Exponential distribution is frequently used as a lifetime distribution in statistics and applied areas; the Lindley distribution has been ignored in the literature since 1958. Lindley distribution originally developed by Lindley [
The rest of the study is organized as follows. Section
The posterior distribution summarizes available probabilistic information on the parameters in the form of prior distribution and the sample information contained in the likelihood function. The likelihood principle suggests that the information on the parameter should depend only on its posterior distribution. Bayesian scientist’s job is to assist the investigator to extract features of interest from the posterior distribution. In this section, we will use the Lindley model as sampling distribution mingles with noninformative priors for the derivation of posterior distribution. A random variable
It is obvious from Figure
The Lindley distribution.
The likelihood function for a random sample
An argument in the favor of uniform prior is that when the data are sufficiently informative, so that likelihood function is sharply peaked, then it really does not matter what prior is used, since all reasonably smooth prior densities will lead to approximately the same posterior density. The uniform density in most cases is convenient to simplify calculations of the posterior. This argument supports the uniform prior only in those cases where it produces approximately the same conclusions as the highly imprecise prior constructed from a sufficiently large class of prior densities. If the data are highly informative, the uniform prior may produce reasonable inferences. The uniform prior for
The posterior distribution of parameter
Jeffreys was motivated by invariance requirements and suggested a solution to provide a noninformative prior. He used differential geometry method. The requirements are invariance under 1-1 transformations and invariance under sufficient statistics. One-dimensional version of Jeffreys prior has been justified from many different viewpoints. Jeffreys [
In case of an informative prior, the use of prior information is equivalent to adding a number of observations to a given sample size and, therefore, leads to a reduction of the variance/posterior risk of the Bayes estimates. Bansal [
Posterior distribution using informative gamma prior (GP) is
The likelihood matching prior (LMP) for Lindley distribution is
and the posterior distribution using LMP is
This section spotlight is on the derivation of the Bayes estimator (BE) under different loss functions and their respective posterior risk (PR). The results are compared for noninformative as well as informative priors. If the decision is a choice of an estimator, then, the Bayes decision is a Bayes estimator. The Bayes estimators are evaluated under squared error loss function (SELF), weighted squared error loss function (WSELF), precautionary loss function (PLF), modified (quadratic) squared error loss function (M/Q SELF), logarithmic loss function (SLLF), entropy loss function (ELF), and K-Loss function. K-loss function proposed by Wasan [
Bayes estimator and posterior risk under different loss functions.
Loss function | Bayes estimator (BE) | Posterior risk (PR) |
---|---|---|
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Var |
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2 |
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exp |
Var |
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E |
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2 |
