Ch. 5. Reliability and hazard based on finite mixture models

Handbook of Statistics

Volumn 20, Issue , 2001, Pages 139-183

  Al Hussaini, Essam K ^a   Sultan, Khalaf S ^a  

^a NONE

Author keywords

[No Author keywords available]

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EID: 4944233044     PISSN: 01697161     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S0169-7161(01)20007-8     Document Type: Review

References (152)

1. Abd-El-Hakim N.S. Finite mixture of the Inverse Gaussian and Weibull distributions. Ph.D. thesis (1985), University of Assiut, New York
2. Adham S.A. On finite mixture of two-component Gompertz model. M.Sc. thesis (1996), King Abdulaziz University, Assiut, Egypt
3. Ahmad K.E. Mixtures of Inverse Gaussian distributions. Ph.D. thesis (1982), University of Assiut, Jeddah, Saudi Arabia
4. Ahmad K.E. Identifiability of finite mixtures using a new transform. Ann. Inst. Stat. Math. 40 (1988) 261-265
5. Ahmad K.E., and Al-Hussaini E.K. Remarks on the non-identifiability of finite mixtures. Ann. Inst. Stat. Math. 34 (1982) 543-544
6. Aitchison J., and Brown J.A.C. The Lognormal Distribution with Special Reference to its Uses in Economics (1966), Cambridge University Press, Assiut, Egypt
7. Al-Hussaini E.K. Predicting observables from a general class of distributions. J. Stat. Plann. Inf. 79 (1999) 79-91
8. Al-Hussaini E.K. Bayesian prediction under a mixture of two exponential components model based on type 1 censoring. J. Appl. Stat. Sci. 8 (1999) 173-185
9. (Submitted). Al-Hussaini E.K. On Bayes prediction of future median (2001)
10. (Submitted). Al-Hussaini E.K. Bayesian predictive density of order statistics based on finite mixture models (2001)
11. Al-Hussaini E.K., and Abd-El-Hakim N.S. Bivariate Inverse Gaussian distribution. Ann. Inst. Stat. Math. 33 (1981) 57-66
12. Al-Hussaini E.K., and Abd-El-Hakim N.S. Failure rate of the Inverse Gaussian-Weibull model. Ann. Inst. Stat. Math. 41 (1989) 617-622
13. Al-Hussaini E.K., and Abd-El-Hakim N.S. Estimation of parameters of the Inverse Gaussian-Weibull model. Commun. Stat. - Theory Meth. 19 (1990) 1607-1622
14. Al-Hussaini E.K., and Abd-El-Hakim N.S. Efficiency of schemes of sampling from the Inverse Gaussian-Weibull mixture model. Commun. Stat. - Theory Meth. 21 (1992) 3143-3169
15. Al-Hussaini E.K., and Ahmad K.E. On the identifiability of finite mixtures of distributions. IEEE Tram. Inf. Theory. 27 (1981) 664-668
16. Al-Hussaini E.K., and Ahmad K.E. Information matrix for a mixture of two Inverse Gaussian distributions. Commun. Stat. - Simul. Comput. 13 (1984) 785-800
17. Al-Hussaini E.K., Al-Dayian G.R., and Adham S.A. On finite mixture of two-component Gompertz life time model. J. Stat. Comp. Simul. 67 (2000) 1-20
18. Al-Hussaini E.K., and Fakhry M.E. On characterization of finite mixtures of distributions. J. Appl. Stat. Sci. 2 (1995) 249-261
19. Al-Hussaini E.K., Mousa M.A.M.A., and Sultan K.S. Parametric and nonparametric estimation of P(Y < X) for finite mixtures of lognormal components. Commun. Stat. - Theory Meth. 26 (1997) 1269-1289
20. Al-Hussaini E.K., and Osman M. On the median of a finite mixture. J. Stat. Comput. Simul. 58 (1997) 121-144
21. Amoh R.K. Classification procedures associated with the Inverse Gaussian distribution. Ph.D. thesis (1983), University of Manitoba, London
22. Andrews D., Bickel P.J., Hampel F., Huber P., Rogers W., and Tukey J.W. Robust Estimates of Location: Survey and Advances (1972), Princeton University Press, Winnipeg, Canada
