcbaqcLbsacaaIXaGaey OeI0IabmOuayaajaqcfa4aaSbaaKqaGeaajugWaiaaigdacaGGSaGa amOAaaWcbeaaaOGaayjkaiaawMcaaaWcbeaaaaaaaa@7585@

Stage 2 through L:

At stage j, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGQb Gaaiilaaaa@3824@ for partition the test cases are distributed by the following way:

2jL, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaaikdacqGHKjYOcaWGQbGaeyizImQaamitaiaacYcaaaa@3D3C@ for partition i=1,2,,k1, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMgacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaeyOj GWRaaiilaiaadUgacqGHsislcaaIXaGaaiilaaaa@40F7@

the test cases are distributed by the following way:

S j,i =( l=1 j S l ) C ^ i ( S ¯ j1 ); MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadofajuaGpaWaaSbaaKqaGeaajugWa8qacaWGQbGaaiil aiaadMgaaSWdaeqaaKqzGeWdbiabg2da9Kqbaoaabmaak8aabaqcfa 4dbmaawahakeqajeaipaqaaKqzadGaamiBaiabg2da9iaaigdaaSqa aKqzGeWdbiaadQgaaKazdakapaqaaKqzahWdbiabgQYiXdaajugibi aadofajuaGpaWaaSbaaKqaGeaajugWa8qacaWGSbaal8aabeaaaOWd biaawIcacaGLPaaajugibiqadoeagaqcaKqba+aadaWgaaqcbasaaK qzadWdbiaadMgaaSWdaeqaaKqba+qadaqadaGcpaqaaKqzGeGabm4u ayaaraqcfa4aaSbaaKqaGeaajugWa8qacaWGQbGaeyOeI0IaaGymaa Wcpaqabaaak8qacaGLOaGaayzkaaqcLbsacaGG7aaaaa@5E65@

and

S j,k = l=1 j S l i=1 k1 S j,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadofajuaGpaWaaSbaaKqbGeaajugWa8qacaWGQbGaaiil aiaadUgaaKqba+aabeaajugib8qacqGH9aqpjuaGdaGfWbqabKqbG8 aabaqcLbmacaWGSbGaeyypa0JaaGymaaqcfasaaKqzadWdbiaadQga aKqba+aabaqcLbsapeGaeyOkJepaaiaadofajuaGpaWaaSbaaKqbGe aajugWa8qacaWGSbaajuaGpaqabaqcLbsapeGaeyOeI0scfa4aaybC aeqajuaipaqaaKqzadGaamyAaiabg2da9iaaigdaaKqbGeaajugWa8 qacaWGRbGaeyOeI0IaaGymaaqcfa4daeaajugib8qacqGHQms8aaGa am4uaKqba+aadaWgaaqcfasaaKqzadWdbiaadQgacaGGSaGaamyAaa qcfa4daeqaaaaa@619B@

where

S ¯ j1 = y=1 j1 S y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabm4uay aaraWaaSbaaKqbGeaaqaaaaaaaaaWdbiaadQgacqGHsislcaaIXaaa juaGpaqabaWdbiabg2da9maawahabeqcfaYdaeaacaWG5bGaeyypa0 JaaGymaaqaa8qacaWGQbGaeyOeI0IaaGymaaqcfa4daeaapeGaeyOk JepaaiaadofapaWaaSbaaKqbGeaapeGaamyEaaqcfa4daeqaaaaa@47B7@

At the final stage , the cumulative number of tests allocated to partition  is:

N i =min{ N j=1,ji k1 S L1,j ,max( N C ^ i ( S ¯ L1 ), S L1,i ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadMgaaKqba+aabeaapeGa eyypa0JaciyBaiaacMgacaGGUbWaaiWaa8aabaWdbiaad6eacqGHsi sldaGfWbqabKqbG8aabaGaamOAaiabg2da9iaaigdacaGGSaGaamOA aiabgcMi5kaadMgaaeaapeGaam4AaiabgkHiTiaaigdaaKqba+aaba WdbiabgQYiXdaacaWGtbWdamaaBaaajuaibaqcLbmapeGaamitaiab gkHiTiaaigdacaGGSaGaamOAaaqcfa4daeqaa8qacaGGSaGaciyBai aacggacaGG4bWaaeWaa8aabaWdbiaad6eaceWGdbGbaKaapaWaaSba aKqbGeaapeGaamyAaaqcfa4daeqaa8qadaqadaWdaeaaceWGtbGbae badaWgaaqcfasaa8qacaWGmbGaeyOeI0IaaGymaaqcfa4daeqaaaWd biaawIcacaGLPaaacaGGSaGaam4ua8aadaWgaaqcfasaa8qacaWGmb GaeyOeI0IaaGymaiaacYcacaWGPbaajuaGpaqabaaapeGaayjkaiaa wMcaaaGaay5Eaiaaw2haaaaa@6B37@

