Annotated Bibliography

There are hundreds of good papers in various journals on statistical reli­ability. This annotated bibliography lists only some important textbooks in which the reader can obtain further information on the various topics of the present book. A few research papers are mentioned too. 1. Ascher, H. and Feingold, H., Repairable Systems Reliability: Mod­

eling, Inference, Misconceptions and Their Causes. Lecture Notes in Statistics, Vol. 7, Marcel Dekker, New York, 1984. In this book the reader will find discussion of important issues connected with the availability, maintainability and readiness of repairable systems.

2. Bain, L.J., Statistical Analysis of Reliability and Life-Testing Models, Marcel Dekker, New York, 1978. Provides a comprehensive statistical analysis of life distributions in the uncensored and censored cases.

3. Barlow, R.E. and Proschan, F., Mathematical Theory of Reliabil­ity, John Wiley, New York, 1965. Advanced mathematical treatment of maintenance policies and redun­dancy optimization.

4. Barlow, R.E. and Preschan, F., Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, New York, 1975. An excellent advanced introduction to the theory of reliability. This book treats probabilistic models connected with lifetime of complex systems. The book includes treatment of the case where the components of a system are dependent.

5. Bartholomew, D.J., The sampling distribution of an estimate arising in life testing, Technometrics, 5: 361-374 (1963).

200 Annotated Bibliography

Derives the CDF of the MLE, 8, in the case of Type I censoring from an exponential distribution.

6. Beyer, W.H., Standard Mathematical Tables, CRC Press, West Palm Beach, FL, 1978. Qne can find in this collection tables of Laplace transforms and their inverses.

7. Box, G.E.P. and Tiao, G.C., Bayesian Inference in Statistical Anal­ysis, Addison-Wesley, Reading, MA, 1973. Comprehensive textbook on Bayesian classical statistical analysis. Com­parison of means, variances, linear models, block designs, components of variance and regression analysis, are redone in the Bayesian framework.

8. Chow, Y.S., Robbins, H. and Siegmund, D., Great Expectations: The Theory of Optimal Stopping, Houghton Mifflin, Boston, 1971. Very advanced mathematical presentation of the theory of optimal stop­ping times.

9. Cohen, A.C., Jr., Tables for maximum likelihood estimates; singly trun­cated and single censored samples, Technometrics, 3: 535-541 (1961). Such tables were required when computers were not readily available.

10. DeGroot, M.H., Optimal Statistical Decisions, McGraw-Hill, New York,1970. An excellent introduction to the theory of optimal Bayesian decision making.

11. Gerstbakh, LB., Statistical Reliability Theory, Marcel Dekker, New York,1989. Advanced mathematical treatment of systems with renewable compo­nents and optimal preventive maintenance. Discusses also statistical aspects of lifetime data analysis.

12. Gnedenko, B.V., Belyayev, Yu. K. and Solovyev, A.D., Mathematical Methods of Reliability Theory, Academic Press, New York, 1969. An advanced monograph on reliability of renewable systems. Contains development of sequential tests.

13. Good, I.J., The Estimation of Probability: An Essay on Modern Bayesian Methods, MIT Press, Cambridge, MA, 1965. Skillfully written introduction to Bayesian estimation of probabilities and distributions.

14. Hald, A., Maximum likelihood estimation of the parameters of a normal distribution which is truncated at a known point, Skandinavisk Ak­tuar., 32: 119-134 (1949). This paper develops the theory and methodology of estimating the pa­rameters of a truncated normal distribution.

15. Henley, E.J. and Kumamoto, H., Reliability Engineering and Risk Assessment, Prentice-Hall, Englewood Cliffs, NJ, 1981. Chapter 2 provides an excellent introduction to fault tree construction. Chapter 3 discusses path and cut sets and decision tables.

Annotated Bibliography 201

16. Ireson, W.G., Editor, Reliability Handbook, McGraw-Hill, New York, 1966, reissued 1982. This is a good reference book on various reliability problems: reliability data systems, system effectiveness, mathematical and statistical models, testing, estimation, human factors, etc.

