The Inverse Weighted Lindley Distribution: Properties, Estimation and an Application on a Failure Time Data

11/26/2017
by   Pedro L. Ramos, et al.
Universidade de São Paulo
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In this paper a new distribution is proposed. This new model provides more flexibility to modeling data with upside-down bathtub hazard rate function. A significant account of mathematical properties of the new distribution is presented. The maximum likelihood estimators for the parameters in the presence of complete and censored data are presented. Two corrective approaches are considered to derive modified estimators that are bias-free to second order. A numerical simulation is carried out to examine the efficiency of the bias correction. Finally, an application using a real data set is presented in order to illustrate our proposed distribution.

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1 Introduction

In recent years, several new distributions have been introduced in literature for describing real problems. An important distribution was presented by Lindley [16]

in the context of fiducial statistics and Bayes’ theorem. Ghitany et al.

[11]

argued that the Lindley distribution provides flexible mathematical properties and outlined that in many cases this distribution outperforms the exponential distribution. Since then, new generalizations of Lindley distribution have been proposed such as the generalized Lindley

[26], extended Lindley [3], and Power Lindley [9] distribution.

The study of weight distributions provide new comprehension of standard distributions and contributes in adding more flexibility for fitting data [18]. Ghitany et al. [10]

presented a two-parameter weighted Lindley (WL) distribution which has bathtub and increasing hazard rate. The WL distribution has probability density function (PDF) given by

(1)

for all , and where is the gamma function. Mazucheli et al. [17] compared the finite sample properties of the parameters of the WL distribution numerical simulations using four methods. Wang and Wang [25] presented bias-corrected MLEs and argued that the proposed estimators are strongly recommended over other estimators without bias-correction. Ali [2]

considered a Bayesian approach and derived several informative and noninformative priors under different loss functions. Ramos and Louzada

[20] introduced three parameters generalized weighted Lindley distribution.

In this study, a new two-parameter distribution with upside-down bathtub hazard rate is proposed, hereafter, inverse weighted Lindley (IWL) distribution. This new model can be rewritten as the inverse of the WL distribution. A significant account of mathematical properties for the IWD distribution is presented such as moments, survival properties and entropy functions. The maximum likelihood estimators of the parameters and its asymptotic properties are obtained. Further, two corrective approaches are discussed to derive modified MLEs that are bias-free to second order. The first has an analytical expression derived by Cox and Snell (

12) and the second is based on the bootstrap resampling method (see Efron [8] for more details), which can be used to reduce bias. Similar corrective approaches has been considered by many authors for other distributions, e.g., Cordeiro et al. [5], Lemonte [15], Teimouri and Nadarajah [24], Giles et al. [12], Ramos et al. [19], Schwartz et al. [22] and Reath et al. [21]. In addition, the MLEs in the presence of randomly censored data is presented. Approximated bias-corrected MLEs for censored data are also discussed. A numerical simulation is performed to examine the effect of the bias corrections in the MLEs for complete and censored data.

The new distribution is a useful generalization of the inverse Lindley distribution [23] and can be represented by a two-component mixture model. Mixture models play an important role in statistics for describing heterogeneity (see, Aalen [1]). Therefore, the IWL distribution can be used to describe data sets in the presence of heterogeneity. For instance, we can be interested in describing the lifetime of components that are composed of new and repaired products, however, only the failure time is observed and the groups are latent variables. In this case, the proposed distribution, as a mixture distribution, can express the heterogeneity in the data. In reliability, this model may be used to describe the lifetime of components associated with a high failure rate after short repair time. In studies involving the lifetime of patients this model can be useful to describe the course of a disease, where their mortality rate reaches a peak and then declines as the time increase, i.e., problems where their hazard function has upside-down bathtub shape.

In order to illustrate our proposed methodology, we considered a real data set related to failure time of devices of an airline company. Such study is important in order to prevent customer dissatisfaction and customer attrition, and consequently to avoid customer loss. In this context, the choice of the distribution that fits better this data is fundamental for the company reduces its costs. We showed that the inverse weighted Lindley distribution fits better than other well-known distributions for this data set.

