STEP STRESS TESTS AND SOME EXACT INFERENTIAL RESULTS N. BALAKRISHNAN. McMaster University Hamilton, Ontario, Canada. p.

Transcription

1 p. 1/6 STEP STRESS TESTS AND SOME EXACT INFERENTIAL RESULTS N. BALAKRISHNAN McMaster University Hamilton, Ontario, Canada

2 p. 2/6 In collaboration with Debasis Kundu, IIT, Kapur, India H.K. Tony Ng, Southern Methodist University, Dallas Nandini Kannan, University of Texas, San Antonio Evans Gouno, Université de Britagne Sud, France Ananda Sen, University of Michigan, Ann Arbor Qihao Xie, McMaster University, Hamilton, Canada Dong-hoon Han, McMaster University, Hamilton, Canada

3 p. 3/6 Roadmap 1. Step-stress Problem 2. Models 3. Cumulative Exposure Model 4. Likelihood under Type-II Censoring and MLEs 5. Exact Conditional Distributions of MLEs 6. Confidence Intervals & Bootstrap Intervals 7. Simulation Results & Comments 8. Illustrative Example 9. Time-Constrained Step-Stress Test 10. Progressive Type-II Censoring and Results 11. Other Extensions and Generalizations 12. Further Problems of Interest 13. Bibliography

4 p. 4/6 1. Step-stress Problem In industrial experiments, products that are tested are often extremely reliable with large mean times to failure under normal operating conditions. So, an experimenter may resort to accelerated life-testing (ALT) wherein the units are subjected to higher stress levels than normal. Examples include assessing the effects of temperature, voltage, load, vibration, etc. on the lifetime of a product.

5 p. 4/6 1. Step-stress Problem In industrial experiments, products that are tested are often extremely reliable with large mean times to failure under normal operating conditions. So, an experimenter may resort to accelerated life-testing (ALT) wherein the units are subjected to higher stress levels than normal. Examples include assessing the effects of temperature, voltage, load, vibration, etc. on the lifetime of a product. Step-stress testing is one such ALT.

6 p. 4/6 1. Step-stress Problem In industrial experiments, products that are tested are often extremely reliable with large mean times to failure under normal operating conditions. So, an experimenter may resort to accelerated life-testing (ALT) wherein the units are subjected to higher stress levels than normal. Examples include assessing the effects of temperature, voltage, load, vibration, etc. on the lifetime of a product. Step-stress testing is one such ALT. Some key references on ALT are: Nelson (1990) Meeker and Escobar (1998) Bagdonavicius and Nikulin (2002)

7 1. Step-stress Problem p. 5/6

8 p. 6/6 2. Models There are three important models discussed in the literature: Tampered Random Variable Model [DeGroot and Goel (1979)] Tampered Hazard Model [Bhattacharyya and Zanzawi (1989)] Cumulative Exposure Model [Nelson (1980)]

9 p. 7/6 3. Cumulative Exposure Model The cumulative exposure model [Nelson (1980, 1990)] relates the life distribution of an unit at one stress level to the life distribution of that unit at the next stress level by assuming that the residual life of the unit depends only on the cumulative exposure that unit had experienced, with no memory of how this exposure was accumulated.

10 p. 8/6 3. Cumulative Exposure Model In the case of a simple step-stress model, with the life distributions as Exp(θ 1 ) and Exp(θ 2 ) at stress levels s 0 and s 1, the cumulative exposure distribution (CED) of T (time-to-failure of unit) becomes

11 p. 8/6 3. Cumulative Exposure Model In the case of a simple step-stress model, with the life distributions as Exp(θ 1 ) and Exp(θ 2 ) at stress levels s 0 and s 1, the cumulative exposure distribution (CED) of T (time-to-failure of unit) becomes G(t) = where G 1 (t) = F 1 (t;θ 1 ) G 2 (t) = F 2 ( ) ) 1 θ 2 τ;θ2 θ1 if 0 < t < τ if τ t <, F k (t;θ k ) = 1 e t/θ k, t 0, θ k > 0, k = 1, 2.

12 p. 9/6 3. Cumulative Exposure Model The corresponding PDF is g 1 (t) = 1 θ g(t) = 1 e 1 t θ 1 g 2 (t) = 1 θ 2 e 1 (t τ) 1 τ θ 2 θ 1 if 0 < t < τ if τ t <.

13 p. 10/6 4. Likelihood under Type-II Censoring & MLEs Suppose n units are placed under a step-stress test at an initial stress level of s 0, and at a pre-fixed time τ the stress level will be changed to s 1.

14 p. 10/6 4. Likelihood under Type-II Censoring & MLEs Suppose n units are placed under a step-stress test at an initial stress level of s 0, and at a pre-fixed time τ the stress level will be changed to s 1. Under Type-II censoring, the experiment will terminate when a required number (say, r) of the n units fail. If r is taken as n, then a complete sample would be observed from the step-stress test.

15 p. 10/6 4. Likelihood under Type-II Censoring & MLEs Suppose n units are placed under a step-stress test at an initial stress level of s 0, and at a pre-fixed time τ the stress level will be changed to s 1. Under Type-II censoring, the experiment will terminate when a required number (say, r) of the n units fail. If r is taken as n, then a complete sample would be observed from the step-stress test. Let n 1 denote the (random) number of failures that occur before τ.

16 p. 11/6 4. Likelihood under Type-II Censoring & MLEs The likelihood of the observed Type-II censored data t 1:n < < t r:n is

17 p. 11/6 4. Likelihood under Type-II Censoring & MLEs The likelihood of the observed Type-II censored data t 1:n < < t r:n is L(θ 1, θ 2 ) = 8 >< c 1 θ1 r e θ 1 1 [ P r i=1 t i:n +(n r)t r:n] if n 1 = r h c 1 θ2 r e θ 1 Pri=1 i θ2 θ2 2 θ1 τ+t i:n τ +(n r) θ1 τ+t r:n τ if n 1 = 0 >: c 2 θ n 1 1 θr n 1 2 e 1 θ 1 { P n 1 i=1 t i:n+(n n 1 )τ} e 1 θ 2 n Pri=n1 +1 (t i:n τ)+(n r)(t r:n τ) if 1 n 1 r 1. o

18 p. 12/6 4. Likelihood under Type-II Censoring & MLEs It is evident that the MLEs of θ 1 and θ 2 exist only when 1 n 1 r 1, and they are:

19 p. 12/6 4. Likelihood under Type-II Censoring & MLEs It is evident that the MLEs of θ 1 and θ 2 exist only when 1 n 1 r 1, and they are: ˆθ 1 = ˆθ 2 = n1 i=1 t i:n + (n n 1 )τ, n 1 r i=n 1 +1 (t i:n τ) + (n r)(t r:n τ) r n 1.