and

N k =N i=1 k1 N i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadUgaaKqba+aabeaapeGa eyypa0JaamOtaiabgkHiTmaawahabeqcfaYdaeaacaWGPbGaeyypa0 JaaGymaaqaa8qacaWGRbGaeyOeI0IaaGymaaqcfa4daeaapeGaeyOk Jepaaiaad6eapaWaaSbaaKqbGeaapeGaamyAaaqcfa4daeqaaaaa@47AE@

Among several ways of determining the number of cases at each sampling stage, the simplest one is to select:

S 1 = S 2 ==N/L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGtbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaapeGa eyypa0Jaam4ua8aadaWgaaqcfasaa8qacaaIYaaajuaGpaqabaWdbi abg2da9iabgAci8kabg2da9iaad6eacaGGVaGaamitaaaa@42F9@

However, choosing stage sizes, especially the  initial stage size, can be done by following some general criteria,  a good initial stage size can be N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGcaaWdaeaajugib8qacaWGobaajuaGbeaaaaa@38C3@  for when a  two stage sampling scheme is considered, and more generally, Rekab [9] shows that for  a two stage procedure,  must be chosen such that:

lim N S 1 =,and lim N S 1 N =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeae aaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaqcfaYdaeaapeGaamOt aiabgkziUkabg6HiLcqcfa4daeqaa8qacaWGtbWdamaaBaaajuaiba WdbiaaigdaaKqba+aabeaapeGaeyypa0JaeyOhIuQaaiilaiaadgga caWGUbGaamiza8aadaWfqaqaa8qaciGGSbGaaiyAaiaac2gaaKqbG8 aabaWdbiaad6eacqGHsgIRcqGHEisPaKqba+aabeaapeWaaSaaa8aa baWdbiaadofapaWaaSbaaKqbGeaapeGaaGymaaqcfa4daeqaaaqaa8 qacaWGobaaaiabg2da9iaaicdaaaa@551C@

Accelerated Sampling Scheme

By adopting a fully Bayesian approach with Beta priors, Rekab, Thompson and Wei [6] proposed an accelerated sequential design shown to be first order optimal. The accelerated sampling scheme combines the fully sequential sampling scheme and the multistage sampling scheme. It is defined as follows:

Stage 1:

The procedure starts with an initial sample S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGtbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaaaaa@3943@ , which satisfies the conditions specified in (4.1). Then, allocate S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGtbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaaaaa@3943@  equally among   partitions   

Stage 2 through L1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeacq GHsislcaaIXaaaaa@38F2@

At stage j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGQbaaaa@3794@ 2jL1, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIYaGaeyizImQaamOAaiabgsMiJkaadYeacqGHsislcaaI XaGaaiilaaaa@3EE3@ for partition i=1,2,,k1, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGPbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiabgAci 8kaacYcacaWGRbGaeyOeI0IaaGymaiaacYcaaaa@40F6@ the test cases are distributed by the following way:

S j,i =( l=1 j S l ) C ^ i ( S ¯ j1 ); MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGtbWdamaaBaaajuaibaWdbiaadQgacaGGSaGaamyAaaqc fa4daeqaa8qacqGH9aqpdaqadaWdaeaapeWaaybCaeqajuaipaqaai aadYgacqGH9aqpcaaIXaaabaWdbiaadQgaaKqba+aabaWdbiabgQYi XdaacaWGtbWdamaaBaaajuaibaWdbiaadYgaaKqba+aabeaaa8qaca GLOaGaayzkaaGabm4qayaajaWdamaaBaaajuaibaWdbiaadMgaaKqb a+aabeaapeWaaeWaa8aabaGabm4uayaaraWaaSbaaKqbGeaapeGaam OAaiabgkHiTiaaigdaaKqba+aabeaaa8qacaGLOaGaayzkaaGaai4o aaaa@5175@