The paper is organized as follows. Section 2 introduces the inverse weighted Lindley distribution. Section 3 presents the properties of the IWL distribution such as moments, survival properties and entropy. Section 4 discusses the inferential procedure based on MLEs for complete and censored data. A bias correction approach is also presented for complete and censored data. Section 5 describes two corrective approaches to reduce the bias in the MLEs for complete and censored data. Section 6 presents a simulation study to verify the performance of the proposed estimators. Section 7 illustrates the relevance of our proposed methodology in a real lifetime data. Section 8 summarizes the present study.

2 Inverse Weighted Lindley distribution

A non-negative random variable T follows the IWL distribution with parameters

and if its PDF is given by

(2)

Note that if , the IWL distribution reduces to the inverse Lindley distribution [23]. The IWL distribution can be expressed as a two-component mixture

where and , for , i.e.,

is Inverse Gamma distribution, given by

Therefore, the IWL distribution is a mixture distribution and can express the heterogeneity in the data.

Proposition 2.1.

Let then follows a weighted Lindley distribution [10].

Proof.

Define the transformation then the resulting transformation is

Figure 1 gives examples from the shapes of the density function for different values of and .

Figure 1: Density function shapes for IWL distribution and considering different values of and .

The cumulative distribution function from the IWL distribution is given by

where is the upper incomplete gamma.

3 Properties of IWL Distribution

In this section, we provide a significant account of mathematical properties of the new distribution.

3.1 Moments

Moments play an important role in statistics. They can be used in many applications, for instance the first moment of the PDF is the well know mean, while the second moment is used to obtain the variance, skewness and kurtosis are also obtained from the moments. In the following, we will derive the moments for the IWL distribution.

Proposition 3.1.

For the random variable with distribution, the r-th moment is given by

(3)
Proof.

Note that if distribution then the r-th moment from the random variable is given by

Since the IWL distribution can be expressed as a two-component mixture, we have

Proposition 3.2.

The r-th central moment for the random variable is given by

(4)
Proof.

The result follows directly from the proposition 3.1.∎

Proposition 3.3.

A random variable with distribution, has the mean and variance given by

Proof.

From (3) and considering , it follows that . The second result follows from (4) considering and with some algebraic operation the proof is completed. ∎

3.2 Survival Properties

Survival analysis has become a popular branch of statistics with wide range of applications. Although many functions related to survival analysis can be derived for this model, in this section we will present the most common functions. The survival function of IWL distribution representing the probability of an observation does not fail until a specified time is given by

where is the lower incomplete gamma function. The hazard function of is given by

(5)

This model has upside-down bathtub hazard rate. The following Lemma is useful to prove such result.

Lemma 3.4.

Glaser [13]

: Let T be a non-negative continuous random variable with twice differentiable PDF

, hazard rate function and . Then if has an upside-down bathtub shape, has an upside-down bathtub shape.

Theorem 3.5.

The hazard function (5) is upside-down bathtub for all and .

Proof.

For IWL distribution we have

it follows that

The study of the behaviour of is not simple. However using the WolframAlpha software, we can check that for all and , is increasing in and decreasing in , i.e., at , where is a very large function computed to the WolframAlpha (available upon request). Therefore, and consequently has upside-down bathtub shape. ∎

This properties make the IWL distribution an useful model for reliability data. Figure 2 gives examples of different shapes for the hazard function.

Figure 2: Hazard function shapes for IWL distribution and considering different values of and .
Proposition 3.6.

The mean residual life function of the distribution is given by

Proof.

Note that, for the Inverse Gamma distribution we have that

Using the following relationship

and after some algebraic manipulations, the proof is completed. ∎

3.3 Entropy

In information theory, entropy has played a central role as a measure of the uncertainty associated with a random variable. Shannon’s entropy is one of the most important metrics in information theory. The Shannon’s Entropy from IWL distribution is given by solving the following equation

(6)
Proposition 3.7.

A random variable with distribution, has the Shannon’s Entropy given by

where .

Proof.

From the equation (6) we have

Since

Then

4 Inference

In this section, we present the maximum likelihood estimator of the parameters and of the IWL distribution. Additionally, MLEs considering randomly censored data are also discussed.