23. Anderson J.E. Multivariate survival analysis using random effects models. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, Princeton NJ 603-622
24. Arnold B.C. Conditional survival models. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, Boca Raton, FL 589-601
25. Arnold B.C., and Balakrishnan N. Relations, Bounds and Approximations for Order Statistics. Lecture Notes in Statistics No. 53 (1989), Springer, Boca Raton, FL
26. Ashour S.K. Estimation of the parameters of mixed Weibull-exponential models from censored samples. Tamkang J. Math. 16 (1985) 103-111
27. Ashour S.K. Bayesian estimation of mixed Weibull exponential in life testing. Appl. Stoch. Models and Data Anal. 3 (1987) 51-57
28. Ashour S.K., and Jones P.W. Maximum likelihood estimation of parameters in mixed Weibull distributions with equal shape parameter from complete and censored type 1 samples. The Egyptian Stat. J. 20 (1976) 1-13
29. Ashton W.D. Distributions for gaps in road traffic. J. Inst. Math. Appl. 7 (1971) 37-46
30. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, New York
31. Balakrishnan N., and Ambagaspitiya R.S. An empirical power comparison of three tests of exponentiality under mixture- and outlier-models. Biometrical J. 31 (1989) 49-66
32. In: Balakrishnan, and Basu A.P. (Eds). The Exponential Distribution: Theory and Applications (1995), Gordon and Breach, Boca Raton, FL
33. Balakrishnan N., and Chan P.S. Maximum likelihood estimation for the log-gamma distribution under Type-II censored samples and associated inference. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, New York, NJ 409-438
34. Balakrishnan N., and Chan P.S. Maximum likelihood estimation for the three-parameter log-gamma distribution under Type-II censoring. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, Boca Raton, FL 439-453
35. In: Balakrishnan N., and Chen W.W.S. (Eds). CRC Handbook of Tables for Order Statistics from Inverse Gaussian Distributions with Applications (1997), CRC Press, Boca Raton, FL
36. Balakrishnan N., and Geyer L. Order Statistics from location-mixture normal distributions and applications to robustness. Report (1995), McMaster University, Boca Raton, FL
37. Bandyopadhyay D., and Basu A.P. A class of tests for exponentiality against decreasing bivariate mean residual life alternatives. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, Hamilton, Canada 575-586
38. Barlow R.L. A Bayes explanation of an apparent failure rate paradox. IEEE Trans. Rel. R-34 (1985) 107-108
39. Barlow R.E., and Proschan F. Mathematical Theory of Reliability (1965), Wiley, Boca Raton, FL
40. Barlow R.E., and Proschan F. Statistical Theory of Reliability and Life Testing (1975), Holt, Reinhart and Winston, New York
41. To Begin With, Silver Spring, MD. Barlow R.E., and Proschan F. Statistical Theory of Reliability and Life Testing (1981)
42. Barndorff-Nielsen O. Identifiability of mixtures of exponential families. J. Math. Anal. Appl. 12 (1965) 115-121
43. Barnett V., and Lewis T. Outliers in Statistical Data. 3rd Edn. (1993), Wiley, New York
44. Behboodian J. On a mixture of two normal distributions. Biometrika 57 (1970) 215-217
45. Behboodian J. Information matrix for a mixture of two normal distributions. J. Stat. Comput. Simul. 1 (1972) 295-314
46. Billinton R., and Allan R.N. Reliability Evaluation of Engineering Systems: Concepts and Techniques (1983), Pitman, Chichester, England
