Bayesian analysis for the Lomax model using noninformative priors

Figures & data

Figure 1. Square root of the mean squared error and coverage probability of the Bayesian estimation of α based on the gamma priors ${\pi }_G$ with $\tau =1$ (circle), $\tau =5$ (cross), and $\tau =9$ (diamond). Panel (a) is for the square root of mean squared error, and panel (b) is for the coverage probability.

Table 1. Empirical MSEs and CPs (within parentheses) of Bayesian estimators based on the priors ${\pi }_J$ and ${\pi }_{R_2}$.

	n	20	50	80	100	150
$\alpha =3,\text{β}=1$
	$P(C{V}_n<1)$	0.250	0.076	0.025	0.013	0.002
${\pi }_J$	α	4.445 (0.983)	7.601 (0.974)	6.743 (0.961)	5.532 (0.959)	3.414 (0.948)
	β	1.021 (0.984)	1.442 (0.972)	1.254 (0.958)	1.007 (0.955)	0.654 (0.950)
${\pi }_{R_2}$	α	3.023 (0.977)	5.957 (0.972)	5.528 (0.959)	4.646 (0.957)	2.965 (0.952)
	β	0.685 (0.980)	1.128 (0.969)	1.027 (0.958)	0.844 (0.956)	0.567 (0.951)
$\alpha =2,\text{β}=1.5$
	$P(C{V}_n<1)$	0.146	0.021	0.003	0.001	0.000
${\pi }_J$	α	3.864 (0.978)	3.312 (0.958)	2.085 (0.938)	1.495 (0.942)	0.565 (0.941)
	β	5.006 (0.979)	3.930 (0.956)	2.412 (0.939)	1.689 (0.938)	0.653 (0.943)
${\pi }_{R_2}$	α	2.538 (0.971)	2.516 (0.960)	1.733 (0.942)	1.272 (0.946)	0.505 (0.944)
	β	3.277 (0.975)	3.080 (0.959)	2.007 (0.943)	1.431 (0.942)	0.582 (0.947)
$\alpha =1.5,\text{β}=2$
	$P(C{V}_n<1)$	0.082	0.005	0.000	0.000	0.000
${\pi }_J$	α	2.898 (0.975)	1.405 (0.947)	0.660 (0.939)	0.346 (0.946)	0.154 (0.946)
	β	14.218 (0.966)	6.280 (0.947)	3.015 (0.943)	1.456 (0.944)	0.639 (0.946)
${\pi }_{R_2}$	α	1.927 (0.972)	1.113 (0.951)	0.572 (0.942)	0.304 (0.947)	0.143 (0.947)
	β	9.361 (0.971)	4.926 (0.952)	2.593 (0.942)	1.280 (0.948)	0.590 (0.949)

Table 2. Summary of the parametric bootstrap and the Bayesian estimates.

Method	Parameter	Estimator	SD	95% CI
Bootstrap	α	0.504	0.047	(0.422, 0.606)
	β	132.654	27.649	(86.717, 194.908)
${\pi }_J$	α	0.497	0.043	(0.420, 0.587)
	β	128.406	24.125	(89.679, 182.416)
${\pi }_{R_2}$	α	0.495	0.043	(0.419, 0.586)
	β	127.067	23.545	(89.755, 180.693)