双定数混合截尾下两参数Pareto分布的统计分析

摘要/Abstract

摘要：

在传统的定时和定数截尾试验的基础上，该文首次提出了一种新的截尾试验方案：双定数混合截尾.基于这类截尾数据求出了两参数Pareto分布参数的极大似然估计及θ的置信区间.当α已知时，取Gamma先验分布的情况下，求出了三种不同损失函数下参数θ、可靠度函数以及失效率函数的Bayes估计；当α，θ都未知时，分别取无信息先验分布和指数先验分布，在平方损失函数下分别计算出α，θ、可靠度函数以及失效率函数的Bayes估计.利用Monte-Carlo方法模拟出双定数混合截尾样本，进而得到了两参数Pareto分布的参数及可靠性指标的估计，计算出相对误差并把各种估计的精度进行了比较.最后对一个数值例子进行了分析.

关键词: 两参数Pareto分布, 双定数混合截尾, 极大似然估计, 先验分布, Bayes估计

Abstract:

On the basis of the traditional type-Ⅰ and type-Ⅱ censoring tests, a new type of censoring test scheme, double type-Ⅱ hybrid censoring, is proposed for the first time in this paper. Based on this kind of censored data, the maximum likelihood estimates of the parameters and confidence interval of θ are obtained for two-parameter Pareto distribution. The Bayesian estimates of θ, reliability function and failure rate function under three different loss functions are obtained when α is known and the Gamma prior distribution is selected. When both α and θ are unknown, we take the noninformation prior distribution and exponential prior distribution respectively, and calculate the Bayesian estimates of α and θ, the reliability function and failure rate function under the squared loss function. The Monte-Carlo method is used to simulate double type-Ⅱ hybrid censored samples, the estimates of the parameter and reliability indexes for two-parameter Pareto distribution are obtained. The relative errors are calculated, and the accuracy of various estimates is compared. Finally, a numerical example is analyzed.

Key words: Two-parameter Pareto distribution, Double type-Ⅱ hybrid censoring, Maximum likelihood estimate, Prior distribution, Bayesian estimate

中图分类号:

O213

龙兵,张忠占. 双定数混合截尾下两参数Pareto分布的统计分析[J]. 数学物理学报, 2022, 42(1): 269-281.

Bing Long,Zhongzhan Zhang. Statistical Analysis of Two-Parameter Pareto Distribution Under Double Type-Ⅱ Hybrid Censoring Scheme[J]. Acta mathematica scientia,Series A, 2022, 42(1): 269-281.

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n	(α, θ)	m₁	m₂	t₀	$\hat{\theta}_M$	$\hat{\theta}_{B1}$	$\hat{\theta}_{B2}$	$\hat{\theta}_{B3}$
20	(6, 2)	9	12	8	2.1378	2.0942	1.9327	1.9298
					(0.2663)	(0.2178)	(0.2021)	(0.2188)
	(7, 3)	11	16	9	3.1269	2.8987	2.6540	2.7132
					(0.2270)	(0.1880)	(0.1951)	(0.2026)
30	(6, 2)	13	18	8	2.0730	2.0559	1.9437	1.9417
					(0.2170)	(0.1903)	(0.1816)	(0.1922)
	(7, 3)	16	23	9	3.0643	2.9136	2.7320	2.7799
					(0.1851)	(0.1640)	(0.1696)	(0.1742)
50	(6, 2)	22	30	8	2.0394	2.0342	1.9634	1.9623
					(0.1612)	(0.1492)	(0.1473)	(0.1527)
	(7, 3)	26	39	9	3.0282	2.9415	2.8256	2.8588
					(0.1465)	(0.1366)	(0.1393)	(0.1418)

n	(α, θ)	m₁	m₂	t₀	$\hat{R}(x) $	$\hat{R}_{B1}(x)$	$\hat{R}_{B2}(x)$	$\hat{R}_{B3}(x)$
20	(6, 2)	9	12	8	0.5505	0.5614	0.5481	0.5460
					(0.1438)	(0.1185)	(0.1196)	(0.1230)
	(7, 3)	11	16	9	0.6628	0.6848	0.6792	0.6783
					(0.0891)	(0.0767)	(0.0769)	(0.0761)
30	(6, 2)	13	18	8	0.5587	0.5648	0.5557	0.5542
					(0.1189)	(0.1046)	(0.1055)	(0.1064)
	(7, 3)	16	23	9	0.6663	0.6814	0.6781	0.6767
					(0.0743)	(0.0670)	(0.0671)	(0.0667)
50	(6, 2)	22	30	8	0.5605	0.5640	0.5595	0.5574
					(0.0918)	(0.0851)	(0.0852)	(0.0855)
	(7, 3)	26	39	9	0.6701	0.6789	0.6774	0.6760
					(0.0574)	(0.0545)	(0.0546)	(0.0542)

