在統計品質管制中，若有興趣的問題是查驗產品以決定是否接受或者拒絕此產品，我們將這類的查驗程序稱之為允收抽樣。如果產品的品質特性是產品壽命，則允收抽樣的問題就轉變成壽命試驗的問題。壽命試驗的目地在於當壽命試驗顯示產品壽命超過期望標準時，我們選擇接受此批產品；相對地，當壽命試驗顯示產品壽命未到達標準時，我們將會拒絕此批產品。假如產品的壽命很長，則等待直到全部產品壽命結束以完成壽命試驗可能將會相當費時。因此，在本篇論文中，我們規劃出當產品壽命服從Inverse Gaussian分配與Birnbaum-Saunders分配時的截略壽命試驗，以降低完成試驗所需的時間，並說明如何推算出此允收抽樣計劃的相關參數值。在每個抽樣計劃之後，本論文也提供部份實用的表列值以供當產品壽命服從Inverse Gaussian分配與Birnbaum-Saunders分配時，使用者可以直接使用本文建議的允收抽樣計劃。

In statistical quality control, the type of inspection procedure is usually called acceptance sampling when the inspection of products is for the purpose of acceptance or rejection a product. If the quality characteristic is the lifetime of product, the problem of accepance sampling becomes the life test. One objective of the test is to accept the lot when the test shows that the mean life of products exceeds the standard; otherwise, we reject the lot if the test shows that the mean life of products below the standard. Frequently, it might time consuming to wait until all the products fail in a life test if the lifetimes of products are high. In this thesis, a truncated life test is proposed to the Inverse Gaussian and Birnbaum-Saunders life data, which can save the testing time. An algorithm is provided to obtain the sampling plans based on the truncated life test. Moreover, some useful tables are provided for the sampling plans under those two lifetime distributions.

1 Introduction 1
2 Sampling Plan Based on Inverse Gaussian Distribution 3
2.1 Inverse Gaussian Distribution and Its Moments . . . . . . . . 3
2.2 The Sampling Plan . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Steps for Constructing Tables . . . . . . . . . . . . . . . . . . . . 11
2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Sampling Plan Based on Birnbaum-Saunders Distribution 26
3.1 Birnbaum-Saunders Distribution and Its Moments . . . . . . 26
3.2 The Sampling Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Steps for Constructing Tables . . . . . . . . . . . . . . . . . . . . 33
3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Conclusions 48
Bibliography 49

List of Figures
1 IG distribution c.d.f. for β =1,2,4, and θ =1. . . . . . . . . . . . 6
2 IG distribution p.d.f. for β =1,2,4, and θ =1. . . . . . . . . . . . 7
3 Curves of the IG c.d.f. versus t/θ when β =1,2,3,4. . . . . . . .  . 10
4 BS distribution c.d.f. for γ =0.5,0.6,1.0, and δ =1. . . . . . . . . 28
5 BS distribution p.d.f. for γ =0.5,0.6,1.0, and δ =1. . . . . . . . . 29
6 Curves of the BS c.d.f. versus t/δ when γ =0.5,0.6,0.7,1. . . . . . 32


List of Tables
1 Minimum sample sizes necessary to assert the average life to exceed a
given value, μ0, with probability P_ and the corresponding acceptance
number, c, for the IG lifetime distribution with β =1.. . . . . . . . . 14

2 Minimum sample sizes necessary to assert the average life to exceed a
given value, μ0, with probability P_ and the corresponding acceptance
number, c, for the IG lifetime distribution with β =2.. . . . . . . . . 15

3 Minimum sample sizes necessary to assert the average life to exceed a
given value, μ0, with probability P_ and the corresponding acceptance
number, c, for the IG lifetime distribution with β =3.. . . . . . . . . 16

4 Minimum sample sizes necessary to assert the average life to exceed a
given value, μ0, with probability P_ and the corresponding acceptance
number, c, for the IG lifetime distribution with β =4.. . . . . . . . . 17

5 Operating characteristic values of the sampling plan (n, c, t/μ0) for a
given P_, under the IG lifetime distribution with β =1. . . . . . . . . 18

6 Operating characteristic values of the sampling plan (n, c, t/μ0) for a
given P_, under the IG lifetime distribution with β =2. . . . . . . . . 19

7 Operating characteristic values of the sampling plan (n, c, t/μ0) for a
given P_, under the IG lifetime distribution with β =3. . . . . . . . . 20

8 Operating characteristic values of the sampling plan (n, c, t/μ0) for a
given P_, under the IG lifetime distribution with β =4. . . . . . . . . 21

9 Minimum ratio of true mean life to specified mean life for the acceptability
of a lot, for the IG lifetime distribution with parameter β = 1
and producer’s risk of 0.05. . . . . . . . . . . . . .. . . . . . . . . 22

10 Minimum ratio of true mean life to specified mean life for the acceptability of a lot, for the IG lifetime distribution with parameter β = 2 and producer’s risk of 0.05. . . . . . . . . . . .. . . . . . . . . 23

11 Minimum ratio of true mean life to specified mean life for the acceptability of a lot, for the IG lifetime distribution with parameter β = 3 and producer’s risk of 0.05. . . . .  . . . . . . . . . . . . . . . 24

12 Minimum ratio of true mean life to specified mean life for the acceptability of a lot, for the IG lifetime distribution with parameter β = 4 and producer’s risk of 0.05. . . . . .  . . . . . . . . . . . . . . 25

13 Minimum sample sizes necessary to assert the average life to exceed a
given value, μ0, with probability P_ and the corresponding acceptance
number, c, for the BS lifetime distribution with γ = 0.5. . . . . . . . 36