20 p. 12/6 4. Likelihood under Type-II Censoring & MLEs It is evident that the MLEs of θ 1 and θ 2 exist only when 1 n 1 r 1, and they are: ˆθ 1 = ˆθ 2 = n1 i=1 t i:n + (n n 1 )τ, n 1 r i=n 1 +1 (t i:n τ) + (n r)(t r:n τ) r n 1. These are the conditional MLEs of θ 1 and θ 2, conditional on 1 n 1 r 1.

21 p. 12/6 4. Likelihood under Type-II Censoring & MLEs It is evident that the MLEs of θ 1 and θ 2 exist only when 1 n 1 r 1, and they are: ˆθ 1 = ˆθ 2 = n1 i=1 t i:n + (n n 1 )τ, n 1 r i=n 1 +1 (t i:n τ) + (n r)(t r:n τ) r n 1. These are the conditional MLEs of θ 1 and θ 2, conditional on 1 n 1 r 1. The inference we develop here will be exact and conditional.

22 p. 13/6 5. Exact Conditional Distributions of MLEs Denote the CMGFs of ˆθ 1 and ˆθ 2, given 1 n 1 r 1, by M 1 (t) and M 2 (t), respectively.

23 p. 13/6 5. Exact Conditional Distributions of MLEs Denote the CMGFs of ˆθ 1 and ˆθ 2, given 1 n 1 r 1, by M 1 (t) and M 2 (t), respectively. M 1 (t) = E M 2 (t) = E [ ] e tˆθ 1 1 n 1 r 1, [ ] e tˆθ 2 1 n 1 r 1.

24 p. 13/6 5. Exact Conditional Distributions of MLEs Denote the CMGFs of ˆθ 1 and ˆθ 2, given 1 n 1 r 1, by M 1 (t) and M 2 (t), respectively. M 1 (t) = E M 2 (t) = E [ ] e tˆθ 1 1 n 1 r 1, [ ] e tˆθ 2 1 n 1 r 1. For deriving M k (t), we may write M k (t) = r 1 j=1 E [ ] e tˆθ k n 1 = j P [n 1 = j 1 n 1 r 1].

25 p. 14/6 5. Exact Conditional Distributions of MLEs Lemma 1: The number of failures occurring before τ, viz. n 1, is a binomial random variable with PMF (for j = 0, 1,,n) P [n 1 = j] = ( n j ) ( ) 1 e τ j θ 1 e τ θ (n j) 1 = p j (say).

26 p. 14/6 5. Exact Conditional Distributions of MLEs Lemma 1: The number of failures occurring before τ, viz. n 1, is a binomial random variable with PMF (for j = 0, 1,,n) P [n 1 = j] = ( n j ) ( ) 1 e τ j θ 1 e τ θ (n j) 1 = p j (say). So, P [n 1 = j 1 n 1 r 1] = p j r 1 i=1 p. i

27 p. 15/6 5. Exact Conditional Distributions of MLEs Lemma 2: Let X 1:n < < X n:n be the order statistics of a sample of size n from PDF f(x) and CDF F(x). Let D be the number of order statistics τ (fixed time).

28 p. 15/6 5. Exact Conditional Distributions of MLEs Lemma 2: Let X 1:n < < X n:n be the order statistics of a sample of size n from PDF f(x) and CDF F(x). Let D be the number of order statistics τ (fixed time). The conditional joint PDF of X 1:n,,X D:n, given that D = j, is same as the joint PDF of all order statistics from a sample of size j from the right truncated density f τ (t) = f(t) F(τ) for 0 < t < τ 0 otherwise.

29 p. 16/6 5. Exact Conditional Distributions of MLEs Lemma 3: Let Z be a right-truncated exponential random variable with PDF f Z (z) = 1 θ 1 e z θ 1 1 e for 0 < z < τ. τ θ 1

30 p. 16/6 5. Exact Conditional Distributions of MLEs Lemma 3: Let Z be a right-truncated exponential random variable with PDF f Z (z) = Then, the MGF of Z is 1 θ 1 e z θ 1 1 e for 0 < z < τ. τ θ 1 M Z (t) = E(e tz ) = 1 e τ θ 1 (1 θ 1 t) (1 e τ θ 1 )(1 θ 1 t).

31 p. 17/6 5. Exact Conditional Distributions of MLEs Lemma 4: Let Y be a Gamma(α,λ), i.e., a gamma random variable with shape parameter α and scale parameter λ. The PDF of Y is given by f G (y;α,λ) = λα Γ(α) yα 1 e λy for y > 0.

32 p. 17/6 5. Exact Conditional Distributions of MLEs Lemma 4: Let Y be a Gamma(α,λ), i.e., a gamma random variable with shape parameter α and scale parameter λ. The PDF of Y is given by f G (y;α,λ) = λα Γ(α) yα 1 e λy for y > 0. For any constant A, the MGF of Y + A is ( M Y +A (t) = e ta 1 λ) t α.

33 p. 18/6 5. Exact Conditional Distributions of MLEs By using the Lemmas, it can be shown that E ( ) e tˆθ 1 n 1 = j = ( e t j (n j)τ 1 e τ θ 1 (1 θ 1 t j ) ) j ) (1 e τ j ) j. θ 1 (1 θ 1t j

34 p. 18/6 5. Exact Conditional Distributions of MLEs By using the Lemmas, it can be shown that E ( ) e tˆθ 1 n 1 = j = ( e t j (n j)τ 1 e τ θ 1 (1 θ 1 t j ) ) j ) (1 e τ j ) j. θ 1 (1 θ 1t j So, E ( ) e tˆθ 1 1 n 1 r 1 = r 1 j=1 j k=0 ( c j,k 1 θ ) j 1t e tτ j (n j+k). j

35 p. 19/6 5. Exact Conditional Distributions of MLEs Theorem 1: The PDF of ˆθ 1, conditional on 1 n 1 r 1, is given by fˆθ1 (t) = r 1 j=1 j k=0 c j,k f G ( t τ j,k ;j, j θ 1 ) ;

36 p. 19/6 5. Exact Conditional Distributions of MLEs Theorem 1: The PDF of ˆθ 1, conditional on 1 n 1 r 1, is given by here, fˆθ1 (t) = r 1 j=1 j k=0 c j,k = ( 1)k r 1 i=1 p i τ j,k = τ j c j,k f G ( t τ j,k ;j, j θ 1 ) (n j + k). ( )( ) n j e τ θ (n j+k) 1, j k ;

37 p. 20/6 5. Exact Conditional Distributions of MLEs Lemma 5: Let T 1:n < < T r:n be the first r order statistics from the cumulative exposure density. The conditional joint PDF of T n1 +1:n,,T r:n, given that n 1 = j, where 1 j r 1, is given by

38 p. 20/6 5. Exact Conditional Distributions of MLEs Lemma 5: Let T 1:n < < T r:n be the first r order statistics from the cumulative exposure density. The conditional joint PDF of T n1 +1:n,,T r:n, given that n 1 = j, where 1 j r 1, is given by f Tn1 +1:n,,T r:n (n 1 =j)(t n1 +1:n,,t r:n ) = c 3 θ r j 2 e n tj+1:n τ θ t r:n τ θ 2 +(n r) o t r:n τ θ 2 for τ < t j+1:n < < t r:n <, and c 3 is the normalizing constant.