17. Johnson, N .L. and Kotz, S., Distributions in Statistics: Continuous Univariate Distributions-l, Houghton Mifflin, Boston, 1970. An excellent survey of the important families of statistical distributions and their particular estimation problems. Comprehensive bibliography for each chapter.

18. Klassen, B. and van Peppen, J.C.L., System Reliability: Concepts and Applications, Edward Arnold (A division of Hodder & Stoughton), New York, 1989. Good discussion of availability, maintainability and repairability of sys­tems.

19. Lloyd, D.K. and Lipow, M., Reliability: Management, Methods and Mathematics, 2nd Edition, published by the authors, 1977. Very authoritative textbook. Provides good introduction to reliability management, organization, data systems, etc. Good treatment of sta­tistical methods of estimation and testing (reliability demonstration). Discusses problems and methods of software reliability.

20. Mann, N.R., Schafer, R.E. and Singpurwalla, N.D., Methods for Sta­tistical Analysis of Reliability and Life Data, John Wiley, New York,1974. Includes a good chapter on testing statistical hypotheses on the parame­ters of various life distributions. Also an excellent chapter on accelerated life testing and MLE of the parameters.

2l. Martz, H.F. and Waller, R.A., Bayesian Reliability Analysis, John Wiley, New York, 1982. The book contains important information for Bayesian analysis in relia­bility theory, including a good chapter on empirical Bayes methods.

22. McCormick, N.J., Reliability and Risk Analysis. Academic Press, New York, 1981. The book provides interesting and illuminating presentation of fault tree analysis and lists available computer programs for such analysis. It pro­vides also treatment of the subject of availability of systems with repair.

23. Miller, R.G., Jr., Survival Analysis, John Wiley, New York, 1981. Treats mainly non-parametric methods in a concise manner. Provides many references.

24. Nair, V.N., Confidence Bands for Survival Functions with Censored Data: A Comparative Study, Technometrics, 26: 265-275 (1984). Compares several methods of determining confidence regions (bands) for survival functions with censored data.

25. Nelson, W., Applied Data Analysis, John Wiley, New York, 1982. This book is recommended for additional study of graphical methods

202 Auruuotated Bibliography

in life data analysis, maximum likelihood estimation with censored data and iinear estimation methods.

26. Nelson, W., Accelerated Testing: Statistical Models, Test Plans and Data Analyses, John Wiley, New York, 1990. Very authoritative book on accelerated life testing. The introductory chapter provides important background material on types of engineering applications.

27. Press, S.J., Bayesian Statistics: Principles, Models and Applica­tions, John Wiley, New York, 1989. An excellent introduction to the subject of Bayesian analysis.

28. Tsokos, C.P. and Shimi, LN., The Theory and Applications ofReli­ability with Emphasis on Bayesian and Non-Parametric Meth­ods, Academic Press, New York, 1977. A collection of articles on various topics connected with Bayesian reli­ability analysis. In particular, several articles are devoted to empirical Bayes methods.

Appendix of Statistical Tables

Table A-I: Standard Normal Cumulative Distribution Function. Area Under the Standard Normal Curve Between -00 and z

<p(z)

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.0 0.50 0.50 0.51 0.51 0.52 0.52 0.52 0.53 0.53 0.1 0.54 0.54 0.55 0.55 0.56 0.56 0.56 0.57 0.57 0.2 0.58 0.58 0.59 0.59 0.59 0.60 0.60 0.61 0.61 0.3 0.62 0.62 0.63 0.63 0.63 0.64 0.64 0.64 0.65 0.4 0.66 0.66 0.66 0.67 0.67 0.67 0.68 0.68 0.68 0.5 0.69 0.70 0.70 0.70 0.71 0.71 0.71 0.72 0.72

0.6 0.73 0.73 0.73 0.74 0.74 0.74 0.75 0.75 0.75 0.7 0.76 0.76 0.76 0.77 0.77 0.77 0.78 0.78 0.78 0.8 0.79 0.79 0.79 0.80 0.80 0.80 0.81 0.81 0.81 0.9 0.82 0.82 0.82 0.82 0.83 0.83 0.83 0.83 0.84 1.0 0.84 0.84 0.85 0.85 0.85 0.85 0.86 0.86 0.86