and

S j,k = l=1 j S l l=1 k1 S j,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGtbWdamaaBaaajuaibaWdbiaadQgacaGGSaGaam4Aaaqc fa4daeqaa8qacqGH9aqpdaGfWbqabKqbG8aabaGaamiBaiabg2da9i aaigdaaeaapeGaamOAaaqcfa4daeaapeGaeyOkJepaaiaadofapaWa aSbaaeaapeGaamiBaaWdaeqaa8qacqGHsisldaGfWbqabKqbG8aaba GaamiBaiabg2da9iaaigdaaeaapeGaam4AaiabgkHiTiaaigdaaKqb a+aabaWdbiabgQYiXdaacaWGtbWdamaaBaaajuaibaWdbiaadQgaca GGSaGaamyAaaqcfa4daeqaaaaa@5341@

where

S ¯ j1 = y=1 j1 S y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabm4uay aaraWaaSbaaKqbGeaaqaaaaaaaaaWdbiaadQgacqGHsislcaaIXaaa juaGpaqabaWdbiabg2da9maawahabeqcfaYdaeaacaWG5bGaeyypa0 JaaGymaaqaa8qacaWGQbGaeyOeI0IaaGymaaqcfa4daeaapeGaeyOk JepaaiaadofapaWaaSbaaKqbGeaapeGaamyEaaqcfa4daeqaaaaa@47B7@

At each stage, S j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGtbWdamaaBaaajuaibaWdbiaadQgaaKqba+aabeaaaaa@3977@  must satisfy the two conditions as S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGtbWdamaaBaaajuaqbaWdbiaaigdaaKqba+aabeaaaaa@3963@ .

Stage L:

In the final stage, we adopt a fully sequential approach by allocating one test from partition , if for all ,

n l,i n l,j < p i R ^ l,i ( 1 R ^ l,i ) p j R ^ l,j ( 1 R ^ l,j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaamOBa8aadaWgaaqcfasaa8qacaWGSbGa aiilaiaadMgaaKqba+aabeaaaeaapeGaamOBa8aadaWgaaqcfasaa8 qacaWGSbGaaiilaiaadQgaaKqba+aabeaaaaWdbiabgYda8maalaaa paqaa8qacaWGWbWdamaaBaaajuaqbaqcLbmapeGaamyAaaqcfa4dae qaa8qadaGcaaWdaeaaceWGsbGbaKaadaWgaaqcfasaa8qacaWGSbGa aiilaiaadMgaaKqba+aabeaapeWaaeWaa8aabaWdbiaaigdacqGHsi slceWGsbGbaKaapaWaaSbaaKqbGeaapeGaamiBaiaacYcacaWGPbaa juaGpaqabaaapeGaayjkaiaawMcaaaqabaaapaqaa8qacaWGWbWdam aaBaaajuaibaWdbiaadQgaaKqba+aabeaapeWaaOaaa8aabaGabmOu ayaajaWaaSbaaKqbGeaapeGaamiBaiaacYcacaWGQbaajuaGpaqaba Wdbmaabmaapaqaa8qacaaIXaGaeyOeI0IabmOuayaajaWdamaaBaaa juaibaWdbiaadYgacaGGSaGaamOAaaqcfa4daeqaaaWdbiaawIcaca GLPaaaaeqaaaaaaaa@61B2@

until all the test cases have been allocated. Note that n l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGUbWdamaaBaaajuaibaWdbiaadYgacaGGSaGaamyAaaqc fa4daeqaaaaa@3B32@ , n l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGUbWdamaaBaaajuaibaWdbiaadYgacaGGSaGaamyAaaqc fa4daeqaaaaa@3B32@ are the cumulative test cases allocated to partition  and  after the allocations of a total of  test cases, where

j=1 L1 S j,i l j=1 L1 S j,i +N j=1 L1 S j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGfWbqabKqbG8aabaGaamOAaiabg2da9iaaigdaaeaapeGa amitaiabgkHiTiaaigdaaKqba+aabaWdbiabggHiLdaacaWGtbWdam aaBaaajuaibaWdbiaadQgacaGGSaGaamyAaaqcfa4daeqaa8qacqGH KjYOcaWGSbGaeyizIm6aaybCaeqajuaipaqaaiaadQgacqGH9aqpca aIXaaabaWdbiaadYeacqGHsislcaaIXaaajuaGpaqaa8qacqGHris5 aaGaam4ua8aadaWgaaqaa8qacaWGQbGaaiilaiaadMgaa8aabeaape Gaey4kaSIaamOtaiabgkHiTmaawahabeqcfaYdaeaacaWGQbGaeyyp a0JaaGymaaqaa8qacaWGmbGaeyOeI0IaaGymaaqcfa4daeaapeGaey yeIuoaaiaadofapaWaaSbaaKqbGeaapeGaamOAaaqcfa4daeqaaaaa @61B6@