n	(α, θ)	m₁	m₂	t₀	$\hat{H}(x) $	$\hat{H}_{B1}(x)$	$\hat{H}_{B2}(x)$	$\hat{H}_{B3}(x)$
20	(6, 2)	9	12	8	0.2676	0.2620	0.2593	0.2415
					(0.2677)	(0.2188)	(0.2156)	(0.2186)
	(7, 3)	11	16	9	0.3913	0.3624	0.3582	0.3392
					(0.2316)	(0.1915)	(0.1914)	(0.2064)
30	(6, 2)	13	18	8	0.2591	0.2570	0.2551	0.2428
					(0.2133)	(0.1872)	(0.1856)	(0.1892)
	(7, 3)	16	23	9	0.3823	0.3636	0.3606	0.3469
					(0.1860)	(0.1649)	(0.1650)	(0.1750)
50	(6, 2)	22	30	8	0.2547	0.2540	0.2529	0.2451
					(0.1630)	(0.1509)	(0.1503)	(0.1542)
	(7, 3)	26	39	9	0.3774	0.3667	0.3648	0.3564
					(0.1449)	(0.1355)	(0.1355)	(0.1407)

n	(α, θ)	m₁	m₂	t₀	$\hat{\alpha} $	$\hat{\theta}$	$\hat{\alpha}_{B}$	$\hat{\theta}_{B}$
20	(6, 2)	9	12	8	6.1534	2.3660	6.0033	2.1155
					(0.0256)	(0.3123)	(0.0185)	(0.2516)
	(7, 3)	11	16	9	7.1179	3.3724	7.0012	3.1045
					(0.0168)	(0.2500)	(0.0122)	(0.2174)
30	(6, 2)	13	18	8	6.1017	2.2269	6.0017	2.0765
					(0.0169)	(0.2359)	(0.0122)	(0.2085)
	(7, 3)	16	23	9	7.0784	3.2426	7.0006	3.0715
					(0.0112)	(0.1992)	(0.0082)	(0.1830)
50	(6, 2)	22	30	8	6.0607	2.1080	6.0007	2.0251
					(0.0101)	(0.1674)	(0.0074)	(0.1590)
	(7, 3)	26	39	9	7.0474	3.1085	7.0007	3.0146
					(0.0068)	(0.1466)	(0.0050)	(0.1425)

m₁	m₂	t₀	$\hat{\alpha}$	$\hat{\theta}$	$\hat{\lambda}$	$\hat{\alpha}_B $	$\hat{\theta}_B $	$\hat{R}_B(x) $	$\hat{H}_B(x) $
6	10	0.53	0.5009	4.6144	0.2167	0.4935	3.9552	0.2879	5.6503
		0.60	0.5009	3.2505	0.3076	0.4917	2.9550	0.3709	4.2215
8	12	0.58	0.5009	3.5210	0.2840	0.4920	3.1297	0.3559	4.4711
		0.61	0.5009	3.4805	0.2873	0.4926	3.2128	0.3401	4.5897
11	14	0.62	0.5009	3.3157	0.3016	0.4921	3.0394	0.3599	4.3420
		0.63	0.5009	3.4674	0.2884	0.4928	3.2362	0.3355	4.6232
13	16	0.66	0.5009	3.4647	0.2886	0.4927	3.2172	0.3385	4.5960
		0.68	0.5009	4.1000	0.2439	0.4941	3.8589	0.2751	5.5127
15	18	0.70	0.5009	3.6627	0.2730	0.4933	3.4337	0.3147	4.9053
		0.72	0.5009	3.7957	0.2635	0.4937	3.5960	0.2971	5.1371
17	20	0.78	0.5009	3.7430	0.2672	0.4935	3.5351	0.3034	5.0501
		0.84	0.5009	3.9352	0.2541	0.4940	3.7478	0.2821	5.3541