1.1 0.86 0.87 0.87 0.87 0.87 0.87 0.88 0.88 0.88 1.2 0.88 0.89 0.89 0.89 0.89 0.89 0.90 0.90 0.90 1.3 0.90 0.90 0.91 0.91 0.91 0.91 0.91 0.91 0.92 1.4 0.92 0.92 0.92 0.92 0.93 0.93 0.93 0.93 0.93 1.5 0.93 0.93 0.94 0.94 0.94 0.94 0.94 0.94 0.94

0.09 0.54 0.58 0.61 0.65 0.69 0.72

0.75 0.79 0.81 0.84 0.86

0.88 0.90 0.92 0.93 0.94

204 Appendix of Statistical Tables

Table A-I. (continued)

Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1.6 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 1.7 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 1.8 0.96 0.96 0.97 0.97 0.97 0.97 0.97 0.97 1.9 0.97 0.97 0.97 0.97 0.97 0.97 0.98 0.98 2.0 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98

2.1 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.99 2.2 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 2.3 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 2.4 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 2.5 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99

2.6 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.7 1.00 1.00 1.00 1.00 1.00 1.00 - 1.00 1.00 2.8 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.9 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

For values of Z < 0 use the relationship ~(-z) = 1- ~(z). Computed by use of GAUSS© software.

0.08 0.09 0.95 0.95 0.96 0.96 0.97 0.97 0.98 0.98 0.98 0.98

0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Table A-2. Selected Fractiles of the Standard Normal Distribution, za'

For Other Values of a, Compile the Value of Za from Table A-I

a Zn.

0.50 0.000 0.55 0.125 0.60 0.253 0.65 0.385 0.70 0.524 0.75 0.674 0.80 0.841 0.85 1.036 0.90 1.282 0.95 1.645 0.975 1.960 0.99 2.326 0.995 2.576

For a < .5 use za = -Zl-a. For example, Z.05 = -Z.95 = -1.645. Computed according to formula 26.2.23 of M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1968.

Appendix of Statistical Tables 205

Table A-3. Fractiles of the Chi-Square Distribution X~[v]. Pr{x2 [v] :s; X~[v]} = a

x~[vl a 0.005 0.01 0.025 0.05 0.10 0.25 1 0.04 393 0.03 157 0.03982 0.02393 0.0158 0.102 2 0.0100 0.0201 0.0506 0.103 0.211 0.575 3 0.0717 0.115 0.216 0.352 0.584 1.21 4 0.207 0.297 0.484 0.711 1.06 1.92 5 0.412 0.554 0.831 1.15 1.61 2.67

6 0.676 0.872 1.24 1.64 2.20 3.45 7 0.989 1.24 1.69 2.17 2.83 4.25 8 1.34 1.65 2.18 2.73 3.49 5.07 9 1.73 2.09 2.70 3.33 4.17 5.90

10 2.16 2.56 3.25 3.94 4.87 6.74

11 2.60 3.05 3.82 4.57 5.58 7.58 12 3.07 3.57 4.40 5.23 6.30 8.44 13 3.57 4.11 5.01 5.89 7.04 9.30 14 4.07 4.66 5.63 6.57 7.79 10.2 15 4.60 5.23 6.26 7.26 8.55 11.0

16 5.14 5.81 6.91 7.96 9.31 11.9 17 5.70 6.41 7.56 8.67 10.1 12.8 18 6.26 7.01 8.23 9.39 10.9 13.7 19 6.84 7.63 8.91 10.1 11.7 14.6 20 7.43 8.26 9.59 10.9 12.4 15.5

21 8.03 8.90 10.3 11.6 13.2 16.3 22 8.64 9.54 11.0 12.3 14.0 17.2 23 9.26 10.2 11.7 13.1 14.8 18.1 24 9.89 10.9 12.4 13.8 15.7 19.0 25 10.5 11.5 13.1 14.6 16.5 19.9

26 11.2 12:2 13.8 15.4 17.3 20.8 27 11.8 12.9 14.6 16.2 18.1 21.7 28 12.5 13.6 15.3 16.9 18.9 22.7 29 13.1 14.3 16.0 17.7 19.8 23.6 30 13.8 15.0 16.8 18.5 20.6 24.5

For v > 30, X~[v] = (za + J2v - 1)2/2. Computed with the aid of STATGRAPHICS© software.