Even though many authors have pointed a need for a formal and comprehensive process for elicitation of hyperparameters, there is no standard method. For elicitation, mainly two points are considered; the functional form of the prior distribution and hyperparameter(s), that is why a natural conjugate prior distribution has been generally recommended because its functional form is identical to likelihood function and posterior distribution can be determined by the way of conjugancy. To determine hyperparameter, we adopted the method discussed by Ali et al. [
For a continuous distribution with the density
Bayramoglu and Gurler [
Ghitany et al. [
From
If
where
where
BEs and their respective PRs of MRLF under SELF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
20 | 16.6609 |
1.16041 |
0.108763 |
17.1312 |
1.2008 |
0.113795 |
40 | 16.9256 |
1.17773 |
0.111713 |
17.1644 |
1.1982 |
0.11429 |
60 | 16.9761 |
1.18603 |
0.112294 |
17.1357 |
1.19976 |
0.11402 |
80 | 17.0298 |
1.19015 |
0.112893 |
17.1499 |
1.20048 |
0.114194 |
100 | 17.0348 |
1.19247 |
0.113075 |
17.1309 |
1.20075 |
0.114117 |
| ||||||
|
LMP | GP | ||||
|
7.142857 | 1.2 | 0.114114 | 7.142857 | 1.2 | 0.114114 |
| ||||||
20 | 16.6296 |
1.15994 |
0.110424 |
17.1139 |
1.19828 |
0.115161 |
40 | 16.9097 |
1.17747 |
0.112539 |
17.1556 |
1.19692 |
0.114963 |
60 | 16.9654 |
1.18586 |
0.112844 |
17.1298 |
1.19891 |
0.114467 |
80 | 17.0218 |
1.19002 |
0.113305 |
17.1455 |
1.19984 |
0.114528 |
100 | 17.0284 |
1.19236 |
0.113405 |
17.1274 |
1.20023 |
0.114385 |
BEs and their respective PRs of MRLF under WSELF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
7.142857 | 1.2 | 0.114114 | 7.142857 | 1.2 | 0.114114 |
20 | 16.1511 |
1.11927 |
0.10415 |
16.6062 |
1.1582 |
0.10896 |
40 | 16.6632 |
1.15649 |
0.109296 |
16.8981 |
1.17658 |
0.111815 |
60 | 16.7998 |
1.17169 |
0.110663 |
16.9577 |
1.18525 |
0.112362 |
80 | 16.8969 |
1.17933 |
0.111659 |
17.0159 |
1.18956 |
0.112945 |
100 | 16.9283 |
1.18378 |
0.112084 |
17.0238 |
1.1920 |
0.113117 |
| ||||||
|
LMP | GP | ||||
|
7.142857 | 1.2 | 0.114114 | 7.142857 | 1.2 | 0.114114 |
| ||||||
20 | 16.1217 |
1.11883 |
0.105691 |
16.5891 |
1.15569 |
0.110343 |
40 | 16.6478 |
1.15624 |
0.110091 |
16.8893 |
1.17531 |
0.112492 |
60 | 16.7894 |
1.17152 |
0.111199 |
16.9518 |
1.18440 |
0.112811 |
80 | 16.889 |
1.1792 |
0.112063 |
17.0115 |
1.18892 |
0.11328 |
100 | 16.9220 |
1.18367 |
0.112409 |
17.0202 |
1.19148 |
0.113385 |
BEs and their respective PRs of MRLF under MSELF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
20 | 15.6991 |
1.0835 |
0.100281 |
16.1011 |
1.11726 |
0.104331 |
40 | 16.4163 |
1.13671 |
0.107086 |
16.6369 |
1.1554 |
0.109393 |
60 | 16.6305 |
1.15802 |
0.109126 |
16.7819 |
1.17094 |
0.110729 |
80 | 16.7679 |
1.16888 |
0.110479 |
16.8833 |
1.17875 |
0.111709 |
100 | 16.8244 |
1.17533 |
0.111128 |
16.9174 |
1.18331 |
0.112125 |
| ||||||
|
LMP | GP | ||||
|
|
|
|
|
|
|
| ||||||
20 | 15.6738 |
1.08313 |
0.101543 |
16.0855 |
1.11502 |
0.105556 |
40 | 16.4021 |
1.13648 |
0.107804 |
16.6286 |
1.15419 |
0.11003 |
60 | 16.6207 |
1.15786 |
0.109627 |
16.7763 |
1.17011 |
0.11116 |
80 | 16.7604 |
1.16876 |
0.110863 |
16.8790 |
1.17813 |
0.112035 |
100 | 16.8182 |
1.17523 |
0.11144 |
16.9140 |
1.18281 |
0.112386 |
BEs and their respective PRs of MRLF under PLF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
20 | 16.9211 (0.520576) | 1.1815 (0.042188) | 0.111158 (0.004791) | 17.3777 (0.493072) | 1.2205 (0.039405) | 0.115987 (0.004383) |
40 | 17.0582 (0.265186) | 1.18848 (0.021509) | 0.112945 (0.002464) | 17.2934 (0.258053) | 1.20859 (0.020784) | 0.115468 (0.002356) |
