Abstract
There are many practical situations where the access to conditional distributions are more likely than to their joint distribution. In the present paper we study partial moments in the conditional setup. It is shown that the conditional partial moments determine the corresponding distribution uniquely. The relationships with reliability measures such as conditional hazard rate and mean residual life are obtained. Characterizations results based on conditional partial moments for some well known bivariate lifetime distributions are derived. We study properties of conditional partial moments in the context of weighted models. Characterizations of conditional partial moments using income gap ratio are also obtained. Finally, non parametric estimators for conditional partial moments are introduced which are validated using simulated and real data sets.
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References
Abdul-Sathar EI, Suresh RP, Nair KRM (2007) A vector valued bivariate Gini index for truncated distributions. Stat Pap 48(4):543–557
Arnold BC (1995) Conditional survival models. In: Recent advances in life-testing and reliability. CRC Press, Boca Raton, p 589–601
Arnold BC (1987) Bivariate distributions with Pareto conditionals. Stat Probab Lett 5(4):263–266
Arnold BC, Strauss D (1988) Bivariate distributions with exponential conditionals. J Am Stat Assoc 83(402):522–527
Balakrishnan N, Lai CD (2009) Continuous bivariate distributions. Springer, New York
Belzunce F, Candel J, Ruiz J (1998) Ordering and asymptotic properties of residual income distributions. Sankhyā Indian J Stat B 60:331–348
Cheng Y, Pai JS (2003) On the nth stop-loss transform order of ruin probability. Insur Math Econ 32(1):51–60
Chong KM (1977) On characterizations of the exponential and geometric distributions by expectations. J Am Stat Assoc 72(357):160–161
Cox DR (1959) The analysis of exponentially distributed lifetimes with two types of failure. J R Stat Soc Ser B 21(2):411–421
Gupta RC (2007) Role of equilibrium distribution in reliability studies. Probab Eng Inf Sci 21(02):315–334
Gupta RC (2008) Reliability studies of bivariate distributions with exponential conditionals. Math Comput Model 47(9):1009–1018
Gupta RC (2016) Reliability characteristics of Farlie–Gumbel–Morgenstern family of bivariate distributions. Commun Stat Theory Methods 45(8):2342–2353
Gupta PL, Gupta RC (1983) On the moments of residual life in reliability and some characterization results. Commun Stat Theory Methods 12(4):449–461
Gupta PL, Gupta RC (2001) Failure rate of the minimum and maximum of a multivariate normal distribution. Metrika 53(1):39–49
Gupta RC, Kirmani S (1990) The role of weighted distributions in stochastic modeling. Commun Stat Theory methods 19(9):3147–3162
Gupta RP, Sankaran PG (1998) Bivariate equilibrium distribution and its applications to reliability. Commun Stat Theory methods 27(2):385–394
Hougaard P (2012) Analysis of multivariate survival data. Springer, New York
Johnson NL, Kotz S (1975) A vector multivariate hazard rate. J Multivar Anal 5(1):53–66
Kulkarni HV, Rattihalli RN (2002) Nonparametric estimation of a bivariate mean residual life function. J Am Stat Assoc 97(459):907–917
Kundu C, Sarkar K (2015) Characterizations based on higher order and partial moments of inactivity time. Stat Pap. doi:10.1007/s00362-015-0714-y
Lin GD (2003) Characterizations of the exponential distribution via the residual lifetime. Sankhyā Indian J Stat 65:249–258
Mardia KV (1962) Multivariate Pareto distributions. Ann Math Stat 33:1008–1015
Nair NU, Sankaran PG, Sunoj SM (2013a) Quantile based reliability aspects of partial moments. J Korean Stat Soc 42(3):329–342
Nair NU, Sankaran PG, Sunoj SM (2013b) Quantile based stop-loss transform and its applications. Stat Methods Appl 22(2):167–182
Nair NU, Hitha N (1990) Characterizations of Pareto and related distributions. J Indian Stat Assoc 28:75–79
Rao CR (1965) On discrete distributions arising out of methods of ascertainment. Sankhyā Indian J Stat A 27:311–324
Roy D (1989) A characterization of Gumbel’s bivariate exponential and Lindley and Singpurwalla’s bivariate Lomax distributions. J Appl Probab 27:886–891
Sankaran PG, Nair NU (1991) On bivariate vitality functions. In: Proceedings of the symposium on distribution theory, p 61–71
Sankaran PG, Nair NU (2004) Partial moments for bivariate distributions. Metron Int J Stat 62(3):339–351
Sankaran P, Sreeja V (2007) Proportional hazards model for multivariate failure time data. Commun Stat Theory Methods 36(8):1627–1641
Sankaran PG, Nair NU, Preethi J (2015) Characterizations of a family of bivariate Pareto distributions. Statistica 75(3):275–290
Sunoj SM (2004) Characterizations of some continuous distributions using partial moments. Metron 66:342–353
Sunoj SM, Linu MN (2012) Dynamic cumulative residual Renyi’s entropy. Statistics 46(1):41–56
Sunoj SM, Maya SS (2006) Some properties of weighted distributions in the context of repairable systems. Commun Stat Theory Methods 35(2):223–228
Sunoj SM, Maya SS (2008) The role of lower partial moments in stochastic modeling. Metron 66(2):223–242
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The first author would like to thank the support of University Grants Commission, India, under the Special Assistance Programme. The second author wish to thank Cochin University of Science and Technology, Cochin, India for the financial assistance for carrying out this research work.
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Sunoj, S.M., Vipin, N. Some properties of conditional partial moments in the context of stochastic modelling. Stat Papers 60, 1971–1999 (2019). https://doi.org/10.1007/s00362-017-0904-x
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DOI: https://doi.org/10.1007/s00362-017-0904-x