206 Appendix of Statistical Tables

Table A-3 (continued)

X~[l/J a 0.50 0.75 0.90 0.95 0.975 0.99 0.995 1 0.455 1.32 2.71 3.84 5.02 6.63 7.88 2 1.39 2.77 4.61 5.99 7.38 9.21 10.6 3 2.37 4.11 6.25 7.81 9.35 11.3 12.8 4 3.36 5.39 7.78 9.49 11.1 13.3 14.9 5 4.35 6.63 9.24 11.1 12.8 15.1 16.7

6 5.35 7.84 10.6 12.6 14.4 16.8 18.5 7 6.35 9.04 12.0 14.1 16.0 18.5 20.3 8 7.34 10.2 13.4 15.5 17.5 20.1 22.0 9 8.34 11.4 14.7 16.9 19.0 21.7 23.6

10 9.34 12.5 16.0 18.3 2!)'5 23.2 25.2

11 10.3 13.7 17.3 19.7 21.9 24.7 26.8 12 11.3 14.8 18.5 21.0 23.3 26.2 28.3 13 12.3 16.0 19.8 22.4 24.7 27.7 29.8 14 13.3 17.1 21.1 23.7 26.1 29.1 31.3 15 14.3 18.2 22.3 25.0 27.5 30.6 32.8

16 15.3 19.4 23.5 26.3 28.8 32.0 34.3 17 16.3 20.5 24.8 27.6 30.2 33.4 35.7 18 17.3 21.6 26.0 28.9 31.5 34.8 37.2 19 18.3 22.7 27.2 30.1 32.9 36.2 38.6 20 19.3 23.8 28.4 31.4 34.2 37.6 40.0

21 20.3 24.9 29.6 32.7 35.5 38.9 41.4 22 21.3 26.0 30.8 33.9 36.8 40.3 42.8 23 22.3 27.1 32.0 35.2 38.1 41.6 44.2 24 23.3 28.2 33.2 36.4 39.4 43.0 45.6 25 24.3 29.3 34.4 37.7 40.6 44.3 46.9

26 25.3 30.4 35.6 38.9 41.9 45.6 48.3 27 26.3 31.5 36.7 40.1 43.2 47.0 49.6 28 27.3 32.6 37.9 41.3 44.5 48.3 51.0 29 28.3 33.7 39.1 42.6 45.7 49.6 52.3 30 29.3 34.8 40.3 43.8 47.0 50.9 53.7

Appendix of Statistical Tables

Table A-4. Fractiles of the t-Distribution, ta[v]. Pr{t[v] ::::; ta[v]} = a

ta[v] a 0.60 0.75 0.90 0.95 0.975 0.99 0.995 v

1 0.325 1.000 3.078 6.314 12.706 31.821 63.657 2 0.289 0.816 1.886 2.920 4.303 6.965 9.925 3 0.277 0.765 1.638 2.353 3.182 4.541 5.841 4 0.271 0.741 1.533 2.132 2.776 3.747 4.604 5 0.267 0.727 1.476 2.015 2.571 3.365 4.032

6 0.265 0.718 1.440 1.943 2.447 3.143 3.707 7 0.263 0.711 1.415 1.895 2.365 2.998 3.499 8 0.262 0.706 1.397 1.860 2.306 2.896 3.355 9 0.261 0.703 1.383 1.833 2.262 2.82L 3.250

10 0.260 0.700 1.372 1.812 2.228 2.764 3.169

11 0.260 0.697 1.363 1.796 2.201 2.718 3.106 12 0.259 0.695 1.356 1.782 2.179 2.681 3.055 13 0.259 0.694 1.350 1.771 2.160 2.650 3.012 14 0.258 0.692 1.345 1.761 2.145 2.624 2.977 15 0.258 0.691 1.341 1.753 2.131 2.602 2.947

