Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring

This paper considers the problem of estimating the parameters of Poisson-Exponential (PE) distribution under progressive type-I interval censoring scheme. PE is a two-parameter lifetime distribution having an increasing hazard function. It has been applied in complementary risks problems in latent risks, that is in scenarios where maximum lifetime values are observed but information concerning factors accounting for component failure is unavailable. Under progressive type-I interval censoring, observations are known within two consecutively pre-arranged times and items would be withdrawn at pre-scheduled time points. This scheme is most suitable in those cases where continuous examination is impossible. Maximum likelihood estimates of Poisson-Exponential parameters are obtained via Expectation-Maximization (EM) algorithm. The EM algorithm is preferred as it has been confirmed to be a more superior tool when dealing with incomplete data sets having missing values, or models having truncated distributions. Asymptotic properties of the estimates are studied through simulation and compared based on bias and the mean squared error under different censoring schemes and parameter values. It is concluded that for an increasing sample size, the estimated values of the parameters tend to the true value. Among the four censoring schemes considered, the third scheme provides the most precise and accurate results followed by fourth scheme, first scheme and finally the second scheme.


Introduction
In reliability and life testing studies exponential distribution has proved to be a distribution with a simple, elegant and closed form of solution, Tomazella [1]. However, its usefulness is limited based on the fact that it has a constant hazard function. In order to overcome this drawback, different authors have come up with new lifetime distributions based on modification of Exponential distribution. Gupta and Kundu [2] suggested a Generalized Exponential distribution (GED) that fit the data with decreasing and increasing hazard function. Modifying an Exponential distribution to a distribution with a decreasing hazard function was done by Kus [3]. The distribution proposed by Kus [3] was further modified by the inclusion of a power parameter by Barreto and Cribari [4].
Poisson-Exponential (PE) distribution is a two-parameter lifetime distribution which was first introduced by Cancho et al. [5]. Its failure rate increases with time. The distribution is widely applicable in complementary risk (CR) studies. Louzada-Neto et al. [6] studied statistical properties of PE distribution and discussed Bayes estimators based on Squared Error Loss Function (SELF). Singh et al. [7] obtained the Maximum Likelihood Estimators and Bayes estimators of the parameters of the PE distribution under symmetric and asymmetric loss function and compared the proposed estimators in terms of their risks with the Maximum Likelihood Estimators. Rodrigues et al. [8] considered different estimation methods for parameters of PE distribution. Gitahi et al. [9] obtained the Maximum Likelihood Estimators of the parameters of PE distribution Distribution Under Progressive Type I Interval Censoring based on progressively type II censoring via the EM algorithm. Belaghi et al. [10] considered prediction and estimation of lifetime data following PE distribution under type II censoring.
In this study, we assume that lifetimes have PE distribution. This distribution has been applied in complementary risk problems in latent risks, that is in scenarios where maximum lifetime values are observed but information concerning factors accounting for component failure is unavailable.
A random variable X is said to have a Poisson-Exponential distribution if its probability density function (PDF) and cumulative distribution function (CDF) are respectively given by Where 0 θ > and 0 λ > are respectively, the shape and scale parameters of the distribution. Louzada-Neto et al. [6] noted that the parameters λ and θ can be directly interpreted in terms of CR. That is; λ represents lifetime failure rate while θ denotes the mean number of CR.
In lifetime analysis, censoring occurs when lifetime of an item is not observed. Various types of censoring exist of which Type I and type II censoring schemes are the most common. The time T of termination of the experiment is pre-arranged in type I censoring. On the other hand, the experiment continues until a pre-arranged number of failures occur in type II censoring. However, the two schemes do not permit removal of experimental units at any other point other than the final termination point of the experiment. In many practical situations, there is need for the removal of test items at different points prior to the termination of the experiment. Progressive censoring schemes allow the removal of experimental units at different time points other than the termination point of the experiment as discussed in Cohen [11]. For detailed discussion on progressive type I and type II censoring schemes, the reader may refer to Balakrishnan [12] and Balakrishnan and Cramer [13].
Aggarwala [14] proposed progressive type I (PTI) interval censoring and provided the statistical inference for the exponential distribution. Under PTI interval censoring, observations are known within two consecutively pre-scheduled times and items would be withdrawn at pre-scheduled time points. Ng and Wang [15] dealt with the estimation of Weibull distribution parameters basing on the PTI interval-censored sample. Cheng et al. [16] introduced a new algorithm for maximum likelihood estimation under PTI interval-censored data. Chen and Lio [17] estimated parameters of Generalized exponential distribution under PTI interval censoring. Lio et al. [18] considered estimation of parameters of Generalized Rayleigh distribution based on progressively type I interval-censored data. Lin and Lio [19] estimated the parameters of Weibull and Generalized exponential distribution under PTI interval censoring by Bayesian method. Teimouri and Gupta [20] considered estimation methods for the Gompertz-makeham distribution under PTI interval censoring. Recently, Singh and Tripathi [21] estimated Inverse Weibull distribution parameters under PTI interval censoring in classical and Bayesian frameworks.
In this paper, we consider Maximum Likelihood Estimation of parameters of PE distribution based on the PTI interval censoring scheme. We propose to use EM algorithm to compute the MLEs. We also assess the precision and accuracy of the MLEs of the parameters of PE distribution under different censoring schemes and parameter values using simulation studies.
The rest of this paper is organized as follows: In section 2, we briefly describe progressive type I interval censoring scheme. In addition, we obtain Maximum likelihood estimates of PE distribution based on PTI interval censoring via EM algorithm. In section 3, we conduct simulation study. Finally, the conclusions are given in section 4.