39 p. 21/6 5. Exact Conditional Distributions of MLEs By using the preceding Lemma, it can be shown that the CMGF of ˆθ 2 is r 1 ( p r j M 2 (t) = r 1 i=1 p 1 tθ ) j 2. i j j=1

40 p. 21/6 5. Exact Conditional Distributions of MLEs By using the preceding Lemma, it can be shown that the CMGF of ˆθ 2 is r 1 ( p r j M 2 (t) = r 1 i=1 p 1 tθ ) j 2. i j j=1 Theorem 2: The PDF of ˆθ 2, conditional on 1 n 1 r 1, is given by fˆθ2 (t) = r 1 j=1 p r j r 1 i=1 p i f G ( t;j, j θ 2 ).

41 p. 22/6 5. Exact Conditional Distributions of MLEs Theorem 3: The first two moments of ˆθ 1 and ˆθ 2 are as follows:

42 p. 22/6 5. Exact Conditional Distributions of MLEs Theorem 3: The first two moments of ˆθ 1 and ˆθ 2 are as follows: E E (ˆθ1 ) ) (ˆθ2 1 = = r 1 j=1 r 1 j=1 j c j,k (τ j,k + θ 1 ), k=0 j k=0 r 1 E(ˆθ 2 ) = θ 2 E(ˆθ 2 2 ) = θ2 2 j=1 r 1 j=1 ( ) (j + 1) c j,k θ1 2 + τj,k 2 + 2θ 1 τ j,k, j p r j r 1 i=1 p i p r j r 1 i=1 p i = θ 2, (j + 1) j.

43 p. 23/6 5. Exact Conditional Distributions of MLEs Remark 1: The conditional PDF of ˆθ 2 in Theorem 2 is a true mixture of gamma densities.

44 p. 23/6 5. Exact Conditional Distributions of MLEs Remark 1: The conditional PDF of ˆθ 2 in Theorem 2 is a true mixture of gamma densities. Remark 2: The expressions of the expected values in Theorem 3 clearly reveal that ˆθ 1 is a biased estimator of θ 1, while ˆθ 2 is an unbiased estimator of θ 2.

45 p. 23/6 5. Exact Conditional Distributions of MLEs Remark 1: The conditional PDF of ˆθ 2 in Theorem 2 is a true mixture of gamma densities. Remark 2: The expressions of the expected values in Theorem 3 clearly reveal that ˆθ 1 is a biased estimator of θ 1, while ˆθ 2 is an unbiased estimator of θ 2. Remark 3: The expressions of the second moments in Theorem 3 can be used for finding standard errors of the estimates.

46 p. 24/6 6. Confidence Intervals & Bootstrap Intervals Theorem 4: The tail probability of ˆθ 1 is ) P θ1 (ˆθ1 b = r 1 j=1 j c j,k Γ k=0 ( j, j θ 1 < b τ j,k > ), z where Γ(a,z) = 1 Γ(a) t a 1 e t dt is incomplete gamma, and < x >= max{x, 0}.

47 p. 24/6 6. Confidence Intervals & Bootstrap Intervals Theorem 4: The tail probability of ˆθ 1 is ) P θ1 (ˆθ1 b = r 1 j=1 j c j,k Γ k=0 ( j, j θ 1 < b τ j,k > ), z where Γ(a,z) = 1 Γ(a) t a 1 e t dt is incomplete gamma, and < x >= max{x, 0}. Similarly, the tail probability of ˆθ 2 is ) P θ2 (ˆθ2 b = r 1 j=1 p r j r 1 i=1 p i Γ ( j, bj θ 2 ).

48 p. 25/6 6. Confidence Intervals & Bootstrap Intervals Exact CIs: Using the results in Theorem 4, exact conditional CIs can be constructed for θ 1 and θ 2.

49 p. 25/6 6. Confidence Intervals & Bootstrap Intervals Exact CIs: Using the results in Theorem 4, exact conditional CIs can be constructed for θ 1 and θ 2. Approximate CIs: Using the observed Fisher information matrix, approximate CIs can be constructed for θ 1 and θ 2 using the asymptotic normality of the MLEs.

50 p. 25/6 6. Confidence Intervals & Bootstrap Intervals Exact CIs: Using the results in Theorem 4, exact conditional CIs can be constructed for θ 1 and θ 2. Approximate CIs: Using the observed Fisher information matrix, approximate CIs can be constructed for θ 1 and θ 2 using the asymptotic normality of the MLEs. BCA Bootstrap CIs: Using the bias-corrected and accelerated percentile bootstrap method [Efron and Tibshirani (1982)], bootstrap CIs can be constructed for θ 1 and θ 2.

51 p. 26/6 7. Simulation Results & Comments We simulated the coverage probabilities (CP) of all three CIs for different values of n, r and τ; see Tables 1 and 2.

52 p. 26/6 7. Simulation Results & Comments We simulated the coverage probabilities (CP) of all three CIs for different values of n, r and τ; see Tables 1 and 2. Exact conditional CI has its CP to be quite close to the nominal level. Approximate CI has its CP to be always smaller than the nominal level, and so it will often be unduly narrower. Bootstrap CI has its CP to be close to the nominal level for θ 2, but is not satisfactory for θ 1, and especially worse for small n.

53 p. 27/6 7. Simulation Results & Comments Table 1: Estimated coverage probabilities (in %) based on 1000 simulations with θ 1 = 12.0 and θ 2 = 4.5, n = 20, r = 16, B = 1000 C.I. of θ 1 90% C.I. 95% C.I. τ Boot Approx Exact Boot Approx Exact

54 p. 28/6 7. Simulation Results & Comments Table 2: Estimated coverage probabilities (in %) based on 1000 simulations with θ 1 = 12.0 and θ 2 = 4.5, n = 20, r = 16, B = 1000 C.I. of θ 2 90% C.I. 95% C.I. τ Boot Approx Exact Boot Approx Exact

55 p. 29/6 8. Illustrative Example Let us consider the data presented by Xiong (1998). The data n = 20, r = 16 and τ = 5 are as follows:

56 p. 29/6 8. Illustrative Example Let us consider the data presented by Xiong (1998). The data n = 20, r = 16 and τ = 5 are as follows: Stress level Failure Time θ 1 = e θ 2 = e

57 p. 30/6 8. Illustrative Example In this example, we have n 1 = 4 and n 2 = r n 1 = 12.

58 p. 30/6 8. Illustrative Example In this example, we have n 1 = 4 and n 2 = r n 1 = 12. We then find ˆθ 1 = and ˆθ2 =

59 p. 30/6 8. Illustrative Example In this example, we have n 1 = 4 and n 2 = r n 1 = 12. We then find ˆθ 1 = and ˆθ2 = The confidence intervals for θ 1 and θ 2 are presented in Table 3.