16 0.258 0.690 1.337 1.746 2.120 2.583 2.921 17 0.257 0.689 1.333 1.740 2.110 2.567 2.898 18 0.257 0.688 1.330 1.734 2.101 2.552 2.878 19 0.257 0.688 1.328 1.729 2.093 2.539 2.861 20 0.257 0.687 1.325 1.725 2.086 2.528 2.845

21 0.257 0.686 1.323 1.721 2.080 2.518 2.831 22 0.256 0.686 1.321 1.717 2.074 2.508 2.819 23 0.256 0.685 1.319 1.714 2.069 2.500 2.807 24 0.256 0.685 1.318 1.711 2.064 2.492 2.797 25 0.256 0.684 1.316 1.708 2.060 2.485 2.787

26 0.256 0.684 1.315 1.706 2.056 2.479 2.779 27 0.256 0.684 1.314 1.703 2.052 2.473 2.771 28 0.256 0.683 1.313 1.701 2.048 2.467 2.763 29 0.256 0.683 1.311 1.699 2.045 2.462 2.756 30 0.256 0.683 1.310 1.697 2.042 2.457 2.750

40 0.255 0.681 1.303 1.684 2.021 2.423 2.704 60 0.254 0.679 1.296 1.671 2.000 2.390 2.660

120 0.254 0.677 1.289 1.658 1.980 2.358 2.617 00 0.253 0.674 1.282 1.645 1.960 2.326 2.576

For a < .5, ta[v] = -tl-a[V]; t.5[V] = 0 for all v. Computed with the aid of STATGRAPHICS© software.

207

0.9995

636.619 31.598 12.941 8.610 6.859

5.959 5.405 5.041 4.781 4.587

4.437 4.318 4.221 4.140 4.073

4.015 3.965 3.922 3.883 3.850

3.819 3.792 3.767 3.745 3.725

3.707 3.690 3.674 3.659 3.646

3.551 3.460 3.373 3.291

208 Appendix of Statistical Tables

Table A-5. Fractiles of the F-Distribution

P III 5 10 20 30 40 50 112

0.5 5 1.000 1.073 1.111 1.123 1.130 1.134 0.75 1.889 1.890 1.882 1.878 1.876 1.875 0.90 3.491 3.298 3.207 3.175 3.158 3.148 0.95 5.050 4.736 4.560 4.498 4.466 4.448 0.975 7.146 6.620 6.332 6.232 6.181 6.150 0.5 10 0.932 1.000 1.035 1.047 1.052 1.056 0.75 1.585 1.551 1.523 1.512 1.506 1.502 0.90 2.522 2.327 2.201 2.156 2.132 2.120 0.95 3.326 2.985 2.774 2.699 2.665 2.642 0.975 4.237 3.724 3.419 3.312 3.262 3.229 0.5 20 0.900 0.966 1.000 1.011 1.017 1.020 0.75 1.450 1.399 1.358 1.340 1.330 1.324 0.90 2.158 1.937 1.795 1.738 1.709 1.690 0.95 2.711 2.348 2.125 2.039 1.994 1.966 0.975 3.289 2.774 2.467 2.349 2.207 2.250 0.5 30 0.890 0.955 0.989 1.000 1.006 1.009 0.75 1.407 1.351 1.303 1.282 1.270 1.263 0.90 2.049 1.820 1.667 1.607 1.573 1.552 0.95 2.534 2.165 1.932 1.841 1.792 1.761 0.975 3.027 2.511 2.195 2.075 2.009 1.968 0.5 40 0.885 0.950 0.983 0.994 1.000 1.003 0.75 1.386 1.327 1.276 1.253 1.240 1.231 0.90 1.997 1.763 1.605 1.541 1.506 1.483 0.95 2.450 2.077 1.839 1.744 1.693 1.660 0.975 2.904 2.388 2.068 1.943 1.876 1.832

For p < .05 apply the relationship Fp[Vb V2] = 1/ FI- p[V2, VI]. Computed with the aid of STATGRAPHICS© software.

