×

zbMATH — the first resource for mathematics

The role of weighted distributions in stochastic modeling. (English) Zbl 0734.62093
Summary: We study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original distribution are examined. Their survival functions, failure rates and mean residual life functions are compared and some characterization results are established.

MSC:
62N05 Reliability and life testing
62E10 Characterization and structure theory of statistical distributions
PDF BibTeX Cite
Full Text: DOI
References:
[1] DOI: 10.2307/2287888
[2] Barlow, R. E. and Proschan, F. 1975. ”Statistical theory of reliability and life testing”. New York: Holt, Rinehart and Winston Inc. · Zbl 0379.62080
[3] DOI: 10.1093/biomet/56.2.391 · Zbl 0179.25402
[4] DOI: 10.1007/BF01897826 · Zbl 0584.62030
[5] DOI: 10.1016/0167-7152(87)90052-6 · Zbl 0633.62017
[6] DOI: 10.1016/0378-3758(83)90033-2 · Zbl 0532.62077
[7] DOI: 10.2307/1266418
[8] DOI: 10.1016/S0022-5193(70)80037-6
[9] DOI: 10.1080/03610928508828955 · Zbl 0573.62096
[10] Cox D. R., New Developments in Survey Sampling pp 506– (1969)
[11] David H. A., Order statistics (1970) · Zbl 0223.62057
[12] Gupta R. C., Communications in Statistics 5 pp 45– (1975) · Zbl 0326.60014
[13] Gupta R. C., Scandinavian Journal of.Statistics 3 pp 215– (1976)
[14] DOI: 10.1080/03610927908827786 · Zbl 0444.62017
[15] Gupta R. C., In Statistical distributions in scientific work 5 pp 327– (1981)
[16] Gupta R. C., Mathematische Operations-forschung and Statistics 15 pp 571– (1984)
[17] Gupta R. C., Scand. J. Statist 13 pp 49– (1986)
[18] DOI: 10.1017/S0269964800000073 · Zbl 1133.62356
[19] DOI: 10.1080/15326348708807050 · Zbl 0617.60084
[20] DOI: 10.1016/0378-3758(87)90085-1 · Zbl 0621.62096
[21] Kanchan and Singh, H. 1987. ”Relations for reliability measures of weighted distribution”. India: Technical Report, Panjab University.
[22] DOI: 10.2307/3556181 · Zbl 0516.60063
[23] Kirmani S. N. U. A., Sankhya Series A 36 pp 197– (1974)
[24] Klefsjo B., Scand. J. Statist 9 pp 37– (1982)
[25] DOI: 10.1017/S0269964800000498 · Zbl 1134.60315
[26] Mahfoud, M. and Patil, G. P. 1982. ”On weighted distributions. In Statistics and Probability: Essays in Honor”. Edited by: Rao, C. R. 479–492. North Holland: Amsterdam. · Zbl 0487.62011
[27] DOI: 10.1016/0022-5193(68)90131-8
[28] DOI: 10.1080/03610928608829159 · Zbl 0601.62022
[29] Patil, G. P. and Rao, C. R. 1977. ”Weighted distributions: A survey of their applications. In Applications of Statistics”. Edited by: Krishnaiah, P. R. 383–405. North Holland Publishing Co. · Zbl 0371.62034
[30] DOI: 10.2307/2530008 · Zbl 0384.62014
[31] Patil G. P., Sankhya 38 pp 48– (1976)
[32] Pyke R., J. Royal Statistical Soc. Series B 27 pp 395– (1965)
[33] Pyke R., Sixth Berkeley Symposium on Math. Statist, and Prob 1 pp 417– (1972)
[34] DOI: 10.2307/2682973
[35] Rao C. R., In Classical and Contagious Discrete Distributions pp 320– (1965)
[36] Rao C. R., A celebration of Statistics, The ISI Centenary Volume pp 543– (1985)
[37] Rao, C. R. 1988. ”Weighted and clouded distributions”. University of Pittsburgh. Technical Report No. 88-01, Center for Multivariate Analysis
[38] Rao J. S., In nonparametric Techniques in Statistical Inference pp 267– (1970)
[39] DOI: 10.2307/1267291 · Zbl 0238.62011
[40] DOI: 10.1016/0022-5193(72)90052-5
[41] DOI: 10.1016/0022-5193(71)90175-5
[42] DOI: 10.1016/0167-7152(87)90062-9 · Zbl 0606.62051
[43] DOI: 10.2307/3213483 · Zbl 0481.60022
[44] DOI: 10.1002/1520-6750(198902)36:1<103::AID-NAV3220360108>3.0.CO;2-7 · Zbl 0658.62116
[45] Simon R., American Journal of Epidemiology 11 pp 444– (1980)
[46] DOI: 10.1016/0022-5193(66)90017-8
[47] DOI: 10.1016/0022-5193(68)90161-6
[48] DOI: 10.1080/03610928708829431 · Zbl 0653.62014
[49] DOI: 10.1016/0022-5193(67)90199-3
[50] DOI: 10.1016/0022-5193(68)90132-X
[51] DOI: 10.1214/aos/1176345802 · Zbl 0491.62034
[52] DOI: 10.2307/1266118 · Zbl 0104.12203
[53] DOI: 10.1093/biomet/56.3.601 · Zbl 0184.23703
[54] Zelen M., Reliability and biometry pp 701– (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.