60 p. 31/6 8. Illustrative Example Table 3: Confidence intervals for θ 1 and θ 2 CI for θ 1 90% 95% Bootstrap CI (7.78, 34.05) (6.03, 34.05) Approx CI (0.00, 35.66) (0.00, 39.36) Exact CI (11.70, 72.95) (10.35, 94.78) CI for θ 2 90% 95% Bootstrap CI (5.76, 11.43) (5.51, 12.80) Approx CI (2.66, 7.46) (2.20, 7.92) Exact CI (3.33, 8.80) (3.07, 9.86)

61 p. 32/6 8. Illustrative Example From Table 3, we observe the following:

62 p. 32/6 8. Illustrative Example From Table 3, we observe the following: Exact CIs are wider in general than the other two intervals;

63 p. 32/6 8. Illustrative Example From Table 3, we observe the following: Exact CIs are wider in general than the other two intervals; Approximate CIs are always narrower while Bootstrap CIs are sometimes narrower and sometimes wider;

64 p. 32/6 8. Illustrative Example From Table 3, we observe the following: Exact CIs are wider in general than the other two intervals; Approximate CIs are always narrower while Bootstrap CIs are sometimes narrower and sometimes wider; This is so because the CPs for the approximate method are lower than the nominal level while those of the bootstrap method are sometimes lower and sometimes higher;

65 p. 32/6 8. Illustrative Example From Table 3, we observe the following: Exact CIs are wider in general than the other two intervals; Approximate CIs are always narrower while Bootstrap CIs are sometimes narrower and sometimes wider; This is so because the CPs for the approximate method are lower than the nominal level while those of the bootstrap method are sometimes lower and sometimes higher; CIs for θ 2 are considerably narrower than those for θ 1. This is so since when τ is small relative to θ 1, relatively small (large) numbers of failures would occur before (after) τ.

66 p. 33/6 9. Time-Constrained Step-Stress Test Consider the step-stress testing when there is a time constraint on the experiment.

67 p. 33/6 9. Time-Constrained Step-Stress Test Consider the step-stress testing when there is a time constraint on the experiment. n identical units are tested at an initial stress level s 0. The stress level is changed to s 1 at time τ 1, and the testing is terminated at time τ 2, where 0 < τ 1 < τ 2 < are pre-fixed.

68 p. 33/6 9. Time-Constrained Step-Stress Test Consider the step-stress testing when there is a time constraint on the experiment. n identical units are tested at an initial stress level s 0. The stress level is changed to s 1 at time τ 1, and the testing is terminated at time τ 2, where 0 < τ 1 < τ 2 < are pre-fixed. Let N 1 = no. of units that fail before time τ 1 ; N 2 = no. of units that fail before time τ 2 at stress level s 1.

69 p. 34/6 9. Time-Constrained Step-Stress Test With these notation, we will observe the following data:

70 p. 34/6 9. Time-Constrained Step-Stress Test With these notation, we will observe the following data: { t = t 1:n < < t τ N1 :n 1 < t N1 +1:n < < t N 1 +N 2 :n τ 2 }.

71 p. 34/6 9. Time-Constrained Step-Stress Test With these notation, we will observe the following data: { t = t 1:n < < t τ N1 :n 1 < t N1 +1:n < < t N 1 +N 2 :n τ 2 }. We obtain the likelihood function of θ 1 and θ 2 based on the above Type-I censored sample as follows:

72 p. 35/6 9. Time-Constrained Step-Stress Test If N 1 = n and N 2 = 0, the likelihood is L(θ 1,θ 2 t) = n! n k=1 g 1 ( tk:n ) = n! θ n 1 exp { 1 θ 1 0 < t 1:n < < t n:n < τ 1 ; n t k:n }, k=1

73 p. 35/6 9. Time-Constrained Step-Stress Test If N 1 = n and N 2 = 0, the likelihood is L(θ 1,θ 2 t) = n! n k=1 g 1 ( tk:n ) = n! θ n 1 exp { 1 θ 1 0 < t 1:n < < t n:n < τ 1 ; n t k:n }, k=1 In all other cases, the likelihood function is { } n! L(θ 1,θ 2 t) = (n r)! θ N exp 1θ1 D 1 1 θn 2 1 1θ2 D 2, 2 0 < t 1:n < < t < τ t < < t N1 :n 1 N 1 +1:n r:n < τ 2,

74 p. 36/6 9. Time-Constrained Step-Stress Test where r = N 1 + N 2 (2 r n), D 1 = D 2 = N 1 X k=1 t k:n + `n N 1 τ1, rx k=n 1 +1 `tk:n τ 1 + (n r)(τ2 τ 1 ).

75 p. 36/6 9. Time-Constrained Step-Stress Test where r = N 1 + N 2 (2 r n), D 1 = D 2 = N 1 X k=1 t k:n + `n N 1 τ1, rx k=n 1 +1 `tk:n τ 1 + (n r)(τ2 τ 1 ). We observe that if at least one failure occurs before τ 1 and between τ 1 and τ 2, the MLE of ( θ1,θ 2 ) exists, and ( D1,D 2 ) is joint complete sufficient for (θ 1,θ 2 ).

76 p. 37/6 9. Time-Constrained Step-Stress Test In this situation, the log-likelihood function of θ 1 and θ 2 is l(θ 1,θ 2 t) = log n! (n r)! N 1 log θ 1 N 2 log θ 2 D 1 θ 1 D 2 θ 2.

77 p. 37/6 9. Time-Constrained Step-Stress Test In this situation, the log-likelihood function of θ 1 and θ 2 is l(θ 1,θ 2 t) = log n! (n r)! N 1 log θ 1 N 2 log θ 2 D 1 θ 1 D 2 θ 2. The MLEs of θ 1 and θ 2 are obtained as ˆθ 1 = D 1 N 1 and ˆθ2 = D 2 N 2.

78 p. 37/6 9. Time-Constrained Step-Stress Test In this situation, the log-likelihood function of θ 1 and θ 2 is l(θ 1,θ 2 t) = log n! (n r)! N 1 log θ 1 N 2 log θ 2 D 1 θ 1 D 2 θ 2. The MLEs of θ 1 and θ 2 are obtained as ˆθ 1 = D 1 N 1 and ˆθ2 = D 2 N 2. We can similarly develop here conditional inference (conditioned on N 1 1 and N 2 1), basing it on truncated trinomial distribution.

