Bayesian Approach for Confidence Intervals of Variance on the Normal Distribution

Table 4

The coverage probability (CP) and average width (AW) via 99% confidence interval.

		ML		Bayesian approach with prior distribution						MCMC		Wald
				Gamma		Chi-squared		Exponential
		CP	AW	CP	AW	CP	AW	CP	AW	CP	AW	CP	AW
2	10	0.9913	9.602	0.9841	—	0.9949	—	0.9826	—	0.9771	—	0.9000	—
	50	0.9907	2.338	0.9825	—	0.9868	—	0.9834	—	0.9838	—	0.9697	—
	100	0.9896	1.550	0.9849	—	0.9862	—	0.9852	—	0.9856	—	0.9774	—
	300	0.9888	0.858	0.9862	—	0.9867	0.543	0.9865	0.543	0.9859	0.545	0.9848	—
	500	0.9905	0.660	0.9895	0.654	0.9899	0.656	0.9897	0.656	0.9897	0.657	0.9883	0.651
	1000	0.9886	0.463	0.9892	0.461	0.9893	0.462	0.9891	0.462	0.9895	0.462	0.9888	0.460
6	10	0.9913	28.80	0.9321	—	0.9629	—	0.9636	—	0.9765	—	0.9000	—
	50	0.9907	7.014	0.9718	—	0.9793	—	0.9802	—	0.9840	—	0.9697	—
	100	0.9896	4.651	0.9783	—	0.9824	—	0.9836	—	0.9856	—	0.9774	—
	300	0.9888	1.947	0.9850	—	0.9861	—	0.9860	—	0.9864	—	0.9848	—
	500	0.9905	1.980	0.9882	1.958	0.9894	1.964	0.9895	1.966	0.9896	1.973	0.9883	1.955
	1000	0.9886	1.391	0.9888	1.383	0.9893	1.385	0.9892	1.386	0.9887	1.388	0.9888	1.382
12	10	0.9913	57.61	0.9096	—	0.9476	—	0.9561	—	0.9766	—	0.9000	—
	50	0.9907	14.02	0.9687	—	0.9765	—	0.9793	—	0.9839	—	0.9697	—
	100	0.9896	9.303	0.9765	—	0.9813	—	0.9824	—	0.9853	—	0.9774	—
	300	0.9888	5.153	0.9846	—	0.9857	—	0.9860	—	0.9869	—	0.9848	—
	500	0.9905	3.960	0.9876	—	0.9890	3.925	0.9893	3.931	0.990	3.946	0.9983	3.911
	1000	0.9886	2.782	0.9830	2.765	0.9892	2.770	0.9893	2.772	0.9895	2.778	0.9888	2.765

Note. The appearance in AW denotes CP in the range of the fixed confidence interval between 0.9880 and 0.9919 and italics minimum of AW.