14 Minimum sample sizes necessary to assert the average life to exceed a
given value, μ0, with probability P_ and the corresponding acceptance
number, c, for the BS lifetime distribution with γ = 0.6. . . . . . . . 37

15 Minimum sample sizes necessary to assert the average life to exceed a
given value, μ0, with probability P_ and the corresponding acceptance
number, c, for the BS lifetime distribution with γ = 0.7. . . . . . . . 38

16 Minimum sample sizes necessary to assert the average life to exceed a
given value, μ0, with probability P_ and the corresponding acceptance
number, c, for the BS lifetime distribution with γ = 1.0. . . . . . . . 39

17 Operating characteristic values of the sampling plan (n, c, t/μ0) for a given P_, under the BS lifetime distribution with γ = 0.5. . . . . . 40

18 Operating characteristic values of the sampling plan (n, c, t/μ0) for a given P_, under the BS lifetime distribution with γ = 0.6. . . . . . 41

19 Operating characteristic values of the sampling plan (n, c, t/μ0) for a given P_, under the BS lifetime distribution with γ = 0.7. . . . . . 42

20 Operating characteristic values of the sampling plan (n, c, t/μ0) for a given P_, under the BS lifetime distribution with γ = 1.0. . . . . . 43

21 Minimum ratio of true mean life to specified mean life for the acceptability of a lot for the BS lifetime distribution with parameter γ = 0.5 and producer’s risk of 0.05. . . . . . . . . . . . . . . . . . . 44

22 Minimum ratio of true mean life to specified mean life for the acceptability of a lot for the BS lifetime distribution with parameter γ = 0.6 and producer’s risk of 0.05. . . . . . . . . . . . . . . . . . 45

23 Minimum ratio of true mean life to specified mean life for the acceptability of a lot for the BS lifetime distribution with parameter γ = 0.7 and producer’s risk of 0.05. . . .  . . . . . . . . . . . . . . 46

24 Minimum ratio of true mean life to specified mean life for the acceptability of a lot for the BS lifetime distribution with parameter γ = 1.0 and producer’s risk of 0.05. . . . . . . . . . . . . . . . . . 47

Banerjee, A.K. & Bhattacharyya, G.K. (1976), A Purchase Incidence Model with Inverse
Gaussian Interpurchase Times, Journal of the American Statistical Association
, 71, pp. 823-829.
Birnbaum, Z.W. & Saunders, S.C. (1969), A New Family of Life Distributions, Journal
of Applied Probability, 6, pp. 319-327.
Chhikara, R.S. & Folks, J.L. (1977), The Inverse Gaussian Distribution as a Lifetime
Model, Technometrics , 19, pp. 461-468.
Desmond, A.F. (1985), Stochastic Models of Failure in Random Environments, Canadian
Journal of Statistics, 13, pp. 171-183.
Durham, S.D. & Padgett, W.J. (1997), A Cumulative DamageModel for System Failure
with Application to Carbon Fibers and Composites, Technometrics, 39, pp. 34-44.
Durham, S.D. & Lynch, J.D. & Padgett, W.J. & Horan, T.J. & Owen, W.J. & Surles,
J. (1997), Localized Load-Sharing Rules and Markov-Weibull Fibers: A Comparison
of Micro-composite Failure Data with Monte Carlo Simulations, Journal of
Composite Materials , 31, pp. 1856-1882.
Goode, H.P. & Kao, J.H.K. (1961), Sampling Plans Based on the Weibull Distribution,
Proceedings of Seventh National Symposium on Reliability and Quality Control,
Philadelphia, Pennsyvania, pp. 24-40.
Gupta, S.S. & Groll, P.A. (1961), Gamma Distribution in Acceptance Sampling Based
on Life Tests, Journal of the American Statistical Association, 56, pp. 942-970.
Hasofer, A.M. (1964), A Dam with Inverse Gaussian Input, Proceedings of the Cambridge
Philosophy Society , 60, pp. 931-933.
Kantam, R.R.L. & Rosaiah, K. (1998), Half Logistic Distribution in Acceptance Sampling
Based on Life Tests, IAPQR Transactions, 23(2), pp. 117-125.
Kantam, R.R.L., Rosaiah, K. & Sprinivasa Rao, G. (2001), Acceptance Sampling Based
on Life Tests: Log-logistic Model, Journal of Applied Statistics, 28, pp. 121-128.
Lancaster, A. (1972), A Stochastic Model for the Duration of a Strike, Journal of the
Royal Statistical Society, A135, pp. 257-271.
Mann, N.R., Schafer, R.E. & Singpurwalla, N.D. (1974), Methods for Statistical Analysis
of Reliability and Life Data , New York : Wiley.
Meeker, W.Q. & Escobar, L.A. (1998), Statistical Methods for Reliability Data , New
York : Wiley.
Schr¨odinger, E. (1915), Z¨ur Theorie der Fall-und Steigversuche an Teilchenn mit
Brownsche Bewegung, Physikalische Zeitschrift, 16, pp. 289-295.
Sheppard, C.W. (1962), Basic Principles of the Tracer Method, New York: Wiley.
Sobel, M. & Tischendrof, J.A. (1959), Acceptance Sampling with Sew Life Test Objective,
Proceedings of Fifth National Symposium on Reliability and Quality Control,
Philadelphia, Pennsyvania, pp. 108-118.
Tweedie, M.C.K. (1941), A Mathmatical Investigation of Some Electrophoretic Measurements
on Colloids, M.Sc. Thesis, University of Reading, England.
Wald, A. (1944), On Cumulative Sums of Random Variables, Annal of Mathematical
Statistic, 15, pp. 283-296.
Yule, G.Y. (1944), The Statistical Study of Literary Vocabulary, Cambridge: Cambridge
University Press.