79 p. 38/6 9. Time-Constrained Step-Stress Test Theorem 5: The conditional PDF of ˆθ 1 }, given {1 N 1 n 1 and 1 N 2 n N 1, is n 1 (x) = C fˆθ1 n i=1 i k=0 where f G ( ) is the gamma density as before, C ik f G ( x τ ik ;i, i θ 1 ), τ ik = 1 i (n i + k)τ, p = G 1 1 1(τ 1 ) = 1 e τ 1 /θ 1, p 2 = G 2 (τ 2 ) G 1 (τ 1 ) = `1 o p 1 n1 e (τ 2 τ 1 )/θ 2, 1 p 3 = 1 p 1 p 2, C n = 1 (1 p 1 ) n (1 p 2 ) n + p n 3 C ik = ( 1) k n i n`1 p1 n i p n i o`1 3 p1 k. i k,

80 p. 39/6 9. Time-Constrained Step-Stress Test Theorem 6: The conditional PDF of ˆθ 2 }, given {1 N 1 n 1 and 1 N 2 n N 1, is n 1 (x) = C fˆθ2 n i=1 n i j=1 j k=0 C ijk f G ( x τ ijk ;j, j θ 2 ), where τ ijk = 1 j (n i j + k)(τ 2 τ 1 ), C ijk = ( 1) k n j i, j, n i j k p i j k. `1 1 pn i j+k 3 p1

81 p. 39/6 9. Time-Constrained Step-Stress Test Theorem 6: The conditional PDF of ˆθ 2 }, given {1 N 1 n 1 and 1 N 2 n N 1, is n 1 (x) = C fˆθ2 n i=1 n i j=1 j k=0 C ijk f G ( x τ ijk ;j, j θ 2 ), where τ ijk = 1 j (n i j + k)(τ 2 τ 1 ), C ijk = ( 1) k n j i, j, n i j k p i j k. `1 1 pn i j+k 3 p1 We obtained the following CIs for θ 1 and θ 2 based on earlier data for different choices of the time constraint, viz., τ 2.

82 p. 40/6 9. Time-Constrained Step-Stress Test Table 4: Interval estimation for θ 1 with τ 1 = 5 and different τ 2 τ 2 Method 90% 95% 6.00 Bootstrap (BCa) ( , ) ( , ) Approximation ( , ) ( , ) Exact ( , ) ( , ) 8.00 Bootstrap (BCa) ( , ) ( , ) Approximation ( , ) ( , ) Exact ( , ) ( , ) Bootstrap (BCa) ( , ) ( , ) Approximation ( , ) ( , ) Exact ( , ) ( , )

83 p. 41/6 9. Time-Constrained Step-Stress Test Table 5: Interval estimation for θ 2 with τ 1 = 5 and different τ 2 τ 2 Method 90% 95% 6.00 Bootstrap (BCa) ( , ) ( , ) Approximation ( , ) ( , ) Exact ( , ) ( , ) 8.00 Bootstrap (BCa) ( , ) ( , ) Approximation ( , ) ( , ) Exact ( , ) ( , ) Bootstrap (BCa) ( , ) ( , ) Approximation ( , ) ( , ) Exact ( , ) ( , )

84 p. 42/6 10. Progressive Type-II Censoring and Results Progressive Type-II Censoring

85 p. 43/6 10. Progressive Type-II Censoring and Results Model Description

86 p. 44/6 10. Progressive Type-II Censoring and Results Set-up

87 p. 44/6 10. Progressive Type-II Censoring and Results Set-up Let us consider a simple step-stress model when the observed failure data are progressively Type-II censored. Exp(θ 1 ) and Exp(θ 2 ) are the distributions. Let N 1 = no. of units that fail before time τ at stress level s 0 N 2 = no. of units that fail after time τ at stress level s 1. Observed data are } t = {t 1:r:n < < t N1:r:n τ < t N1+1:r:n < < t r:r:n.

88 p. 45/6 10. Progressive Type-II Censoring and Results Maximum Likelihood Estimation

89 p. 45/6 10. Progressive Type-II Censoring and Results Maximum Likelihood Estimation The likelihood function is = L(θ 1, θ 2 t) = C p 8 >< >: C p θ1 r C p θ r 2 n exp n exp C p θ N 1 1 θn 2 2 ( ry k=1 1 θ 1 P r k=1 1 θ 2 P r k=1 e 1 θ 1 D 1 1 θ 2 D 2 g`t k:r:n h 1 G`t k:r:n ir k ) `Rk + 1 t k:r:n o if N 1 = r and N 2 = 0 `Rk + 1 `t k:r:n τ 1 θ 1 P r k=1 `Rk + 1 τ if N 1 = 0 and N 2 = r otherwise, where r = N 1 + N 2 (2 r n), C p = Q r j=1 P r k=j (R k + 1) and o D 1 = N 1 X `Rk +1 t k:r:n +τ rx `Rk +1, D 2 = rx `Rk +1 `t k:r:n τ. k=1 k=n 1 +1 k=n 1 +1

90 p. 46/6 10. Progressive Type-II Censoring and Results The MLEs of θ 1 and θ 2 exist only if at least one failure occurs before τ and after τ.

91 p. 46/6 10. Progressive Type-II Censoring and Results The MLEs of θ 1 and θ 2 exist only if at least one failure occurs before τ and after τ. They are ˆθ 1 = D 1 N 1 and ˆθ2 = D 2 N 2.

92 p. 46/6 10. Progressive Type-II Censoring and Results The MLEs of θ 1 and θ 2 exist only if at least one failure occurs before τ and after τ. They are ˆθ 1 = D 1 N 1 and ˆθ2 = D 2 N 2. In this case, we can develop exact conditional inference based on the conditional PMF P θ1,θ 2,c{ N1 = i } { = P N 1 = i } 1 N 1 r 1, i = 1,,r.

93 p. 47/6 10. Progressive Type-II Censoring and Results Notation

94 p. 47/6 10. Progressive Type-II Censoring and Results Notation Let us denote M 12 (ν, ω N 1 ) for the joint CMGF of ˆθ1 and ˆθ 2, and M k ( ω N1 ) for the CMGF of ˆθk, k = 1, 2.

95 p. 47/6 10. Progressive Type-II Censoring and Results Notation Let us denote M 12 (ν, ω N 1 ) for the joint CMGF of ˆθ1 and ˆθ 2, and M k ( ω N1 ) for the CMGF of ˆθk, k = 1, 2. Then, M 12 ( ν, ω N1 ) M k ( ω N1 ) { } = E e νˆθ 1 +ωˆθ 2 1 N 1 r 1 = r 1 i=1 } E θ1,θ 2 {e νˆθ 1 +ωˆθ 2 N 1 = i { } = E e ωˆθ k 1 N 1 r 1 = r 1 i=1 } E θ1,θ 2 {e ωˆθ k N 1 = i P θ1,θ 2,c{ N1 = i }, P θ1,θ 2,c{ N1 = i }.

96 p. 48/6 10. Progressive Type-II Censoring and Results Lemma 6: Let T 1:r:n < < T r:r:n denote the progressively Type-II censored sample from the cumulative exposure PDF g(t).