60 | 17.0648 (0.177533) | 1.19326 (0.014463) | 0.11312 (0.001653) | 17.2229 (0.174329) | 1.20683 (0.012135) | 0.114822 (0.001604) |
80 | 17.0966 (0.133627) | 1.1956 (0.010893) | 0.113516 (0.001246) | 17.2158 (0.131813) | 1.20584 (0.010708) | 0.114803 (0.001218) |
100 | 17.0883 (0.10698) | 1.19684 (0.008735) | 0.113575 (0.000999) | 17.1838 (0.105815) | 1.20506 (0.008616) | 0.114608 (0.000981) |
| ||||||
|
LMP | GP | ||||
|
|
|
|
|
|
|
| ||||||
20 | 16.8904 (0.485218) | 1.18104 (0.034205) | 0.112775 (0.004302) | 17.3612 (0.492507) | 1.21811 (0.033652) | 0.117262 (0.004203) |
40 | 17.0425 (0.256431) | 1.18823 (0.020514) | 0.11376 (0.002342) | 17.2848 (0.256428) | 1.20734 (0.020449) | 0.116118 (0.002310) |
60 | 17.0543 (0.173642) | 1.19309 (0.011465) | 0.113666 (0.001602) | 17.2171 (0.173498) | 1.20599 (0.011464) | 0.115259 (0.001583) |
80 | 17.0886 (0.131689) | 1.19547 (0.010704) | 0.113926 (0.001214) | 17.2114 (0.131609) | 1.20520 (0.010704) | 0.115132 (0.001207) |
100 | 17.0819 (0.105019) | 1.19673 (0.008536) | 0.113903 (0.000979) | 17.1803 (0.105017) | 1.20454 (0.008527) | 0.114871 (0.0009735) |
BEs and their respective PRs of MRLF under SLLF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
20 | 16.4002 (0.031314) | 1.13929 (0.036423) | 0.106377 (0.043796) | 16.8701 (0.031313) | 1.17967 (0.036422) | 0.111405 (0.043784) |
40 | 16.7929 (0.015685) | 1.16697 (0.018282) | 0.110483 (0.021993) | 17.0316 (0.015687) | 1.18743 (0.018286) | 0.113059 (0.021026) |
60 | 16.8873 (0.010466) | 1.1788 (0.012203) | 0.111469 (0.014688) | 17.0468 (0.010471) | 1.19253 (0.012205) | 0.113194 (0.014687) |
80 | 16.9629 (0.007851) | 1.1847 (0.009157) | 0.112271 (0.011023) | 17.083 (0.007854) | 1.19503 (0.009158) | 0.113571 (0.011024) |
100 | 16.9813 (0.006283) | 1.1881 (0.007329) | 0.112576 (0.008823) | 17.0774 (0.006282) | 1.19638 (0.007329) | 0.113618 (0.008826) |
| ||||||
|
LMP | GP | ||||
|
|
|
|
|
|
|
| ||||||
20 | 16.3694 (0.031254) | 1.13883 (0.036408) | 0.108011 (0.043296) | 16.8526 (0.031289) | 1.17711 (0.036412) | 0.112809 (0.043172) |
40 | 16.7772 (0.015670) | 1.16672 (0.018277) | 0.111303 (0.021016) | 17.0228 (0.015674) | 1.18614 (0.018204) | 0.113742 (0.021026) |
60 | 16.8767 (0.010459) | 1.17862 (0.012201) | 0.112016 (0.014542) | 17.0410 (0.010454) | 1.19166 (0.012202) | 0.113645 (0.014523) |
80 | 16.955 (0.007847) | 1.18457 (0.009156) | 0.112681 (0.011023) | 17.0786 (0.007845) | 1.19438 (0.009153) | 0.113908 (0.010987) |
100 | 16.9749 (0.006280) | 1.18799 (0.007328) | 0.112905 (0.008822) | 17.0739 (0.006286) | 1.19586 (0.007329) | 0.113887 (0.008800) |
BEs and their respective PRs of MRLF under ELF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
20 | 16.1511 (5.57928) | 1.11927 (0.243079) | 0.10415 (0.150269) | 16.6062 (5.53632) | 1.1582 (0.231109) | 0.10896 (0.150136) |
40 | 16.6632 (5.63416) | 1.15649 (0.299794) | 0.109296 (0.141658) | 16.8981 (5.62276) | 1.17658 (0.254403) | 0.111815 (0.141275) |
60 | 16.7998 (5.64793) | 1.17169 (0.322941) | 0.110663 (0.139528) | 16.9577 (5.64668) | 1.18525 (0.326034) | 0.112362 (0.136368) |
80 | 16.8969 (5.65816) | 1.17933 (0.334435) | 0.111659 (0.137915) | 17.0159 (5.62723) | 1.18956 (0.315758) | 0.112945 (0.136815) |
100 | 16.9283 (5.6611) | 1.18378 (0.341065) | 0.112084 (0.127363) | 17.0238 (5.67237) | 1.1920 (0.324957) | 0.113117 (0.12645) |
| ||||||
|
LMP | GP | ||||
|
|
|
|
|
|
|
| ||||||
20 | 16.1217 (5.53559) | 1.11883 (0.22428) | 0.105691 (0.147276) | 16.5891 (5.53325) | 1.15569 (0.207762) | 0.110343 (0.148622) |
40 | 16.6478 (5.6233) | 1.15624 (0.249369) | 0.110091 (0.149015) | 16.8893 (5.62316) | 1.17531 (0.232234) | 0.112492 (0.137587) |