Therefore, the estimate for the whole system is finally obtained as:

Optimality of Sequential Designs

First order optimality of these three sequential designs was established by Rekab, Thompson and Wu[5][6][7], although the focus here is on minimizing the variance incurred by the sequential designs rather than minimizing the Bayes risk incurred by these designs. For estimating the mean difference of two independent normal populations, Woodroofe and Hardwick [12] adopted a quasi-Bayesian approach. They determined an asymptotic lower bound for the integrated risk and proposed a three-stage design that is second-order optimal.  For estimating the mean difference of two general one -parameter exponential family, Rekab and Tahir [10] adopted a fully Bayesian approach with conjugate priors. They determined an asymptotic second order lower bound for the Bayes risk.

Monte Carlo Simulations

We consider the case where the test domain is partitioned into two subdomains D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGebWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaaaaa@3934@  and D 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGebWaaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@3905@ with reliability R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGsbaaaa@377C@  and R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGsbWdamaaBaaajuaibaWdbiaaikdaaKqba+aabeaaaaa@3943@ respectively with equal usage probabilities p 1 , p 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGWbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaapeGa aiilaiaadchapaWaaSbaaKqbGeaapeGaaGOmaaqcfa4daeqaaiaac6 caaaa@3D8D@ Second order optimality of the three sequential designs is investigated through Monte Carlo simulations.

Table I. N 2 *( Var( Δ )Var( O ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOta8aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcaGGQaWa aeWaa8aabaWdbiaadAfacaWGHbGaamOCamaabmaapaqaa8qacqqHuo araiaawIcacaGLPaaacqGHsislcaWGwbGaamyyaiaadkhadaqadaWd aeaapeGaam4taaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@4763@  by Fully Sequential Scheme

( R 1 , R 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaa8aabaWdbiaadkfapaWaaSbaaKqbGeaapeGaaGymaaqc fa4daeqaa8qacaGGSaGaamOua8aadaWgaaqcfasaa8qacaaIYaaaju aGpaqabaaapeGaayjkaiaawMcaaaaa@3E4D@

N=300 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiodacaaIWaGaaGimaaaa@3AA4@

N=500 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiwdacaaIWaGaaGimaaaa@3AA6@

N=800 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiIdacaaIWaGaaGimaaaa@3AA9@

N=2000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaikdacaaIWaGaaGimaiaaicdaaaa@3B5D@

N=5000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiwdacaaIWaGaaGimaiaaicdaaaa@3B60@

N=8000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiIdacaaIWaGaaGimaiaaicdaaaa@3B63@

0.1,0.9

28.1562

3.2104

0.7390

2.4225

9.3040

0.7133

0.5,0.2

29.0380

6.1020

0.6300

7.9375

4.6700

4.0500

0.5,0.5

1.5972

0.1460

0.2120

0.9125

0.5120

0.4933

0.5,0.9

80.7747

71.9032

19.9510

7.9600

5.7960

1.2666

0.9,0.3

64.0246

53.8008

5.5497

3.2565

15.2546

2.6517

Table II. N 2 *( Var( Δ )Var( O ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOta8aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcaGGQaWa aeWaa8aabaWdbiaadAfacaWGHbGaamOCamaabmaapaqaa8qacqqHuo araiaawIcacaGLPaaacqGHsislcaWGwbGaamyyaiaadkhadaqadaWd aeaapeGaam4taaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@4763@ by Multistage Scheme

( R 1 , R 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaa8aabaWdbiaadkfapaWaaSbaaKqbGeaapeGaaGymaaqc fa4daeqaa8qacaGGSaGaamOua8aadaWgaaqcfasaa8qacaaIYaaaju aGpaqabaaapeGaayjkaiaawMcaaaaa@3E4D@

N=300 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiodacaaIWaGaaGimaaaa@3AA4@

N=500 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiwdacaaIWaGaaGimaaaa@3AA6@

N=800 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiIdacaaIWaGaaGimaaaa@3AA9@

N=2000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaikdacaaIWaGaaGimaiaaicdaaaa@3B5D@

N=5000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiwdacaaIWaGaaGimaiaaicdaaaa@3B60@

N=8000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiIdacaaIWaGaaGimaiaaicdaaaa@3B63@

