双定数混合截尾下两参数Pareto分布的统计分析

Abstract

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

CLC Number:

O213

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.

Trendmd

Figures/Tables 5

References 21

1	Han M . E-Bayesian estimation and its E-posterior risk of the exponential distribution parameter based on complete and Type-Ⅰ censored samples. Commun Statist-Theor M, 2020, 49 (8): 1858- 1872 doi: 10.1080/03610926.2019.1565837
2	Zhu X J , Balakrishnan N , Feng C Z , et al. Exact likelihood-ratio tests for joint Type-Ⅱ censored exponential data. Statistics, 2020, 54 (3): 636- 648 doi: 10.1080/02331888.2020.1764559
3	Hideki N , Balakrishnan N . A consistent method of estimation for the three-parameter lognormal distribution based on Type-Ⅱ right censored data. Commun Statist-Theor M, 2016, 45 (19): 5693- 5708 doi: 10.1080/03610926.2014.948205
4	Indrani B , Balakrishnan N . A note on the prediction of censored exponential lifetimes in a simple step-stress model with Type-Ⅱ censoring. Calc Stat Assoc Bull, 2018, 70 (1): 57- 73
5	Ren J R , Gui W H . A statistical inference for generalized Rayleigh model under Type-Ⅱ progressive censoring with binomial removals. J Syst Eng Electron, 2020, 31 (1): 206- 223
6	Indrani B , Balakrishnan N . Prediction of censored exponential lifetimes in a simple step-stress model under progressive Type-Ⅱ censoring. Computation Stat, 2017, 32 (4): 1665- 1687 doi: 10.1007/s00180-016-0684-0
7	邓严林. 双边定数截尾下Topp-Leone分布的Bayes估计与预测. 江西师范大学学报(自然科学版), 2021, 45 (3): 272- 277
	Deng Y L . Bayesian estimation and prediction for the Topp-Leone distribution under type-Ⅱ doubly censored sample. Journal of Jiangxi Normal University(Natural Science), 2021, 45 (3): 272- 277
8	Rezapour M , Alamatsaz M H , Balakrishnan N . Distribution of the number of observations greater than the ith dependent progressively Type-Ⅱ censored order statistic and its use in goodness-of-fit testing. Commun Statist-Theor M, 2015, 44 (12): 2517- 2529 doi: 10.1080/03610926.2015.1043796
9	Wu S J , Kus C . On estimation based on progressive first-failure-censored sampling. Comput Stat Data An, 2009, 53 (10): 3659- 3670 doi: 10.1016/j.csda.2009.03.010
10	Ahmadi M V , Doostparast M , Ahmadi . Estimating the lifetime performance index with Weibull distribution based on progressive first-failure censoring scheme. J Comput Appl Math, 2013, 239 (1): 93- 102
11	Ahmed E . Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application. J Appl Stat, 2017, 44 (9): 1576- 1608 doi: 10.1080/02664763.2016.1214692
12	Lio Y L , Tsai T R . Estimation of δ=P(X < Y) for Burr XII distribution based on the progressively first failure-censored samples. J Appl Stat, 2012, 39: 309- 322 doi: 10.1080/02664763.2011.586684
13	Gyan P . Empirical estimation based on progressive first failure censored generalized pareto data. Afri Stat, 2018, 13 (2): 1683- 1697
14	Shafay A R , Balakrishnan N . Estimation and prediction for an exponential form distribution based on combined Type-Ⅱ censored samples. Bull Malays Math Sci Soc, 2019, 42 (4): 1535- 1553 doi: 10.1007/s40840-017-0563-z
15	Lin C T , Hsu Y Y , Lee S Y , Balakrishnan N . Inference on constant stress accelerated life tests for log-location-scale lifetime distributions with Type-Ⅰ hybrid censoring. J Stat Comput Sim, 2019, 89 (4): 720- 749 doi: 10.1080/00949655.2019.1571591
16	Debasis K D , Avijit J . Analysis of Type-Ⅱ progressively hybrid censored data. Comput Stat Data An, 2006, 50 (4): 2509- 2528
17	Balakrishnan N , Shafay A R . One- and two-sample Bayesian prediction intervals based on Type-Ⅱ hybrid censored data. Commun Statist-Theor M, 2012, 41 (9): 1511- 1531 doi: 10.1080/03610926.2010.543300
18	茆诗松, 濮晓龙, 刘忠. 寿命数据中的统计模型与方法. 北京: 中国统计出版社, 1998: 101- 103
	Mao S S , Pu X L , Liu Z . Statistical Models and Methods for Lifetime Data. Beijing: China Statistics Press, 1998: 101- 103
19	Arora S , Mahajan K K , Kumari R . Bayes estimators for the reliability and hazard rate functions of Topp-Leone distribution using Type-Ⅱ censored data. Commun Statist-Simul C, 2019, doi: 10.1080/03610918.2019.1602646
20	韩明. 不同损失函数下Poisson分布参数的E-Bayes估计及其E-MSE. 数学物理学报, 2019, 39A (3): 664- 673 doi: 10.3969/j.issn.1003-3998.2019.03.023
	Han M . E-Bayesian estimation and its E-MSE of Possion distribution parameter under different loss functions. Acta Math Sci, 2019, 39A (3): 664- 673 doi: 10.3969/j.issn.1003-3998.2019.03.023
21	Nigm A M , Al-Hussaini E K , Jaheen Z F . Bayesian one-sample prediction of future observations under Pareto distribution. Statistics, 2003, 37 (6): 527- 536 doi: 10.1080/02331880310001598837

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

Statistical Analysis of Two-Parameter Pareto Distribution Under Double Type-Ⅱ Hybrid Censoring Scheme

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 5

References 21

Related Articles 1

Metrics

Comments

Recommended 0