Bayesian inference for the Topp–Leone distribution based on lower k-record values

Abstract

In this paper, the lower k-record values coming from the Topp–Leone distribution are used to construct Bayesian point and interval estimators for the shape parameter, the survival function and the reversed hazard rate function. The Bayes estimators are obtained under symmetric and asymmetric loss functions. We study the problem of predicting future k-records, and reconstructing past unobserved k-records. Bayesian estimation of the stress-strength parameter is also discussed. Finally, a simulation study and a real data example are presented for the purpose of illustration and comparison of the suggested point and interval estimators.

Appendix

See Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

Table 1 The ERs of the MLEs and the Bayes estimators of $\theta$

Table 2 The ERs of the MLEs and the Bayes estimators of $S_t$ and $H_t (t=0.5)$

Table 3 The simulated CPs (and AWs in parentheses) of the $95\,\%$ Bayesian confidence intervals for $\theta$, $S_t$ and $H_t (t=0.5)$

Table 4 The ERs of the Bayes predictors of $R_{s (k)}$ for $s=7,8$

Table 5 The simulated CPs (and AWs in parentheses) of the $95\,\%$ TEB PIs for $R_{s(k)}, s=7,8$

Table 6 The simulated CPs (and AWs in parentheses) of the $95\,\%$ two-sided equi-tailed (TE) and HCD RIs and the ERs of the MLRs under the SEL function for $R_{l(k)}, ~l=1,2,3,~m=3$

Table 7 The ERs of the MLEs and the Bayes estimators of $\delta$

Table 8 The simulated CPs (and AWs in parentheses) of the $95\,\%$ Bayesian confidence intervals for $\delta$

Table 9 Monthly capacities for February and the proportions of the total capacity for Shasta Reservoir

Table 10 The lower k-record values extracted from the Shasta Reservoir data

Table 11 The numerical results for Example 1