0.1,0.9

10.9835

0.0288

4.0718

55.7962

2.3320

1.8422

0.5,0.2

11.8285

19.1720

2.0555

5.5100

0.8432

7.1258

0.5,0.5

0.02749

0.0459

0.1750

0.4568

0.47324

0.1064

0.5,0.9

37.6665

31.6257

17.6417

3.6377

6.8483

9.2952

0.9,0.3

20.2034

2.86352

7.6161

3.1589

11.7869

5.2018

Table III. N 2 *( Var( Δ )Var( O ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOta8aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcaGGQaWa aeWaa8aabaWdbiaadAfacaWGHbGaamOCamaabmaapaqaa8qacqqHuo araiaawIcacaGLPaaacqGHsislcaWGwbGaamyyaiaadkhadaqadaWd aeaapeGaam4taaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@4763@ by Accelerated Scheme

( R 1 , R 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaa8aabaWdbiaadkfapaWaaSbaaKqbGeaapeGaaGymaaqc fa4daeqaa8qacaGGSaGaamOua8aadaWgaaqcfasaa8qacaaIYaaaju aGpaqabaaapeGaayjkaiaawMcaaaaa@3E4D@

N=300 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiodacaaIWaGaaGimaaaa@3AA4@

N=500 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiwdacaaIWaGaaGimaaaa@3AA6@

N=800 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiIdacaaIWaGaaGimaaaa@3AA9@

N=2000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaikdacaaIWaGaaGimaiaaicdaaaa@3B5D@

N=5000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiwdacaaIWaGaaGimaiaaicdaaaa@3B60@

N=8000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaiIdacaaIWaGaaGimaiaaicdaaaa@3B63@

0.1,0.9

30.5713

6.5870

6.4568

17.4023

2.1138

1.0098

0.5,0.2

8.1424

6.1346

7.1683

0.8968

0.1462

2.1009

0.5,0.5

0.03801

0.23284

0.0319

0.7025

0.0640

0.0392

0.5,0.9

37.6665

31.6257

17.6417

3.6377

6.8483

9.2952

0.9,0.3

48.4148

13.8170

7.8473

8.4656

5.8734

1.6708

Table I, II, III seem to indicate that the speed  N 2 *( Var( Δ )Var( O ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOta8aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcaGGQaWa aeWaa8aabaWdbiaadAfacaWGHbGaamOCamaabmaapaqaa8qacqqHuo araiaawIcacaGLPaaacqGHsislcaWGwbGaamyyaiaadkhadaqadaWd aeaapeGaam4taaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@4763@ is bounded.

Conclusion

Second optimal designs are more efficient than the first order optimal designs especially when the total number of cases is very large. This is the main argument that led us to investigate the second order optimality of these three dynamic designs. Simulation studies seem to indicate that these designs are second order optimal.  We conjecture that second order optimally may be obtained theoretically as well.

It is very common in parametric estimation to use the squared error loss. However, in reliability estimation one should distinguish between the cost of overestimating and underestimating the system reliability.    Examples of practical loss functions were presented by Stüger[2]:

l( R ^ ,R )= c o ( R ^ R ) 2 I { R ^ >R } + c u ( R R ^ ) 2 I { R ^ <R } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGSbWaaeWaa8aabaWdbiqackfagaqcaiaacYcacaWGsbaa caGLOaGaayzkaaGaeyypa0Jaam4ya8aadaWgaaqcfasaa8qacaWGVb aajuaGpaqabaWdbmaabmaapaqaa8qaceWGsbGbaKaacqGHsislcaWG sbaacaGLOaGaayzkaaWdamaaCaaabeqcfasaa8qacaaIYaaaaKqbak aadMeapaWaaSbaaeaapeWaaiWaa8aabaWdbiqadkfagaqcaiabg6da +iaadkfaaiaawUhacaGL9baaa8aabeaapeGaey4kaSIaam4ya8aada Wgaaqcfasaa8qacaWG1baajuaGpaqabaWdbmaabmaapaqaa8qacaWG sbGaeyOeI0IabmOuayaajaaacaGLOaGaayzkaaWdamaaCaaabeqcfa saa8qacaaIYaaaaKqbakaadMeapaWaaSbaaeaapeWaaiWaa8aabaWd biqadkfagaqcaiabgYda8iaadkfaaiaawUhacaGL9baaa8aabeaaaa a@5C11@

and by Granger[1]:

l( R ^ ,R )= c o ( R ^ R ) I { R ^ >R } + c u ( R R ^ ) I { R ^ <R } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGSbWaaeWaa8aabaWdbiqackfagaqcaiaacYcacaWGsbaa caGLOaGaayzkaaGaeyypa0Jaam4ya8aadaWgaaqaaKqzadWdbiaad+ gaaKqba+aabeaapeWaaeWaa8aabaWdbiqadkfagaqcaiabgkHiTiaa dkfaaiaawIcacaGLPaaacaWGjbWdamaaBaaabaWdbmaacmaapaqaa8 qaceWGsbGbaKaacqGH+aGpcaWGsbaacaGL7bGaayzFaaaapaqabaWd biabgUcaRiaadogapaWaaSbaaKqbGeaapeGaamyDaaqcfa4daeqaa8 qadaqadaWdaeaapeGaamOuaiabgkHiTiqadkfagaqcaaGaayjkaiaa wMcaaiaadMeapaWaaSbaaeaapeWaaiWaa8aabaWdbiqadkfagaqcai abgYda8iaadkfaaiaawUhacaGL9baaa8aabeaaaaa@599F@

where represents the overestimation and underestimation costs, respectively.

References

  1. Sankaran M (1970) The discrete Poisson-Lindley distribution. Biometrics 26(1): 145-149.
  2. Ghitany ME, Al-Mutairi DK (2009) Estimation Methods for the discrete Poisson-Lindley distribution. Journal of Statistical Computation and Simulation 79(1): 1-9.
  3. Shanker R, Mishra A (2014) A two-parameter Poisson-Lindley distribution. International Journal of Statistics and Systems 9(1): 79-85.
  4. Shanker R, Mishra A (2013 a) A two-parameter Lindley distribution. Statistics in Transition new Series 14(1): 45-56.
  5. Shanker R, Mishra A (2015) A quasi Poisson-Lindley distribution (submitted).
  6. Shanker R, Mishra A (2013 b) A quasi Lindley distribution. African journal of Mathematics and Computer Science Research 6(4): 64-71.
  7. Shanker R, Sharma S, Shanker R (2012) A Discrete two-Parameter Poisson Lindley Distribution. Journal of Ethiopian Statistical Association 21: 15-22.
  8. Shanker R, Sharma S, Shanker, R (2013) A two-parameter Lindley distribution for modeling waiting and survival times data, Applied Mathematics 4: 363-368.
  9. Shanker R, Tekie AL (2014) A new quasi Poisson-Lindley distribution. International Journal of Statistics and Systems 9(1): 87-94.
  10. Shanker R, Amanuel AG (2013) A new quasi Lindley distribution, International Journal of Statistics and Systems 8(2): 143-156.
  11. Johnson NL, Kotz S, Kemp AW (1992) Univariate Discrete Distributions, 2nd edition, John Wiley & sons Inc.
  12. Fisher RA, Corpet AS, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology 12(1): 42-58.
  13. Kempton RA (1975) A generalized form of Fisher’s logarithmic series. Biometrika 62 (1): 29-38.
  14. Tripathi RC, Gupta RC (1985) A generalization of the log-series distribution. Comm. in Stat. (Theory and Methods) 14(8): 1779-1799.
  15. Mishra A, Shanker R (2002) Generalized logarithmic series distribution-Its nature and applications, Proceedings of the Vth International Symposium on Optimization and Statistics 28-30: 155-168.
  16. Loeschke V, Kohler W (1976) Deterministic and Stochastic models of the negative binomial distribution and the analysis of chromosomal aberrations in human leukocytes. Biometrische Zeitschrift 18: 427-451.
  17. Janardan KG, Schaeffer DJ (1977) Models for the analysis of chromosomal aberrations in human leukocytes. Biometrical Journal (8)599-612.
  18. Mc Guire JU, Brindley TA, Bancroft TA (1957) The distribution of European corn-borer larvae pyrausta in field corn. Biometrics 13: 65-78.
  19. Catcheside DG, Lea DE, Thoday JM (1946) Types of chromosome structural      change induced by the irradiation on Tradescantia microspores. Journal of Genetics 47: 113-136.
  20. Catcheside DG, Lea DE, Thoday JM (1946) The production of chromosome        structural changes in Tradescantia microspores in relation to dosage, intensity and temperature. Journal of Genetics l.47: 137-149.
  21. Lindley DV (1958) Fiducial distributions and Bayes theorem. Journal of Royal Statistical Society Ser. B Vol.20: 102-107.
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