97 p. 48/6 10. Progressive Type-II Censoring and Results Lemma 6: Let T 1:r:n < < T r:r:n denote the progressively Type-II censored sample from the cumulative exposure PDF g(t). Then, the joint density function of T 1:r:n,, T r:r:n is [see Balakrishnan and Aggarwala (2000)] f`t 1,..., t r = C p ( N1 Y k=1 g 1 `tk h 1 G 1`tk ir k ) ( ry k=n 1 +1 g 2`tk h 1 G 2`tk ir k ), 0 < t 1 < < t N1 τ < t N1 +1 < < t r ;

98 p. 48/6 10. Progressive Type-II Censoring and Results Lemma 6: Let T 1:r:n < < T r:r:n denote the progressively Type-II censored sample from the cumulative exposure PDF g(t). Then, the joint density function of T 1:r:n,, T r:r:n is [see Balakrishnan and Aggarwala (2000)] f`t 1,..., t r = C p ( N1 Y k=1 g 1 `tk h 1 G 1`tk ir k ) ( ry k=n 1 +1 g 2`tk h 1 G 2`tk ir k ), 0 < t 1 < < t N1 τ < t N1 +1 < < t r ; further, the probability of the event N 1 = i, i = 1,..., r 1 is P N 1 = i = C p ix k=0 r i 1 X l=0 C k,i (S i )C l,r i 1 (S i+l ) B l.r i (S i+l ) exp ( τ θ 1 rx j=i k+1 S j ),

99 p. 49/6 10. Progressive Type-II Censoring and Results where S j = R j + 1, S i = (S 1,..., S i ), S i+l = (S i+1,..., S i+l ), B l.r i (S i+l ) = r i X j=r i l S i+j with 0X A j 0, j=i C k,i (S i ) = n Qk j=1 ( 1) k P i k+j m=i k+1 S m on Qi k j=1 P o, i k m=j S m C l,r i 1 (S i+l ) = n Ql j=1 P r i l+j 1 m=r i l ( 1) l S i+m on Qr i l 1 j=1 P r i l 1 m=j S i+m o, with Q 0 j=1 A j 1 and C p is as given earlier.

100 p. 50/6 10. Progressive Type-II Censoring and Results Lemma 7: The joint conditional density of T 1:r:n,...,T r:r:n, given N 1 = i, is given by [see Balakrishnan and Aggarwala (2000)] = f ( t 1,...,t r N 1 = i ) C p P { N 1 = i } { i k=1 g 1 ( tk ) [ 1 G 1 ( tk ) ] R k } { r k=i+1 g 2 ( tk ) [ 1 G 2 ( tk ) ] R k }, 0 < t 1 < < t i τ < t i+1 < < t r.

101 p. 51/6 10. Progressive Type-II Censoring and Results Conditional MGFs

102 p. 51/6 10. Progressive Type-II Censoring and Results Conditional MGFs Using Lemmas 6 and Lemma 7, it can be shown that M 12 (ν, ω N 1 ) = D M 1 (ω N 1 ) = D M 2 (ω N 1 ) = D r 1 X ix i=1 k=0 r 1 X i=1 k=0 r 1 X ix ix i=1 k=0 r i 1 X l=0 r i 1 X l=0 r i 1 X l=0 D ikl D ikl e τ i D ikl e τ i P rj=i k+1 S j ν `1 θ 1i ν i`1 θ 2 P rj=i k+1 S j ω `1 θ 1i ω i 1 `1 θ 2 ω r i, r i r i ω r i,, where D = C p P r 1 j=1 P N 1 = j, D ikl = C k,i(s i )C l,r i 1(S i+l) B l.r i (S i+l ) and C p is as defined earlier. exp ( τ θ 1 rx j=i k+1 S j )

103 p. 52/6 10. Progressive Type-II Censoring and Results Theorem 7: The joint conditional PDF of ˆθ 1 and ˆθ 2, given 1 N 1 r 1, is fˆθ1,ˆθ 2 (x,y) = D r 1 i=1 i k=0 r i 1 l=0 ( D ikl f G x τ ik ;i, θ 1 i ) θ 2, r i f G (y;r i, where τ ik = τ i r j=i k+1 (R j + 1). )

104 p. 53/6 10. Progressive Type-II Censoring and Results Theorem 8: The conditional PDF of ˆθ 1, given 1 N 1 r 1, is fˆθ1 (x) = D r 1 i=1 i k=0 r i 1 l=0 ( D ikl f G x τ ik ;i, θ ) 1. i

105 p. 53/6 10. Progressive Type-II Censoring and Results Theorem 8: The conditional PDF of ˆθ 1, given 1 N 1 r 1, is fˆθ1 (x) = D r 1 i=1 i k=0 r i 1 l=0 ( D ikl f G x τ ik ;i, θ ) 1. i Theorem 9: The conditional PDF of ˆθ 2, given 1 N 1 r 1, is fˆθ2 (x) = D r 1 i=1 i k=0 r i 1 l=0 D ikl f G (x;r i, ) θ 2. r i

106 p. 54/6 10. Progressive Type-II Censoring and Results Theorem 10: The mean and variance of ˆθ 1 and ˆθ 2 are

107 p. 54/6 10. Progressive Type-II Censoring and Results Theorem 10: The mean and variance of ˆθ 1 and ˆθ 2 are E (ˆθ1 ) = θ1 + D Var (ˆθ1 ) r 1 = D E (ˆθ2 ) = θ2, i=1 Var (ˆθ2 ) r 1 = D i=1 r 1 i=1 i k=0 i k=0 i k=0 r i 1 l=0 r i 1 ( r i 1 l=0 l=0 D ikl D r 1 i=1 D ikl D ikl τ ik, ( ) τ 2 + θ2 1 ik i i k=0 θ 2 2 r i. r i 1 l=0 D ikl τ ik ) 2,

108 p. 55/6 10. Progressive Type-II Censoring and Results Remark 4: We observe that ˆθ 1 is a biased estimator of θ 1 while ˆθ 2 is an unbiased estimator of θ 2.

109 10. Progressive Type-II Censoring and Results Remark 4: We observe that ˆθ 1 is a biased estimator of θ 1 while ˆθ 2 is an unbiased estimator of θ 2. Furthermore, from the joint density of ˆθ 1 and ˆθ 2 in Theorem 7, we obtain E (ˆθ1ˆθ2 ) r 1 = D = D i=1 r 1 i=1 i k=0 i k=0 = E (ˆθ1 ) E (ˆθ2 ) r i 1 l=0 r i 1 l=0 D ikl ( τ ik + i θ )[ 1 (r i) θ ] 2 i r i D ikl (τ ik + θ 1 ) θ2 so that Cov (ˆθ1, ˆθ 2 ) = 0. p. 55/6

110 p. 56/6 10. Progressive Type-II Censoring and Results Theorem 11: The tail probabilities of ˆθ 1 and ˆθ 2, given 1 N 1 r 1, are

111 p. 56/6 10. Progressive Type-II Censoring and Results Theorem 11: The tail probabilities of ˆθ 1 and ˆθ 2, given 1 N 1 r 1, are } P θ1 {ˆθ1 > ξ } P θ2 {ˆθ2 > ξ = D = D r 1 i=1 r 1 i=1 i k=0 i k=0 where w = max {0,w}, and r i 1 l=0 r i 1 l=0 ( ) i D ikl Γ ξ τik ;i, θ 1 ( ) r i D ikl Γ ξ ;r i, θ 2 Γ(w;α) = w f G ( x;α,1 ) dx = w 1 Γ(α) xα 1 e x dx.