60 | 16.7894 (5.64668) | 1.17152 (0.322650) | 0.111199 (0.148555) | 16.9518 (5.61599) | 1.1844 (0.314585) | 0.112811 (0.145671) |
80 | 16.889 (5.62725) | 1.1792 (0.314213) | 0.112063 (0.138788) | 17.0115 (5.62175) | 1.18892 (0.310674) | 0.11328 (0.139026) |
100 | 16.9220 (5.66035) | 1.18367 (0.320887) | 0.112409 (0.128663) | 17.0202 (5.65197) | 1.19148 (0.35406) | 0.113385 (0.124952) |
BEs and their respective PRs of MRLF under KLF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
20 | 16.404 (0.063125) | 1.13965 (0.073520) | 0.106432 (0.088576) | 16.8667 (0.063124) | 1.17931 (0.073519) | 0.111351 (0.088753) |
40 | 16.7939 (0.031494) | 1.16706 (0.036732) | 0.110498 (0.044231) | 17.0307 (0.031159) | 1.18734 (0.036731) | 0.113046 (0.044226) |
60 | 16.8877 (0.020986) | 1.17884 (0.024480) | 0.111475 (0.029485) | 17.0464 (0.020987) | 1.19249 (0.024481) | 0.113188 (0.029405) |
80 | 16.9632 (0.015733) | 1.18473 (0.018357) | 0.112274 (0.022107) | 17.0828 (0.015732) | 1.19501 (0.018359) | 0.113568 (0.022108) |
100 | 16.9815 (0.012586) | 1.18811 (0.014684) | 0.112579 (0.017685) | 17.0773 (0.012585) | 1.19636 (0.014685) | 0.113616 (0.017682) |
| ||||||
|
LMP | GP | ||||
|
|
|
|
|
|
|
| ||||||
20 | 16.3737 (0.063005) | 1.1392 (0.073491) | 0.108032 (0.088577) | 16.8494 (0.063076) | 1.17679 (0.073498) | 0.112726 (0.087329) |
40 | 16.7783 (0.031144) | 1.16681 (0.036724) | 0.111309 (0.044177) | 17.0220 (0.031132) | 1.18606 (0.036726) | 0.113721 (0.043938) |
60 | 16.8772 (0.020973) | 1.17867 (0.024476) | 0.112019 (0.029403) | 17.0406 (0.020103) | 1.19163 (0.024469) | 0.113636 (0.029356) |
80 | 16.9553 (0.015725) | 1.1846 (0.018355) | 0.112683 (0.022108) | 17.0784 (0.015723) | 1.19436 (0.018358) | 0.113902 (0.022035) |
100 | 16.9751 (0.012581) | 1.18801 (0.014683) | 0.112906 (0.017624) | 17.0737 (0.012582) | 1.19585 (0.014681) | 0.113883 (0.017639) |
Using SELF for
Lorenz curve graph.
For a positive random variable
For the Lindley distribution, the Lorenz curve is
The comparison of Lorenz curve for exponential and Lindley distribution is given in Figures
Empirical CDF of bank data.
Let
where
This quandary has a long narration starting with the revolutionary work of Birnbaum [
Suppose that
where
where
Bayes estimator and posterior risk of stress and strength parameter under SELF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
20, 40 | 0.028511 (0.000077) | 0.064361 (0.000288) | 0.998763 ( |
0.027985 (0.000078) | 0.063217 (0.000282) | 0.998728 ( |
60, 80 | 0.027029 (0.000030) | 0.061592 (0.000109) | 0.998756 ( |
0.026962 (0.000030) | 0.061386 (0.000109) | 0.998749 ( |
80, 100 | 0.026832 (0.000023) | 0.061133 (0.000084) | 0.998754 ( |
0.026797 (0.000023) | 0.061068 (0.000084) | 0.998749 ( |
80, 60 | 0.026663 (0.000029) | 0.060702 (0.000105) | 0.998735 ( |
0.026796 (0.000029) | 0.061019 (0.000105) | 0.998738 ( |
100, 60 | 0.026575 (0.000026) | 0.060422 (0.000095) | 0.998733 ( |
0.026766 (0.000026) | 0.060865 (0.000095) | 0.99874 ( |
| ||||||
|
LMP | GP | ||||
|
0.026364 | 0.0604 | 0.998771 | 0.026364 | 0.0604 | 0.998771 |
| ||||||
20, 40 | 0.028581 (0.000076) | 0.064860 (0.000281) | 0.99874 ( |
0.027987 (0.000078) | 0.063762 (0.000280) | 0.99871 ( |
60, 80 | 0.027051 (0.000029) | 0.061760 (0.000109) | 0.998748 ( |
0.026956 (0.000029) | 0.061625 (0.000108) | 0.998743 ( |
80, 100 | 0.026848 (0.000022) | 0.061323 (0.000083) | 0.998748 ( |
0.026792 (0.000022) | 0.061255 (0.000083) | 0.998745 ( |
80, 60 | 0.026676 (0.000028) | 0.061014 (0.000105) | 0.998729 ( |
0.026781 (0.000028) | 0.061306 (0.000104) | 0.998733 ( |
100, 60 | 0.026585 (0.000026) | 0.060732 (0.000094) | 0.998728 ( |
0.026748 (0.000026) | 0.061143 (0.000094) | 0.998736 ( |