4.1 Maximum Likelihood Estimation

Among the statistical inference methods, the maximum likelihood method is widely used due to its better asymptotic properties. Under the maximum likelihood method, the estimators are obtained from maximizing the likelihood function. Let be a random sample such that . In this case, the likelihood function from (2) is given by

The log-likelihood function is given by

(7)

From the expressions , , we get the likelihood equations

where is the digamma function. After some algebraic manipulation the solution of is given by

where and can be obtained solving the nonlinear system

(8)

These results are a simple modification of the results obtained for Ghitany et al. [11]

for the WL distribution. Under mild conditions the ML estimates are asymptotically normal distributed with a bivariate normal distribution given by

where the elements of the Fisher information matrix I are given by

and is the trigamma function. An interesting property of the IWL distribution is that the observed matrix information is equal to the expected information matrix.

4.2 Random Censoring

In survival analysis and industrial lifetime testing, random censoring schemes have been received special attention. Suppose that the th individual has a lifetime and a censoring time , moreover the random censoring times s are independent of s and their distribution does not depend on the parameters, then the data set is , where and . This type of censoring have as special case the type I and II censoring mechanism. The likelihood function for is given by

Let be a random sample of IWL distribution, the likelihood function considering data with random censoring is given by

(9)

The logarithm of the likelihood function (9) is given by

From and , the likelihood equations are given as follows

(10)
(11)

where can be computed numerically. Numerical methods are required in order to find the solution of these non-linear equations.

5 Bias correction for the maximum likelihood estimators

In this section, we discuss modified MLEs based on two corrective approaches that are bias-free to second order. Firstly a corrective analytical approach is presented than the bootstrap resampling method is presented.

5.1 A corrective approach

Consider the likelihood function with a

-dimensional vector of parameters

. Thus, the joint cumulants of the derivatives of can be written by

Consequently, the derivatives of such cumulants are given by

The bias of studied by Cox and Snell [7] for independent sample without necessarily be identically distributed can be written by

(12)

where is the -th element of the inverse of Fisher’s information matrix of , . Cordeiro and Klein [6] proved that even if the data are dependent the expression (12) can be re-written as

(13)

Let and define the matrix with , for . Thus, the expression for the bias of can be expressed as

(14)

A bias corrected MLE for is obtained as

(15)

where is the MLE of the parameter , and . The bias of is unbiased . For the IWL distribution the higher-order derivatives can be easily obtained since they do not involve , thus, we have

where . The matrix is given by

To obtain the matrix of (14), we present the elements of

and the elements of are

Thus, the matrix is expressed by

Finally, the bias-corrected maximum likelihood estimators are given by

(16)

where and . It is important to point out that, since the higher-order do not involve , they are the same of the WL distribution [25].

A bias corrected approach can be considered for censored data. Although the Fisher information matrix related to the MLEs (9) does not present closed-form expressions, we can consider the bias corrected presented in (5.1). In this case, approximated bias-corrected maximum likelihood estimates (ACMLE) are archived by

where , and and are the solutions of and . However, the bias of

is not an unbiased estimator with

.

5.2 Bootstrap resampling method

In what follows we consider the bootstrap resampling method proposed by Efron [8] to reduce the bias of the MLEs. Such method consists in generating pseudo-samples from the original sample to estimate the bias of the MLEs. Thus, the bias-corrected MLEs is given by subtraction of the estimated bias with the original MLEs.

Let be a sample with observations randomly selected from the random variable in which has the distribution function . Thus, let the parameter be a function of given by . Finally, let be an estimator of based on , i.e., . The pseudo-samples is obtained from the original sample through resampling with replacement. The bootstrap replicates of is calculated, where and the empirical cdf (ecdf) of is used to estimate (cdf of ). Let be the bias of the estimator given by

Note that the subscript of the expectation indicates that is taken with respect to . The bootstrap estimators of the bias were obtained by replacing with , where generated the original sample. Therefore, the bootstrap bias estimate is given by

If we have bootstrap samples which are generated independently from the original sample and the respective bootstrap estimates are calculated, then it is achievable to determine the bootstrap expectations approximately by

Therefore, the bootstrap bias estimate based on replications of is , which results in the bias corrected estimators obtained through by bootstrap resampling method that is given by

In our case, we have denoted by .

6 Simulation Analysis

In this section a simulation study is presented to compare the efficiency of the maximum likelihood method and the bias correction approaches in the presence of complete and censored data. The proposed comparisons are performed by computing the mean relative errors (MRE) and the relative mean square errors (RMSE) given by

where is the number of estimates obtained through the MLE, CMLE and the bootstrap approach. The

coverage probability of the asymptotic confidence intervals are also evaluated. Considering this approach, we expected that the most efficient estimation metho