47. Birnbaum Z.W., and McCarty R.C. A distribution-free upper confidence bound for P(Y < X), based on independent samples of X and Y. Ann. Math. Stat. 29 (1958) 558-562
48. Buckley D.J. Road traffic headwey distributions. Proceedings of First Conference of Australian Road Research Board (1962)
49. Chandra S. On the mixtures of probability distributions. Scand. J. Stat. 4 (1977) 105-112
50. Cheng S.W., and Fu J.C. Estimation of mixed Weibull parameters in life testing. IEEE Trans. Rel. R-31 (1982) 377-381
51. Chhikara R.S., and Folks J.L. The Inverse Gaussian distribution as a lifetime model. Technometrics 19 (1977) 461-468
52. Chhikara R.S., and Folks J.L. The Inverse Gaussian Distribution: Theory, Methodology and Applications (1989), Marcel Dekker, New York
53. In: Crow E.L., and Shimizu K. (Eds). Lognormal Distributions - Theory and Applications (1988), Marcel Dekker, New York
54. David H.A. Order Statistics. 2nd Edn. (1981), Wiley, New York
55. Davis D.J. An analysis of some failure data. J. Amer. Stat. Assoc. 47 (1952) 113-150
56. Day N.E. Estimating the components of a mixture of normal distributions. Biometrika 56 (1969) 463-474
57. Dempster A.P., Laird N.M., and Rubin D.B. Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Series B. 39 (1977) 1-38
58. Doganaksoy N. Likelihood ratio confidence intervals in life-data analysis. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, New York 359-376
59. Ebrahimi N. On the dependence of structure of multivariate processes and corresponding hitting times (A survey). In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, Boca Raton, FL
60. Elandt-Johnson R.C., and Johnson N.L. Survival Models and Data Analysis (1980), Wiley, Boca Raton, FL
61. Elsayed A.E. Reliability Engineering (1996), Addison Wesley, New York
62. Engelhardt M. Reliability estimation and applications. In: Balakrishnan N., and Basu A.P. (Eds). The Exponential Distribution: Theory, Methods and Applications (1995), Gordon and Breach, New York
63. Everitt B.S., and Hand D.J. Finite Mixtue Distributions (1981), Chapman & Hall, Newyork, NJ
64. Epstein B., and Sobel M. Life testing. J. Amer. Stat. Assoc. 48 (1953) 486-502
65. Falls L.W. Estimation of parameters in compound Weibull distributions. Technometrics 12 (1970) 339-407
66. Fisher R.A. The Mathematical Theory of Probabilities and its Applications to Frequency-curves and Statistical Methods Vol. 1 (1936), Macmillan, London
67. Fowlks E.B. Some methods for studying the mixture of two normal (lognormal) distributions. J. Amer. Stat. Assoc. 74 (1979) 561-575
68. Franck J.R. A simple explanation of the Weibull distribution and its applications. Reliab. Rev. 8 (1988) 6-9
69. Gastwirth J.L., and Cohen M.L. Small sample behaviour of robust linear estimators of location. J. Amer. Stat. Assoc. 65 (1970) 946-973
70. Gilliom R.J., and Helsel D.R. Estimation of distributional parameters for censored trance level water quality data, 1. Estimation techniques. Water Resour. Res. 22 (1986) 135-146
71. Gordon N.H. Maximum likelihood estimation for mixtures of two Gompertz distributions when censoring occurs. Commun. Statist - Simul. Comput. 19 (1990) 733-747
72. Govindarajulu Z. Distribution-free confidence bounds for P(X < Y). Ann. Stat. Math. 20 (1968) 229-238
73. Govindarajulu Z. A note on distribution-free confidence bounds for P(X < Y) when X and Y are independent. Ann. Stat. Math. 28 (1976) 307-308
74. Hathaway R.J. A constrained formulation of maximum likelihood estimation for normal mixture distributions. Ann. Stat. 13 (1985) 795-800