120

1.142 1.872 3.139 4.435 6.135 1.064 1.492 2.085 2.586 3.149 1.028 1.308 1.644 1.897 2.157 1.017 1.242 1.499 1.684 1.867 1.011 1.208 1.425 1.576 1.724

Appendix of Statistical Tables 209

Table A-6. Gamma Function

r(x),O.Ol :s; x :s; 1.00

x r(x) x r(x) x r(x) x r(x) 0.01 99.4326 0.26 3.4785 0.51 1.7384 0.76 1.2123

0.02 .49.4422 0.27 3.3426 0.52 1.7058 0.77 1.1997

0.03 32.7850 0.28 3.2169 0.53 1.6747 0.78 1.1875

0.04 24.4610 0.29 3.1001 0.54 1.6448 0.79 1.1757

0.05 19.4701 0.30 2.9916 0.55 1.6161 0.80 1.1642

0.06 16.1457 0.31 2.8903 0.56 1.5886 0.81 1.1532

0.07 13.7736 0.32 2.7958 0.57 1.5623 0.82 1.1425

0.08 11.9966 0.33 2.7072 0.58 1.5369 0.83 1.1322 0.09 10.6162 0.34 2.6242 0.59 1.5126 0.84 1.1222 0.10 9.5135 0.35 2.5461 0.60 1.4892 0.85 1.1125

0.11 8.6127 0.36 2.4727 0.61 1.4667 0.86 1.1031

0.12 7.8633 0.37 2.4036 0.62 1.4450 0.87 1.0941

0.13 7.2302 0.38 2.3383 0.63 1.4242 0.88 1.0853

0.14 6.6887 0.39 2.2765 0.64 1.4041 0.89 1.0768

0.15 6.2203 0.40 2.2182 0.65 1.3848 0.90 1.0686

0.16 5.8113 0.41 2.1628 0.66 1.3662 0.91 1.0607

0.17 5.4512 0.42 2.1104 0.67 1.3482 0.92 1.0530

0.18 5.1318 0.43 2.0605 0.68 1.3309 0.93 1.0456

0.19 4.8468 0.44 2.0132 0.69 1.3142 0.94 1.0384

0.20 4.5908 0.45 1.9681 0.70 1.2981 0.95 1.0315

0.21 4.3599 0.46 1.9252 0.71 1.2825 0.96 1.0247

0.22 4.1505 0.47 1.8843 0.72 1.2675 0.97 1.0182

0.23 3.9598 0.48 1.8453 0.73 1.2530 0.98 1.0119

0.24 3.7855 0.49 1.8081 0.74 1.2390 0.99 1.0059

0.25 3.6256 0.50 1.7725 0.75 1.2254 1.00 1.0000

Index

accelerated life testing, 195,203 single, 13 accuracy, 111 Type I, 13, 133 actuarial estimator, 102 Type II, 13, 135 administrative time, 3 censoring fraction, 136 angular transformation, 139 central limit theorem, 30 Arrhenius reaction model, 204 conditional consumer's risk, 199 asymptotic conditional producer's risk, 199

availability, 81 confidence interval, 116 covariance matrix, 129 conjugate prior distribution, 171 operational reliability, 82 convolution, 74 standard deviation, 132 coverage probability, 113

availability, 3,8 credibility intervals, 174 intrinsic, 3 cumulative distribution, 5

availability function, 79 cut set, 56 average prediction risk, 200 minimal, 56 average sample number, 192 algorithm for, 63 Ayring model, 203 death and birth process, 83 Bayesian reliability decomposition method, 52 demonstration test, 199 decreasing failure rate

sequential, 201 distribution, 87 Bernoulli trials, 36 distributions beta function, 169 beta, 169 binomial testing, 199 binomial, 36 censoring discrete, 35

left, 14 F, 34 multiple, 13 hypergeometric, 39 right, 14 inverse gamma, 170

Index 211

normal, 28 truncated normal, 28 Poisson, 36 Weibull, 24 t, 34 likelihood function, 120

down time, 2 likelihood kernel, 171 empirical Bayes, 180 likelihood ratio, 190 empirical CDF, 7,92 likelihood score function, 123 estimator likelihood score vector, 125