Bayes estimator and posterior risk of stress and strength parameter under WSELF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
0.026364 | 0.0604 | 0.998771 | 0.026364 | 0.0604 | 0.998771 |
20, 40 | 0.025616 (0.002896) | 0.059646 (0.004714) | 0.998762 ( |
0.025146 (0.002839) | 0.058595 (0.004622) | 0.998728 ( |
60, 80 | 0.025913 (0.001116) | 0.059791 (0.001801) | 0.998756 ( |
0.025849 (0.001113) | 0.059591 (0.001795) | 0.998749 ( |
80, 100 | 0.025971 (0.000860) | 0.059748 (0.001384) | 0.998754 ( |
0.025938 (0.000859) | 0.059685 (0.001383) | 0.998749 ( |
80, 60 | 0.025571 (0.001092) | 0.058960 (0.001742) | 0.998735 ( |
0.025699 (0.001091) | 0.059267 (0.001742) | 0.998738 ( |
100, 60 | 0.025581 (0.000993) | 0.058846 (0.001576) | 0.998733 ( |
0.025765 (0.009991) | 0.059276 (0.001568) | 0.99874 ( |
| ||||||
|
CP | GP | ||||
|
0.026364 | 0.0604 | 0.998771 | 0.026364 | 0.0604 | 0.998771 |
| ||||||
20, 40 | 0.025679 (0.002801) | 0.060117 (0.004543) | 0.99874 ( |
0.025148 (0.002839) | 0.059091 (0.004571) | 0.99871 ( |
60, 80 | 0.025934 (0.001111) | 0.059954 (0.001786) | 0.998748 ( |
0.025843 (0.001110) | 0.059822 (0.001702) | 0.998743 ( |
80, 100 | 0.025987 (0.000860) | 0.059934 (0.001382) | 0.998748 ( |
0.025933 (0.000859) | 0.059869 (0.001382) | 0.998745 ( |
80, 60 | 0.025584 (0.001092) | 0.059262 (0.001741) | 0.998729 ( |
0.025684 (0.001091) | 0.059548 (0.001737) | 0.998733 ( |
100, 60 | 0.025591 (0.000993) | 0.059146 (0.001558) | 0.998728 ( |
0.025747 (0.00990) | 0.059549 (0.001553) | 0.998736 ( |
Bayes estimator and posterior risk of stress and strength parameter under MSELF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
0.026364 | 0.0604 | 0.998771 | 0.026364 | 0.0604 | 0.998771 |
20, 40 | 0.023332 (0.999986) | 0.055666 (0.999815) | 0.998762 (0.003708) | 0.023021 (0.999987) | 0.054887 (0.999813) | 0.998728 (0.003812) |
60, 80 | 0.024906 (0.999984) | 0.058118 (0.999798) | 0.998756 (0.003728) | 0.024851 (0.999984) | 0.057935 (0.999800) | 0.998749 (0.003729) |
80, 100 | 0.025178 (0.999984) | 0.058442 (0.999796) | 0.998754 (0.003734) | 0.025149 (0.999984) | 0.058385 (0.999797) | 0.998749 (0.003734) |
80, 60 | 0.024596 (0.999985) | 0.057357 (0.999806) | 0.998735 (0.003790) | 0.024705 (0.999984) | 0.057630 (0.999803) | 0.998738 (0.003780) |
100, 60 | 0.024689 (0.999984) | 0.057390 (0.999806) | 0.998733 (0.003796) | 0.024846 (0.999984) | 0.057777 (0.999802) | 0.99874 (0.003776) |
| ||||||
|
LMP | GP | ||||
|
0.026364 | 0.0604 | 0.998771 | 0.026364 | 0.0604 | 0.998771 |
| ||||||
20, 40 | 0.023375 (0.999986) | 0.056022 (0.999811) | 0.99874 (0.003706) | 0.023023 (0.999987) | 0.055252 (0.999802) | 0.998709 (0.003666) |
60, 80 | 0.024924 (0.999984) | 0.058258 (0.999797) | 0.998748 (0.003721) | 0.024847 (0.999984) | 0.058141 (0.999796) | 0.998743 (0.003727) |
80, 100 | 0.025191 (0.999984) | 0.058613 (0.999794) | 0.998748 (0.003732) | 0.025144 (0.999984) | 0.058552 (0.999795) | 0.998745 (0.003731) |
80, 60 | 0.024607 (0.999985) | 0.057626 (0.999803) | 0.998728 (0.003780) | 0.024692 (0.999984) | 0.057881 (0.99980) | 0.998733 (0.003759) |
100, 60 | 0.024697 (0.999984) | 0.057659 (0.999803) | 0.998728 (0.003712) | 0.024831 (0.999984) | 0.058025 (0.99980) | 0.998736 (0.003708) |
Bayes estimator and posterior risk of stress and strength parameter under PLF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
0.026364 | 0.0604 | 0.998771 | 0.026364 | 0.0604 | 0.998771 |
20, 40 | 0.029827 (0.002631) | 0.066557 (0.004393) | 0.998763 ( |
0.029348 (0.002628) | 0.065490 (0.004345) | 0.998728 ( |
60, 80 | 0.027570 (0.001082) | 0.062474 (0.001764) | 0.998756 ( |