112 p. 57/6 10. Progressive Type-II Censoring and Results Simulation Results and Comments

113 p. 57/6 10. Progressive Type-II Censoring and Results Simulation Results and Comments The coverage probabilities (in %) of CIs for θ 1 and θ 2 based on 1000 simulations and M = 1000 replications with n = 20, r = 8, θ 1 = e 2.5 and θ 2 = e 1.5 are presented in the following Tables 6 and 7.

114 Table 6: Coverage Probabilities of CIs for θ 1 90% C.I. 95% C.I. PCS τ Bootstrap App. Exact Bootstrap App. Exact P St BCa P St BCa (7 0, 12) (12, 7 0) (6 0, 6, 6)

115 Table 7: Coverage Probabilities of CIs for θ 2 90% C.I. 95% C.I. PCS τ Bootstrap App. Exact Bootstrap App. Exact P St BCa P St BCa (7 0, 12) (12, 7 0) (6 0, 6, 6)

116 p. 58/6 10. Progressive Type-II Censoring and Results Comments:

117 p. 58/6 10. Progressive Type-II Censoring and Results Comments: The approximate CIs and the Studentized-t bootstrap CIs are both unsatisfactory in terms of coverage probabilities. The percentile bootstrap method seems to be sensitive for small values of τ 1 and τ 2, the method does improve for larger sample size. Among all three bootstrap methods, the adjusted percentile method seems to be the one with somewhat satisfactory coverage probabilities (not so for θ 1 when τ 1 is small). Use the exact method whenever possible, and use the adjusted percentile method in case of large sample size when the computation of the exact CIs becomes difficult.

118 p. 59/6 10. Progressive Type-II Censoring and Results Optimal Sampling Scheme:

119 p. 59/6 10. Progressive Type-II Censoring and Results Optimal Sampling Scheme: With (R 1,..., R r ) as the progressive censoring scheme, we may consider the optimal choice of R = (R 1,..., R r ), denoted by R = (R 1,..., R r).

120 p. 59/6 10. Progressive Type-II Censoring and Results Optimal Sampling Scheme: With (R 1,..., R r ) as the progressive censoring scheme, we may consider the optimal choice of R = (R 1,..., R r ), denoted by R = (R 1,..., R r). Variance Optimality r 1 ψ`r = Var`ˆθ1 + 2 ˆθ X = D ix i=1 k=0 r i 1 X l=0 D D ikl r 1 X ix i=1 k=0 τ 2 ik + θ2 1 i r i 1 X l=0 «+ θ2 2 r i D ikl τ ik! 2.

121 p. 59/6 10. Progressive Type-II Censoring and Results Optimal Sampling Scheme: With (R 1,..., R r ) as the progressive censoring scheme, we may consider the optimal choice of R = (R 1,..., R r ), denoted by R = (R 1,..., R r). Variance Optimality r 1 ψ`r = Var`ˆθ1 + 2 ˆθ X = D MSE Optimality ix i=1 k=0 r i 1 X l=0 D D ikl r 1 X ix i=1 k=0 τ 2 ik + θ2 1 i r i 1 X l=0 «+ θ2 2 r i D ikl τ ik! 2. r 1 X ϕ`r = MSE`ˆθ1 + Var`ˆθ2 = D ix i=1 k=0 r i 1 X l=0 D ikl τ 2 ik + θ2 1 i «+ θ2 2. r i

122 p. 60/6 10. Progressive Type-II Censoring and Results Determination of Optimal Censoring Schemes:

123 p. 60/6 10. Progressive Type-II Censoring and Results Determination of Optimal Censoring Schemes: For θ 1 = e 1.5 and θ 2 = e 0.5, we present in Tables 8 and 9 the best and worst censoring schemes determined under variance optimality and mean square error optimality, respectively, for different choices of n, r and τ.

124 p. 60/6 10. Progressive Type-II Censoring and Results Determination of Optimal Censoring Schemes: For θ 1 = e 1.5 and θ 2 = e 0.5, we present in Tables 8 and 9 the best and worst censoring schemes determined under variance optimality and mean square error optimality, respectively, for different choices of n, r and τ. The relative efficiency values of worst to best censoring schemes presented in these two tables reveal the distinct advantage of adopting an optimal censoring scheme in the simple step-stress life-test.

125 Table 8: Optimal Censoring Schemes under Variance Optimality n r τ Best PCS Var Worst PCS Var RE (6, 3 0) 6.67 (0, 6, 2 0) % 3 (0, 6, 2 0) (3 0, 6) % 5 (2 0, 6, 0) (3 0, 6) % 7 (2 0, 6, 0) 9.58 (3 0, 6) % 9 (2 0, 6, 0) 9.43 (3 0, 6) % 6 1 (4, 5 0) 6.97 (0, 4, 4 0) % 3 (0, 4, 4 0) (4, 5 0) % 5 (3 0, 4, 2 0) 8.50 (4, 5 0) % 7 (3 0, 4, 2 0) 6.99 (4, 5 0) % 9 (3 0, 3, 0, 1) 6.74 (4, 5 0) % 8 1 (2, 7 0) 7.90 (0, 2, 6 0) % 3 (2 0, 2, 5 0) (2, 7 0) % 5 (3 0, 2, 4 0) 7.40 (2, 7 0) % 7 (4 0, 1, 2 0, 1) 5.71 (2, 7 0) % 9 (7 0, 2) 5.28 (1, 5 0, 1, 0) % (6, 5 0) 8.87 (0, 6, 4 0) % 3 (2 0, 6, 3 0) (6, 5 0) % 5 (3 0, 6, 2 0) 7.22 (6, 5 0) % 7 (3 0, 5, 0, 1) 6.70 (6, 5 0) % 9 (3 0, 4, 0, 2) 6.66 (6, 5 0) % 8 1 (4, 7 0) 9.80 (0, 4, 6 0) % 3 (2 0, 4, 5 0) (4, 7 0) % 5 (4 0, 3, 2 0, 1) 6.05 (4, 7 0) % 7 (4 0, 2, 2 0, 2) 5.23 (4, 7 0) % 9 (4 0, 1, 2 0, 3) 5.08 (3, 5 0, 1, 0) % 10 1 (2, 9 0) (2 0, 2, 7 0) % 3 (3 0, 2, 6 0) (2, 9 0) % 5 (9 0, 2) 5.40 (2, 9 0) % 7 (9 0, 2) 4.46 (8 0, 2, 0) % 9 (9 0, 2) 4.29 (8 0, 2, 0) %