Bayes, 173 likelihood statistic, 178 maximum likelihood, 120 logistic time, 3 moment equations, 182 loss function, 171 unbiased, 112 maintainability, 8

Euler constant, 150 Markov process, 83 event tree, 61 maximum likelihood estimator failure intensity, 75 asymptotic variance, 124 failure rate function, 6, 17 censored data, 133 fault tree, 60 Erlang distribution, 147 Fisher information function, 123 exponential distribution, 143 Fisher information matrix, 128 extreme value distributions, 159 FIT, 12 gamma distribution, 149 fixed closeness, 113 invariance property, 122 fractile, 16 Kaplan-Meier, 138 free time, 2 lognormal distribution, 161 gamma function, 22 normal distribution, 161 gates, and/or, 61 shifted exponential, 145 graphical analysis, 91 system reliability function, 130 hazard function, 6,17 truncated normal, 163 hypothesis Weibull distribution, 154

alternative, 185 mean time till failure, 6, 16 null, 185 composite systems, 57

interquartile range, 41 median, 16 Jeffrey's prior, 171 moments, 16,44 Kaplan-Meier PL normal approximations, 30 estimator, 100,139 normal score, 93

keystone, 52 operating characteristic Laplace transform, 77 function, 186 life distributions operating time, 2

chi-square, 20,22 operational reliability Erlang, 20 function, 82 exponential, 18 parallel structure function, 49 extreme value, 27 parameter space, 120 gamma, 20,23 path set, 55 Gumbel, 28 minimal, 55 lognormal, 33 plotting positions, 93 Rayleigh, 20 Poisson processes, 196 shifted exponential, 19 posterior risk, 172

212 Index

power rule model, 204 Poisson processes, 196 precision, 112 sequential testing, 189 prediction intervals, 119 series structure function, 48

B~yesian, 176 shape parameter, 20 predictive acceptance significance level, 185 probability, 199 skewness, 44

preyentive maintenance, 84 standard deviation, 17 probability density function, 5 standard normal integral, 29

discrete variables, 35 standby unit, 59,83 posterior, 169 prior, 168 stationary availability

symmetric, 17 coefficient, 82

probability papers, 104 steepness, 44

probability plotting, 91 stopping rule, 201

censored data, 98 storage time, 2

proportional closeness, 113 structUre function, 55

quartiles, 16 sufficient statistic, 178 random sample, 91, 108 survival function, 14 readiness, 2 system effectiveness, 2

operational, 3 system failure function, 66 relative efficiency, 125 system reliability, reliability conditional, 52

demonstration, 185 system structure function, 5 bridge, 55 mission, 1 crosslinked, 52 operational, 1 double-crosslinked, 53

renewal k out ofn, 50 cycle, 73 module, 49 density, 76 function, 76 parallel, 48

process, 75 sequential, 58

repair intensity, 75 series, 46

repair time, 3 time categories, 2

repairability, 8 time censored, 13

residual time, 8 time till failure, 73 ruT, 12 time till repair, 73 root mean square error, 112 total time on test, 103 sampling distribution, 110 plot, 103 scale parameter, 19 up time, 2 sequential probability ratio variance, 17 test, 190 weakest link, 26

Springer Texts in Statistics (continued from p. ii)

Mandansky Prescriptions for Working Statisticians

McPherson Statistics in Scientific Investigation: Its Basis, Application, and Interpretation

Nguyen and Rogers Fundamentals of Mathematical Statistics: Volume I: Probability for Statistics

Nguyen and Rogers Fundamentals of Mathematical Statistics: Volume II: Statistical Inference

Noether Introduction to Statistics: The Nonparametric Way

Peters Counting for Something: Statistical Principles and Personalities

Pfeiffer Probability for Applications

Santner and Duffy The Statistical Analysis of Discrete Data

Saville and Wood Statistical Methods: The Geometric Approach

Sen and Srivastava Regression Analysis: Theory, Methods, and Applications

Zacks Introduction to Reliability Analysis: Probability Models and Statistical Methods