Parameter Estimation of Lindley Distribution Under Type Ⅱ Censored Samples

LONG Bing

(School of Mathematics and Physics, Jingchu University of Technology, Jingmen 448000, China)

Parameter Estimation of Lindley Distribution Under Type Ⅱ Censored Samples

LONG Bing

(School of Mathematics and Physics, Jingchu University of Technology, Jingmen 448000, China)

Maximum likelihood estimation of the parameter from the Lindley distribution was discussed under Ⅱ censored data. The interval estimation and inverse moment estimation were given, and the random simulation method was used to analyze the parameter. An example was provided to find out the two kinds of point estimation and interval estimation about the parameter under different type Ⅱ censored samples.Comparison between the two point estimations was discussed.

Lindley distribution; Chi-quare distribution; interval estimation; inverse moment estimation; maximum likelihood estimation

Lindley distribution was proposed by Lindley in the literature[1-2] in 1958, which is an important distribution in reliability study. For some life data, the use of Lindley distribution model to fit the effect will be better. At present, there are a lot of statistical scholars who have discussed its properties, and obtained a lot of research results. The properties and applications of compound Lindley distribution were studied in the literatures [3-5]. The empirical Bayes one-sided test was discussed in the literature [6] in the case of independent identically distributed sample. The EB test function was constructed by using the recursive kernel estimation of density function. The optimality of the test function was proved, and the convergence rate was obtained. The empirical Bayes test function of the parameter from Lindley distribution was discussed based on NA random sample sequence in the literature [7]. The interval estimation and hypothesis testing of the parameter in Lindley distribution were studied in the full sample in the literature [8]. The failure model of incomplete data with Lindley distribution was discussed in the literature[9],and maximum likelihood estimation method was used to obtain point estimation and asymptotic confidence interval estimation of the model. An example was given to show that the Lindley distribution have better adaptability compared with exponential distribution and Weibull distribution.

In this paper, we will discuss the maximum likelihood estimation, interval estimation and inverse moment estimation of the parameter from Lindley distribution in Ⅱ censored samples, the parameter will be estimated by the method of stochastic simulation, and the average deviation will be calculated. Finally, an example will be given to illustrate the feasibility of the proposed method.

The probability density function of Lindley distribution is

(1)

Its distribution function is

(2)

with parameterθ>0.

1 Maximum likelihood estimation of the parameter

The data in life test for some data will usually be censored, leading to incomplete data, namely, censored way are type I censored, type Ⅱ censored and so on.

It is assumed that there arenproducts that are independent of each other and are subject to Lindley distribution, with type Ⅱ censored test. When observed withmfailure samples, the remainingn-msamples have been withdrawn from the test. The failure time of themsamples have been observed to meetX(1)≤X(2)≤…≤X(m). For the sake of convenience, we will omit the parenthesis of the subscript numbers. TheXirepresents the minimumiobservation value, which is the full sample case whenm=n.

According to the above test, the likelihood function is

(3)

Substituting (1) and (2) to (3) gives

Log likelihood function is

Obviouslyh1(θ)>0,h2(θ)>0. Whenθ→0+,h1(θ)>h2(θ)

So,h1(θ) is strictly monotone decreasing lower convex function on (0，+∞) and

Therefore,h2(θ) is also strictly monotone decreasing lower convex function on (0，+∞),andh1(θ)

(4)

2 Interval estimation and inverse moment estimation of the parameter

According to the distribution function of Lindley distribution, we can get the following lemma.

Proof：LetFX(x),FY(y) denote the distribution function of random variableXandY, respectively, then

The above expression is just the distribution function of standard exponential distribution, proving Lemma 1.

According to the above test, letX(1),X(2),…，X(m)be a type Ⅱ censored sample from Lindley distribution (2) with a capacity ofn. One can obtain from Lemma 1 that

is a type Ⅱ censored sample from standard exponentialy distribution (2) with a capacity ofn.

Set up

……

According to literature [10],W1,W2,…，Wmare independent and identically distributed and follow standard exponential distribution.

Proof: BecauseWj,j=1,2,…，mfollows standard exponential distribution, its characteristic function is

The above formula is the characteristic function of the Chi-square variable with 2mdegree of freedom. Lemma 2 is proved by the unique decision of characteristic function and the distribution of random variable.

It can be proved thatWj,j=1,2,…，mis a monotonically increasing function ofθ, soSmis a monotonically increasing function ofθ.