0.027506 (0.001087) | 0.062271 (0.001769) | 0.998749 ( |
80, 100 | 0.027252 (0.000841) | 0.061814 (0.001363) | 0.998754 ( |
0.027219 (0.000843) | 0.061751 (0.001361) | 0.998749 ( |
80, 60 | 0.027198 (0.001070) | 0.061565 (0.001725) | 0.998735 ( |
0.027326 (0.001059) | 0.061873 (0.001707) | 0.998738 ( |
100, 60 | 0.027064 (0.000979) | 0.061206 (0.001568) | 0.998733 ( |
0.027248 (0.000966) | 0.061638 (0.001546) | 0.99874 ( |
| ||||||
|
LMP | GP | ||||
|
0.026364 | 0.0604 | 0.998771 | 0.026364 | 0.0604 | 0.998771 |
| ||||||
20, 40 | 0.0298893 (0.002617) | 0.067019 (0.004317) | 0.99874 ( |
0.029351 (0.002627) | 0.066001 (0.004417) | 0.99871 ( |
60, 80 | 0.027591 (0.001080) | 0.062633 (0.001748) | 0.998748 ( |
0.027500 (0.001080) | 0.062503 (0.001746) | 0.998743 ( |
80, 100 | 0.027268 (0.000839) | 0.061999 (0.001354) | 0.998748 ( |
0.027214 (0.000834) | 0.061934 (0.001347) | 0.998745 ( |
80,60 | 0.027211 (0.001058) | 0.061868 (0.001707) | 0.998729 ( |
0.027311 (0.001060) | 0.062150 (0.001689) | 0.998733 ( |
100, 60 | 0.027074 (0.000958) | 0.061508 (0.001543) | 0.998728 ( |
0.027231 (0.000957) | 0.061908 (0.001529) | 0.998736 ( |
Bayes estimator and posterior risk of stress and strength parameter under SLLF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
0.026364 | 0.0604 | 0.998771 | 0.026364 | 0.0604 | 0.998771 |
20, 40 | 0.027033 (0.11001) | 0.061970 (0.077534) | 0.998762 ( |
0.026509 (0.109867) | 0.060832 (0.077376) | 0.998728 ( |
60, 80 | 0.026465 (0.042628) | 0.060684 (0.029901) | 0.998756 ( |
0.026398 (0.042621) | 0.060480 (0.029899) | 0.998749 ( |
80, 100 | 0.026398 (0.032862) | 0.060436 (0.023035) | 0.998754 ( |
0.026363 (0.032859) | 0.060372 (0.023033) | 0.998749 ( |
80, 60 | 0.026109 (0.042241) | 0.059822 (0.029329) | 0.998735 ( |
0.026243 (0.042241) | 0.060138 (0.029323) | 0.998738 ( |
100, 60 | 0.026071 (0.038466) | 0.059625 (0.026609) | 0.998733 ( |
0.026262 (0.038458) | 0.060068 (0.026605) | 0.99874 ( |
| ||||||
|
LMP | GP | ||||
|
|
|
|
|
|
|
| ||||||
20, 40 | 0.027104 (0.109771) | 0.062474 (0.077367) | 0.99874 ( |
0.026511 (0.109817) | 0.061372 (0.077337) | 0.99871 ( |
60, 80 | 0.026487 (0.042620) | 0.060853 (0.029893) | 0.998748 ( |
0.026392 (0.042619) | 0.060718 (0.029804) | 0.998743 ( |
80, 100 | 0.026414 (0.032860) | 0.060626 (0.023029) | 0.998748 ( |
0.026358 (0.032858) | 0.060559 (0.023030) | 0.998745 ( |
80, 60 | 0.026123 (0.042240) | 0.060133 (0.029322) | 0.998729 ( |
0.026227 (0.042235) | 0.060425 (0.029307) | 0.998733 ( |
100, 60 | 0.026081 (0.038459) | 0.059934 (0.026602) | 0.998728 ( |
0.026244 (0.038455) | 0.060347 (0.026580) | 0.998736 ( |
Bayes estimator and posterior risk of stress and strength parameter under ELF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
20, 40 | 0.025616 |
0.059646 |
0.998762 |
0.025146 |
0.058595 |
0.998728 |
60, 80 | 0.025913 |
0.059791 |
0.998756 |
0.025849 |
0.059591 |
0.998749 |
80, 100 | 0.025971 |
0.059748 |
0.998754 |
0.025938 |
0.059685 |
0.998749 |
80, 60 | 0.025571 |
0.058960 |
0.998735 |
0.025699 |
0.059267 |
0.998738 |
100, 60 | 0.025581 |
0.058846 |
0.998733 |
0.025765 |
0.059276 |
0.99874 |
| ||||||
|
LMP | GP | ||||
|
|
|
|
|
|
|
| ||||||
20, 40 | 0.025679 |
0.060117 |
0.99874 |
0.025148 |
0.059091 |
0.99871 |
60, 80 | 0.025934 |
0.059954 |
0.998748 |
0.025843 |
0.059822 |
0.998743 |
80, 100 | 0.025987 |
0.059934 |
0.998748 |
0.025933 |
0.059868 |
0.998745 |
80, 60 | 0.025584 |
0.059262 |
0.998729 |
0.025684 |
0.059548 |
0.998733 |
100, 60 | 0.025591 |
0.059146 |
0.998728 |
0.025747 |
0.059549 |
0.998736 |
Bayes estimator and posterior risk of stress and strength parameter under KLF.