75. Hosmer D.W. A comparison of iterative maximum likelihood estimates of parameters of a mixture of two normal distributions under three different types of sample. Biometrics 29 (1973) 761-770
76. Huber P.J. Robust Statistics (1981), Wiley, New York
77. Jiang R., and Murthy D.N. Modeling failure-data by mixture of 2 Weibull distributions: a graphical approach. IEEE Trans. Reliab. R-44 (1995) 477-488
78. Johnson R.A., Evans J., and Green D. Predictive distributions of order statistics. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, New York 505-521
79. Johnson N.L., and Kotz S. A vector valued multivariate hazard rate. J. Multiv. Anal. 5 (1975) 53-66
80. Johnson N.L., Kotz S., and Balakrishnan N. Continuous Univariate Distributions. 2nd Edn. Vol. 1 (1994), Wiley, Boca Raton, FL
81. Johnson N.L., Kotz S., and Balakrishnan N. Continuous Univariate Distributions. 2nd Edn Vol. 2 (1995), Wiley, New York
82. Kalbfleish J.D., and Prentice R.L. The Analysis of Failure Time Data (1980), Wiley, New York
83. Kao J.H.K. A graphical estimation of mixed Weibull parameters in life testing of electron tubes. Technometrics 1 (1959) 389-407
84. Kimber A.S. A Weibull based on score test for heterogeneity. Lifetime Data Anal. 2 (1996) 63-71
85. Kleyle R.M., and Dahiya R.C. Estimation of parameters of mixed failure time distribution from censored samples. Commun. Stat. - Theory Meth. 4 (1975) 873-882
86. Kocherlakota S., and Balakrishnan N. Tolerance limits which control percentages in both tails: sampling from mixtures of normal distributions. Biom. J. 28 (1986) 209-217
87. Kotz S., Balakrishnan N., and Johnson N.L. Continuous Multivariate Distributions 1: Models and Applications. 2nd Edn (2000), Wiley, New York
88. Krysicki W. Application de la méthode des moments a l'estimation des paramétres d'un melange de deux distribtions de Rayleigh. Revue de Statistique Appliquée 11 (1963) 25-45
89. Kulasekera K.B., and Nelson P.R. Choosing a model from among four families of distributions. In: Balakrishnan N. (Ed). ecent Advances in Life-Testing and Reliability (1995), CRC Press, New York 491-504
90. Larsen R.I. A new mathematical model of air pollutant concentration averaging time and frequency. J. Air Pollut. Control Assoc. 19 (1969) 24-30
91. Larsen R.I. An air quality data analysis system for interrelating effects, standards, and needed source reduction: part 4. A three-parameter averaging-time model. J. Air Pollut. Control Assoc. 27 (1977) 454-459
92. Lawless J.F. Statistical Models and Methods for Lifetime Data (1982), Wiley, Boca Raton, FL
93. Lepeltier C. A simplified statistical treatment of geochemical data by graphical representation. Econ. Geol. 64 (1969) 538-550
94. Lindley D.V. Approximate Bayesian method. Trabajos de Estadistica 31 (1980) 223-237
95. Lindsay B.G. Mixture Models: Theory, Geometry and Applications (1995), The Institute of Mathematical Statistics, New York
96. Magder L.S., and Zeger S.L. A smooth nonparametric estimate of a mixing distribution using mixtures of Gaussians. J. Amer. Stat. Assoc. 91 (1996) 1141-1151
97. Makino T. Mean hazard rate and its applications to normal approximation of Weibull distribution. Nav. Res. Logist. Q. 31 (1984) 1-8
98. Malik M.A. A note on the physical meaning of the Weibull distribution. IEEE Trans. Rel. R-24 (1975) 95
99. Mann N.R., Schafer R.E., and Singpurwalla N.D. Methods for Statistical Analysis of Reliability and Life Data (1974), Wiley, Hayward, CA