Theorem1LetX(1),X(2),…，X(m)be a type Ⅱ censored sample from Lindley distribution (2) with a capacity ofn, for any 0<α<1, under the confidence level of 1-α,the confidence interval ofθis

BecauseW1,W2,…，Wmare independent and identically distributed and follow standard exponential distribution, it can be used as a quasi sample. Thus, inverse moment estimation of the parameterθis determined by the following expression

(5)

3 Stochastic simulation

Setθ=1. A simple random sample is randomly generated with a capacity ofnfrom Lindley distribution (2). Differentmvalues are used to obtain type Ⅱ censored samples. Maximum likelihood estimation and inverse moment estimation of the parameterθcan be obtained by expressions (4) and (5). Repeat the above process 1 000 times to obtain the mean and relative deviation of the parameter estimations. The simulation results are shown in Table 1.

Tab.1 Results of stochastic simulation

The simulation results show that the deviation of inverse moment estimation is smaller than maximum likelihood estimation in small sample case, with the increase of sample size, the deviation of inverse moment estimation is larger than maximum likelihood estimation. In general, the mean of inverse moment estimation is larger than maximum likelihood estimation, therefore, the inverse moment estimation is more effective in small sample situations.

4 Example

Setθ=1, a simple random sample is generated with a capacity of 20 from Lindley distribution (2) as follows 0.111 8，0.291 5, 0.327 9, 0.394 8， 0.531 5, 0.628 4, 0.779 1, 0.848 4, 0.939 5, 1.332 3, 1.475 2, 1.719 0, 1.862 5, 1.900 4, 2.061 6, 2.098 8, 3.130 7, 3.303 7, 4.295 6, 6.593 2. Differentmvalues are used to obtain different type Ⅱ censored samples, and carried on the statistical analysis. The results are listed in Table 2.

Tab.2 Results of statistical analysis (α=0.1)

We can see from Table 2 that when sample sizenis fixed, the interval length increases with the decrease ofm. Inverse moment estimation is closer to the true value than maximum likelihood estimation.

[1] LINDLEY D. Introduction to probability and statistics from a Bayesian viewpoint, part II: inference[M]. Cambridge: Cambridge University Press, 1965.

[2] LINDLEY D. Fiducial distributions and Bayes’ theorem[J].J Royal Stat Soc, 1958,20(1):102-107.

[3] GHITANY M E, ATIEH B, NADARAJAH S. Lindley distribution and its application[J]. Math Comput Simula, 2008,78(4):493-506.

[4] GHITANY M E, ALMDK, NADARAJAH S. Zero-truncated Poisson-Lindley distribution and its application[J]. Math Comput Simula, 2008,79(3):279-287.

[5] ZAMANI H, ISMAIL N. Negative binomial-Lindley distribution and its application[J].J Math Stat, 2010,6(1):4-9.

[6] 杜偉娟，彭家龍，李體政.Lindley分布參數(shù)的經(jīng)驗(yàn)Bayes檢驗(yàn)的收斂速度[J].統(tǒng)計(jì)與決策，2012,21:23-26.

[7] 范梓淼，周菊玲. NA 樣本下Lindley 分布參數(shù)的經(jīng)驗(yàn)Bayes 檢驗(yàn)[J].貴州大學(xué)學(xué)報(bào)(自然科學(xué)版)，2016,34(2):68-70.

[8] 龍 兵. Lindley分布中參數(shù)的區(qū)間估計(jì)和假設(shè)檢驗(yàn)[J].廣西民族大學(xué)學(xué)報(bào)(自然科學(xué)版)，2014,20(1):59-62.

[9] 黃文平，周經(jīng)倫，寧菊紅,等.基于競爭失效數(shù)據(jù)的Lindley分布參數(shù)估計(jì)[J].系統(tǒng)工程與電子技術(shù)，2016,38(2):464-469.

[10] LAWLESS J F. Statistical models and methods for lifetime data[M].New York: Wiley, 2003.

2017-03-31

國家自然科學(xué)基金資助項(xiàng)目 (61374080)*通訊作者,E-mail：qh-longbing@163.com

O212.2

A

1000-2537(2017)06-0071-05

Ⅱ型刪失下Lindley分布的參數(shù)估計(jì)

龍 兵*

(荊楚理工學(xué)院數(shù)理學(xué)院，中國 荊門 448000)

在Ⅱ型刪失數(shù)據(jù)下，討論了Lindley分布參數(shù)的最大似然估計(jì).給出了參數(shù)的區(qū)間估計(jì)和逆矩估計(jì)，運(yùn)用隨機(jī)模擬的方法對參數(shù)進(jìn)行了統(tǒng)計(jì)分析.通過一個(gè)例子求出了在不同Ⅱ型刪失樣本下參數(shù)的兩種點(diǎn)估計(jì)及區(qū)間估計(jì)，并進(jìn)行了比較.

Lindley分布；χ2分布；區(qū)間估計(jì)；逆矩估計(jì)；最大似然估計(jì)

10.7612/j.issn.1000-2537.2017.06.012

(編輯 HWJ)