|
UP | JP | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
20, 40 | 0.027025 |
0.061958 |
0.998762 ( |
0.026527 |
0.060863 |
0.998728 ( |
60, 80 | 0.026465 |
0.060685 |
0.998756 ( |
0.026399 |
0.060482 |
0.998749 ( |
80, 100 | 0.026398 |
0.060436 |
0.998754 ( |
0.026364 |
0.060373 |
0.998749 ( |
80, 60 | 0.026111 |
0.059825 |
0.998735 ( |
0.026242 |
0.060137 |
0.998738 ( |
100, 60 | 0.026073 |
0.059629 |
0.998733 ( |
0.026260 |
0.060066 |
0.99874 ( |
| ||||||
|
LMP | GP | ||||
|
|
|
|
|
|
|
| ||||||
20, 40 | 0.027091 |
0.062444 |
0.99874 ( |
0.026529 |
0.061382 |
0.99871 ( |
60, 80 | 0.026486 |
0.06085 |
0.998748 ( |
0.026393 |
0.060717 |
0.998743 ( |
80, 100 | 0.026414 |
0.060625 |
0.998748 ( |
0.026359 |
0.060558 |
0.998745 ( |
80, 60 | 0.026125 |
0.060132 |
0.998729 ( |
0.026227 |
0.060420 |
0.998733 ( |
100, 60 | 0.026083 |
0.059934 |
0.998728 ( |
0.026243 |
0.060341 |
0.998736 ( |
The Bayes estimates of stress and strength under SELF, MSELF, and ELF are underestimated. By comparing symmetric and asymmetric loss functions, it is noted that posterior risk of SELF is smaller than asymmetric loss functions. In case of asymmetric loss functions, WSELF and PLF have smaller posterior risk than other available loss functions.
Evaluating the performance of informative and noninformative priors, one can easily observe that informative priors have smaller posterior risk due to the availability of compact information. LM and gamma priors both have approximately the same behaviour depending upon the choice of hyperparameters value. More compact information will lead to correct hyperparameters which will lead to definitely better results and smaller posterior risk than noninformative priors. Although there are some depicts where informative priors have posterior risks greater than noninformative priors which is just due to random generation. Increasing sample size in case of SLLF has an inverse effect.
Ghitany et al. [
Waiting time (in minutes) before customer service in Bank A.
0.8 | 2.9 | 4.3 | 5.0 | 6.7 | 8.2 | 9.7 | 11.9 | 14.1 | 19.9 |
0.8 | 3.1 | 4.3 | 5.3 | 6.9 | 8.6 | 9.8 | 12.4 | 15.4 | 20.6 |
1.3 | 3.2 | 4.4 | 5.5 | 7.1 | 8.6 | 10.7 | 12.5 | 15.4 | 21.3 |
1.5 | 3.3 | 4.4 | 5.7 | 7.1 | 8.6 | 10.9 | 12.9 | 17.3 | 21.4 |
1.8 | 3.5 | 4.6 | 5.7 | 7.1 | 8.8 | 11.0 | 13.0 | 17.3 | 21.9 |
1.9 | 3.6 | 4.7 | 6.1 | 7.1 | 8.8 | 11.0 | 13.1 | 18.1 | 23.0 |
1.9 | 4.0 | 4.7 | 6.2 | 7.4 | 8.9 | 11.1 | 13.3 | 18.2 | 27.0 |
2.1 | 4.1 | 4.8 | 6.2 | 7.6 | 8.9 | 11.2 | 13.6 | 18.4 | 31.6 |
2.6 | 4.2 | 4.9 | 6.2 | 7.7 | 9.5 | 11.2 | 13.7 | 18.9 | 33.1 |
2.7 | 4.2 | 4.9 | 6.3 | 8.0 | 9.6 | 11.5 | 13.9 | 19.0 | 38.5 |
Waiting time (in minutes) before customer service in Bank B.