100. Maritz J.S., and Lwin T. Empirical Bayes Methods. 2nd Edn (1989), Chapman & Hall, New York
101. McClean S. Estimation for the mixed exponential distribution using grouped follow-up data. Appl. Stat. 15 (1986) 31-37
102. McLachlan G.J., and Basford K.E. Mixture Models: Applications to Clustering (1988), Marcel Dekker, London
103. McLachlan G.J., and Krishnan T. The EM Algorithm and Extensions (1997), Wiley, New York
104. Medgyessy P. Decomposition of Superpositions of Distribution Functions (1961), Hungarian Academy of Sciences, New York
105. Mendenhall W., and Hader R.J. Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45 (1958) 504-520
106. Mi J. A new explanation of decreasing failure rate of a mixture of exponentials. IEEE Trans. Reliab. R-47 (1998) 460-462
107. Mudholkar G.S., Srivastava D.K., and Freimer M. The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37 (1995) 436-445
108. Neiburger M., and Weickmann H.K. The meteorological background for weather modification. In: Hess W.N. (Ed). Weather and Climate Modification (1974), Wiley, Budapest, Hungary
109. Nelson W. Applied Life Data Analysis (1982), Wiley, New York
110. Nelson W., and Doganaksoy N. Statistical analysis of life or strength data from sepecimens of various sizes using the power-(log)normal model. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, New York 377-408
111. Newcomb S. A generalized theory of the combination of observations so as to obtain the best result. Amer. J. Math. 8 (1886) 343-366
112. Olssen D.M. Estimation for mixtures of distributions by direct maximization of the likelihood function. J. Q. Technol. 11 (1979) 153-159
113. Owen D.B., Craswell K.J., and Hanson D.L. Nonparametric upper confidence bounds for P(Y < X) and confidence limits for P(Y < X) when X and Y are normal. J. Amer. Stat. Assoc. 59 (1964) 906-924
114. Ott W.R. Environmental Indices: Theory and Practice (1978), Ann Arbor Science Publishers, Boca Raton, FL
115. Patterson E.M., and Gillete D.A. Commonalities in measured size distributions for aerosols having a soil-derived component. J. Geophys. Res. 82 (1977) 2074-2082
116. Patil G.P., and Bildikar S. Identifiability of countable mixtures of discrete probability distributions using methods of finite matrices. Proc. Camb. Phil. Soc. 62 (1966) 485-494
117. Papadopoulos A.S., and Padgett W.J. On Bayes estimation for mixtures of two exponential-life-distributions from right-censored samples. IEEE Trans. Reliab. R-35 (1986) 102-104
118. Pearson K. Contributions to the mathematical theory of evolution. Phil. Trans. A. 185 (1894) 71-110
119. Peña E.A. Residuals from type II censored samples. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, Ann Arbor, Michigan 523-543
120. Prakasa Rao B.L.S. On distributions with periodic failure rate and related inference problems. In: Panchapakesan S., and Balakrishnan N. (Eds). Advances in Statistical Decision Thoery and Applications (1997), Birkhäuser, Boca Raton, FL 65-78
121. Proschan F. Theoretical explanation of observed decreasing failure rate. Technometrics 5 (1963) 375-383
122. Quandt R.E., and Ramsey J.B. Estimating mixture of normal distributions and switching regression (with discussion). J. Amer. Stat. Assoc. 73 (1978) 730-752
123. Raiffa H., and Schlaifer R. Applied Statistical Decision Theory (1961), Harvard University Press, Boston
124. Rider P.R. The method of moments applied to a mixture of two exponential distributions. Ann. Math. Stat. 32 (1961) 143-147