126 Table 9: Optimal Censoring Schemes under MSE Optimality n r τ Best PCS MSE Worst PCS MSE RE (%) (6, 3 0) 6.69 (0, 6, 2 0) % 3 (0, 6, 2 0) (3 0, 6) % 5 (2 0, 6, 0) (3 0, 6) % 7 (2 0, 6, 0) (3 0, 6) % 9 (2 0, 6, 0) (3 0, 6) % 6 1 (4, 5 0) 7.06 (0, 4, 4 0) % 3 (2 0, 4, 3 0) (5 0, 4) % 5 (3 0, 4, 2 0) 9.26 (5 0, 4) % 7 (3 0, 4, 2 0) 8.54 (5 0, 4) % 9 (3 0, 4, 2 0) 9.62 (5 0, 4) % 8 1 (2, 7 0) 8.24 (0, 2, 6 0) % 3 (2 0, 2, 5 0) (2, 7 0) % 5 (3 0, 2, 4 0) 7.74 (2, 7 0) % 7 (4 0, 2, 3 0) 6.31 (7 0, 2) % 9 (4 0, 2, 3 0) 6.53 (7 0, 2) % (6, 5 0) 9.21 (2 0, 6, 3 0) % 3 (2 0, 6, 3 0) (5 0, 6) % 5 (3 0, 6, 2 0) 7.94 (5 0, 6) % 7 (3 0, 6, 2 0) 8.37 (5 0, 6) % 9 (2, 2 0, 4, 2 0) 9.63 (5 0, 6) % 8 1 (4, 7 0) (2 0, 4, 5 0) % 3 (2 0, 4, 5 0) (4, 7 0) % 5 (4 0, 4, 3 0) 6.31 (7 0, 4) % 7 (4 0, 4, 3 0) 5.97 (7 0, 4) % 9 (1, 2 0, 1, 2, 3 0) 6.51 (7 0, 4) % 10 1 (2, 9 0) (2 0, 2, 7 0) % 3 (3 0, 2, 6 0) (2, 9 0) % 5 (4 0, 2, 5 0) 5.57 (2, 9 0) % 7 (5 0, 2, 4 0) 4.83 (9 0, 2) % 9 (5 0, 2, 4 0) 5.03 (9 0, 2) %

127 10. Progressive Type-II Censoring and Results p. 61/6

128 p. 62/6 11. Other Extensions and Generalizations Some other extensions and generalizations have been carried out:

129 p. 62/6 11. Other Extensions and Generalizations Some other extensions and generalizations have been carried out: Other forms of censoring, such as hybrid censoring, have been considered; Extensions of these results to the multiple-step stress model have been done.

130 p. 63/6 12. Further Problems of Interest Generalization to k-step stress model and discuss exact conditional inference for the full-parameter model;

131 p. 63/6 12. Further Problems of Interest Generalization to k-step stress model and discuss exact conditional inference for the full-parameter model; With a link function (connecting the mean lifetimes to the stress levels), discuss inference for the parameters in this reduced-parameter model and compare efficiency;

132 p. 63/6 12. Further Problems of Interest Generalization to k-step stress model and discuss exact conditional inference for the full-parameter model; With a link function (connecting the mean lifetimes to the stress levels), discuss inference for the parameters in this reduced-parameter model and compare efficiency; Test for the suitability of this reduced-parameter model;

133 p. 63/6 12. Further Problems of Interest Generalization to k-step stress model and discuss exact conditional inference for the full-parameter model; With a link function (connecting the mean lifetimes to the stress levels), discuss inference for the parameters in this reduced-parameter model and compare efficiency; Test for the suitability of this reduced-parameter model; Generalizations to other life-time models such as Weibull and lognormal.

134 p. 64/6 This talk is based on [1] Balakrishnan, N., Kundu, D., Ng, H. K. T. and Kannan, N. (2006). Point and interval estimation for a simple step-stress model with Type-II censoring, Journal of Quality Technology (to appear). [2] Balakrishnan, N., Xie, Q. and Kundu, D. (2006). Exact inference for a simple step-stress model from the exponential distribution under time constraint, Ann. Inst. Statist. Math. (to appear). [3] Balakrishnan, N. and Xie, Q. (2006a). Exact inference for a simple step-stress model with Type-I hybrid censored data from the exponential distribution, J. Statist. Plann. Inf. (to appear). [4] Balakrishnan, N. and Xie, Q. (2006b). Exact inference for a simple step-stress model with Type-II hybrid censored data from the exponential distribution, J. Statist. Plann. Inf. (to appear). [5] Balakrishnan, N., Xie, Q. and Han, D. (2006). Exact inference and optimal censoring scheme for a simple step-stress model under progressive Type-II censoring, (Submitted for publication). [6] Gouno, E., Sen, A. and Balakrishnan, N. (2004). Optimal step-stress test under progressive Type-I censoring, IEEE Trans. on Reliab., 53,

135 p. 65/6 13. Bibliography Arnold, B. C., Balakrishnan, N. and Nagaraja, H.N.(1992). A First Course in Order Statistics, John Wiley & Sons, New York. Bagdonavicius, V. and Nikulin, M. (2002). Accelerated Life Models: Modeling and Statistical Analysis, Chapman and Hall/CRC Press, Boca Raton, Florida. Bai, D. S., Kim, M. S. and Lee, S. H. (1989). Optimum simple step-stress accelerated life test with censoring, IEEE Transactions on Reliability, 38, Balakrishnan, N. and Aggarwala, R. (2000). Progressive Censoring: Theory, Methods, and Applications. Birkhäuser, Boston. Bhattacharyya, G. K. and Zanzawi, S. (1989). A tampered failure rate model for step-stress accelerated life test, Communications in Statistics Theory and Methods, 18, DeGroot, M. H. and Goel, P. K. (1979). Bayesian estimation and optimal design in partially accelerated life testing, Naval Research Logistics Quarterly, 26, Efron, B. and Tibshirani, R. (1993). An Introduction to the Booststrap, Chapman and Hall, New York.

136 p. 66/6 13. Bibliography Khamis, I. H. and Higgins, J. J. (1998). A new model for step-stress testing, IEEE Transactions on Reliability, 47, Madi, M. T. (1993). Multiple step-stress accelerated life test: the tampered failure rate model, Communications in Statistics Theory and Methods, 22, Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data, John Wiley & Sons, New York. Nelson, W. (1980). Accelerated life testing: step-stress models and data analysis, IEEE Transactions on Reliability, 29, Nelson, W. (1990). Accelerated Testing: Statistical Models, Test Plans, and Data Analyses, John Wiley & Sons, New York. Xiong, C. (1998). Inference on a simple step-stress model with Type-II censored exponential data, IEEE Transactions on Reliability, 47, Xiong, C. and Milliken, G.A. (1999). Step-stress life-testing with random stress change times for exponential data, IEEE Transactions on Reliability, 48,