0.1 | 1.2 | 2.3 | 2.9 | 3.5 | 5.3 | 6.8 | 8.0 | 8.5 | 13.2 |
0.2 | 1.8 | 2.3 | 3.1 | 3.9 | 5.6 | 7.3 | 8.5 | 11.0 | 13.7 |
0.3 | 1.9 | 2.5 | 3.1 | 4.0 | 5.6 | 7.5 | 8.7 | 12.1 | 14.5 |
0.7 | 2.0 | 2.6 | 3.2 | 4.2 | 6.2 | 7.7 | 9.5 | 12.3 | 16.0 |
0.9 | 2.2 | 2.7 | 3.4 | 4.5 | 6.3 | 7.7 | 10.7 | 12.8 | 16.5 |
1.1 | 2.3 | 2.7 | 3.4 | 4.7 | 6.6 | 8.0 | 10.9 | 12.9 | 28.0 |
We fit on both data sets Kolmogorov-Smirnov test and found that Lindley distribution is good fitted. The values of K-S test along
K-S distances and associated
Data set | K-S test |
|
MLE |
---|---|---|---|
Bank A | 0.065 | 0.721 | 0.187 |
Bank B | 0.083 | 0.826 | 0.280 |
The Bayes estimates of stress and strength reliability under different priors and their posterior risk are evaluated in Tables
BEs and respective PRs of stress and strength under different prior of real data where
|
UP | JP | LMP | GP |
---|---|---|---|---|
SELF | 0.354572 (0.001590) | 0.35578 (0.001588) | 0.354518 (0.001589) | 0.355719 (0.001588) |
WSELF | 0.350132 (0.004439) | 0.351317 (0.004436) | 0.35008 (0.004438) | 0.351257 (0.004432) |
MSELF | 0.345899 (0.958108) | 0.347017 (0.957694) | 0.34585 (0.957126) | 0.346961 (0.957115) |
PLF | 0.356806 (0.004470) | 0.358004 (0.004448) | 0.356754 (0.004470) | 0.357944 (0.004449) |
SLLF | 0.352333 (0.012639) | 0.353537 (0.012664) | 0.35228 (0.012637) | 0.353476 (0.012663) |
ELF | 0.350132 (2.09262) | 0.351317 (2.08584) | 0.35008 (2.09292) | 0.351257 (2.08618) |
KLF | 0.352345 (0.025358) | 0.353541 (0.023508) | 0.352292 (0.025354) | 0.353481 (0.025307) |
BEs and respective PRs of mean residual life function under different prior of real data of Bank A.
|
UP | JP | LMP | GP |
---|---|---|---|---|
SELF | 8.40917 (0.477926) | 8.45888 (0.475225) | 8.40618 (0.477931) | 8.45692 (0.475245) |
WSELF | 8.35267 (0.056499) | 8.40204 (0.056847) | 8.34973 (0.056459) | 8.40007 (0.056485) |
MSELF | 8.29763 (574.087) | 8.34567 (584.203) | 8.29478 (573.489) | 8.34375 (583.797) |
PLF | 8.43754 (0.056738) | 8.48693 (0.056088) | 8.43456 (0.056735) | 8.48498 (0.056124) |
SLLF | 8.38079 (0.006753) | 8.43049 (0.006754) | 8.37781 (0.006750) | 8.42852 (0.006754) |
ELF | 8.35267 (4.24852) | 8.40204 (4.26033) | 8.34973 (4.24782) | 8.40007 (4.23986) |
KLF | 8.38088 (0.013528) | 8.43041 (0.013531) | 8.37791 (0.013523) | 8.42845 (0.013524) |
Since the Lindley distribution belongs to the exponential family so the natural conjugate prior is Gamma distribution.
The posterior risks of LMP and GP are approximately the same as compared to noninformative priors. There are some posterior risk values which are greater than noninformative priors. These are just due to hyperparameters value effect that is, more accurate values will lead to the smaller posterior risk. PLF and WSELF loss functions have smaller posterior risk as compared to other loss functions.
We consider the Bayesian analysis of the Lindley model via informative and informative priors under different loss functions. Based on posterior distribution, different properties, we conclude that informative priors (LMP, GP) performance approximately equal and have smaller posterior risk’s as compared to the noninformative priors; also Jeffreys prior results are more precised than uniform prior. In other words, we can summarize result as
The choice of loss function as concerned, one can easily observe based on evidence (different properties as discussed above) that PLF, SLLF, and WSELF are suitable than other asymmetrical loss functions. One thing is common as we increase sample size posterior risk comes down. In future, this work can be extended using censored data.