Progressive Type I Interval Censoring Scheme
On a lifetime experiment, let n units be placed on a life testing simultaneously at time 0 0 t = and under inspection at Hence, the PTI interval-censored sample is

Maximum Likelihood Estimation Based on Progressive Type I Interval Censoring Scheme
, the likelihood function is derived as follows, according to Aggarwala [14].
The likelihood function (3) reduces to the likelihood function for the conventional type I censored case if By substituting (2) in (3) and simplifying, we obtain ( ) Obtaining log-likelihood of (4) and simplifying, yields ( ) Differentiating (5) partially with respect to θ and λ and equating to zero, we obtain the following normal equations 1 1 Clearly, normal equations (6) and (7) cannot yield solution to θ and λ in a closed-form. We therefore introduce the Expectation-Maximization (EM) algorithm to obtain the MLEs of θ and λ .

EM Algorithm
The EM Algorithm was introduced by Dempster et al. [22] to deal with incomplete data problems. PTI interval censoring can be considered as an incomplete data problem and therefore EM Algorithm is used as the most suitable method in obtaining the MLEs of the unknown parameters. EM is an iterative procedure whereby each iteration consists of two steps that is the Expectation step (E-step) and the Maximization step (M-step ; ; Introducing logs on (8) Simplification of equation (9) yields The E-step requires the construction of pseudo-log-likelihood function. This is achieved by computing the conditional expectations and then replacing them in the log-likelihood function (10) Replacing the conditional expectations 1i (10) completes the E-step. We get the pseudo-log-likelihood function as After substituting the conditional expectations, partial derivatives of (12) with respect to parameters are derived in order to maximize the pseudo-log-likelihood function as follows.

( )
and ( ) Suppose that The M-step requires solutions to equations (15) and (16)

Simulation Study
Simulation study was conducted to investigate the behavior of the proposed MLEs of PE distribution parameters under PTI interval censoring scheme via EM algorithm on simulated data. The simulation was conducted in R language, a software package that was designed by Ihaka and Gentleman [23].

Simulation Algorithm
According to the algorithm proposed in Aggarwala [14], a progressively type I interval-censored data { } , , Where floor () returns the greatest integer equal to or less than the argument.
Clearly, if becomes a simulated sample from the conventional type I interval censoring. The above algorithm is an improvement from the one suggested by Kemp and Kemp [24].
To investigate the behavior of the MLEs, we consider the schemes below, similar to schemes used in Ng and Wang [15], Chen and Lio [17] and Lio et al. [18]. In this paper, convergence is assumed to occur when the absolute difference between successive estimates is less than 0.0001.

A summary of results from Tables 1-4 is provided below:
i. The MSE and bias of the estimates decrease as the sample size increase from n=20 to n=200 for each censoring scheme. These imply that the estimates become more precise and accurate as the sample size increases. ii. Among the four censoring schemes, the third scheme  p . Similar performance among the four schemes is observed when the sample size is increased from n=20 to n=200. The results of the performance comparisons among these four censoring schemes are similar to the results obtained in Aggarwala [14], Ng and Wang [15], Chen and Lio [17] and Lio et al. [18].
These phenomena are expected since the third censoring scheme ( )  p . Intuitively, these are also consistent with the statistical theory that the larger the "sample size" is the more accurate the parameter estimate is.
iii. As for the performance among the assumed true parameters, it is observed that when the values are varied from  Table 1 and Table 4.

Conclusion
The study has addressed the problem of the Maximum Likelihood Estimation of parameters for Poisson-Exponential distribution based on progressive type I interval-censored data. The maximum likelihood estimates were obtained using the EM algorithm.
A comparison of the MLEs obtained was made by simulation under four different censoring schemes and various parameter values. It was observed that: i. For an increasing sample size, the estimated values of the parameters became closer to the true values. ii. Among the four censoring schemes considered, the third scheme provided the most accurate and precise results then schemes four, one and lastly scheme two.