125. Redner R.A., and Walker H.F. Mixture densities, maximum likelihood and the EM algorithm. SIAM Rev. 26 (1984) 195-239
126. Rennie R.R. On the interdependence of the identifiability of finite multivariate mixtures and identifiability of the marginal mixtures. Sankhyā, Series A. 34 (1972) 449-452
127. Rodrigues F.W., and Wechsler S. A discrete Bayes explanation of a failure-rate paradox. IEEE Trans. Reliab. R-42 (1993) 132-133
128. Roeder K., and Wasserman L. Practical Bayesian density estimation using mixtures of normals. J. Amer. Stat. Assoc. 92 (1997) 894-902
129. Sankaran M. Mixtures by the Inverse Gaussian distribution. Sankhyā, Series B. 30 (1968) 455-458
130. Seshadri V. The Inverse Gaussian Distribution: A Case Study in Exponential Families (1993), Clarendon Press, Cambridge, MA
131. Seshadri V. The Inverse Gaussian Distribution: Statistical Theory and Applications (1999), Springer, Oxford
132. In: Shaked M., and Shanthikumar J.G. (Eds). Stochastic Orders and Their Applications (1994), Academic Press, New York
133. Sinha S.K. Bayesian estimation of the parameters and reliability function of a mixture of Weibull life distributions. J. Stat. Plann. Inf. 16 (1987) 377-387
134. Smith R.L. Likelihood and modified likelihood estimation for distributions with unknown endpoints. In: Balakrishnan N. (Ed). Recent Advances in Life-Testing and Reliability (1995), CRC Press, San Diego, CA 455-474
135. Sultan K.S. Mixtures of lognormal distributions. M.Sc. thesis (1992), University of Assiut, Boca Raton, FL
136. Tallis G.M. The identifiability of mixtures of distributions. J. Appl. Prob. 6 (1969) 389-398
137. Tallis G.M., and Chesson P. Identifiability of mixtures. Aust. Math. Soc. Series A. 32 (1982) 339-348
138. Teicher H. On the mixture of distributions. Ann. Math. Stat. 31 (1960) 55-73
139. Teicher H. Identifiability of mixtures. Ann. Math. Stat. 32 (1961) 244-248
140. Teicher H. Identifiability of finite mixtures. Ann. Math. Stat. 34 (1963) 1265-1269
141. Teicher H. Identifiability of finite mixtures of product measures. Ann. Math. Statist. 38 (1967) 1300-1302
142. Tiku M.L., Tan W.Y., and Balakrishnan N. Robust Inference (1986), Marcel Dekker, Assiut, Egypt
143. Titterington D.M., Smith A.F.M., and Makov U.E. Statistical Analysis of Finite Mixture Distributions (1985), Wiley, New York
144. Tsukatani T. Statistical aspects of air pollution quality. Ann. Report, Res. Reactor Inst. 14 (1981) 129-152
145. Tweedie M.C.K. Statistical properties of Inverse Gaussian distribution: I. Ann. Math. Stat. 28 (1957) 362-377
146. Tweedie M.C.K. Statistical properties of Inverse Gaussian distribution: II. Ann. Math. Stat. 28 (1957) 696-705
147. Vod V.Gh. Inferential procedures on a generalized Rayleigh variate, II. Aplikace Matematiky 21 (1976) 413-419
148. Wald A. Sequential Analysis (1947), Wiley, Chichester
149. Wasan M.T. First passage time distribution of Brownian motion with positive drift (Inverse Gaussian distribution). Queen's Papers in Pure and Applied Mathematics No. 19 (1969), Queen's University, New York
150. Whitby K.T. The physical effects of sulfur aerosols. Atmos. Environ. 12 (1978) 135-159
151. Willamain T.R., Allahvardi A., De Sautels P., Eldridge J., Gur O., Miller M., Panos G., Srivinasan A., Surithadi J., and Topal E. Robust estimation methods for exponential data: A Monte Carlo comparison. Commun. Stat. - Simul. Comput. 21 (1992) 1043-1075
152. Yakowitz S.J., and Spragins J.D. On identifiability of finite mixtures. Ann. Math. Stat. 39 (1968) 209-214