A powerful goodness-of-fit test for Lindley distribution with application to real data

The Lindley distribution may serve as a useful reliability model. Applications of this distribution are presented in statistical literature. In this article, a powerful goodness of fit test for the Lindley distribution is proposed. In order to compute the proposed test statistic, we use the maximum likelihood estimate (MLE) suggested by Ghitany et al. (2008), which is simple explicit estimator. By Monte Carlo simulation, critical points of the proposed test statistic for different sample sizes are obtained. Power values of the proposed test are compared with the competing tests against various alternatives via simulations. Finally, two real data are presented and analyzed.

and Alizadeh Noughabi and Balakrishnan (2015). The main contribution of the paper can express as follows. In this paper, we propose a goodness of fit test statistic for the Lindley distribution based on an estimate of Kullback-Leibler divergence. The properties of the proposed test and critical points are presented. We show through extensive simulation studies that the proposed goodness-of-fit test is more powerful, or at least as good as the classical EDF-tests for different choices of sample sizes and alternatives. We also investigate the behavior of the proposed test for the Lindley model with real data. In Section 2, a summary of the Lindley distribution is presented and then we introduce a goodness of fit test statistic based on the Kullback-Leibler divergence for the Lindley distribution. In Section 3, the critical values of the test statistic are obtained by Monte Carlo simulations. Then power values of the proposed test are computed and then compared with the competing tests. All simulations were carried out by using R 4.0.1 and with 100,000 replications. Section 4 contains applications of the suggested test in real examples.

The Lindley distribution and test Statistic
In this section, we express some properties of the Lindley distribution and then construct a goodness of fit test statistic for this distribution.

The Lindley Distribution
If the density function of the random variable X be as follows, then we say that X has a Lindley distribution.  Lindley (1958) in the context of Bayesian statistics, as a counter example of fiducial statistics. The cumulative distribution function of the Lindley distribution is as In the literature of survival analysis and reliability theory, the exponential distribution is widely used as a model of lifetime data. However, the exponential distribution only provides a reasonable fit for modeling phenomenon with constant failure rates. Distributions like gamma, Weibull and lognormal have become suitable alternatives to the exponential distribution in many practical situations. Ghitany et al. (2008) found that the Lindley distribution can be a better model than one based on the exponential distribution. The Lindley distribution belongs to an exponential family and it can be written as a mixture of an exponential with parameter  and a gamma distribution with parameters (2, ) Shanker et al. (2015) discussed a comparative study of Lindley and exponential distributions for modelling various lifetime data sets from biomedical science and engineering, and concluded that there are lifetime data where exponential distribution gives better fit than Lindley distribution and in majority of data sets Lindley distribution gives better fit than exponential distribution.
For computing the suggested test statistic, we need to estimate the unknown parameter  . We apply the maximum likelihood estimate (MLE) approach to estimate the unknown parameter.
A powerful goodness-of-fit test for Lindley distribution with application to real data 763 Suppose 1 ,..., n XX is a random sample from the Lindley distribution, the estimator for both maximum likelihood estimate (MLE) and method of moments estimate of the parameter  is Ghitany et al. (2008) showed that the estimator  of  is positively biased: ( ) 0 E  −   , and it is consistent and asymptotically normal ( ) .

The Proposed Goodness-of-Fit Test
Assuming that 1 ,..., n XX is the sample from a distribution F , we wish to assess whether the unknown () Fx can be satisfactorily approximated by a Lindley model 0 () Fx.
The Kullback-Leibler (KL) discrimination has been widely studied in the literature as a central index for measuring quantitative similarity between two probability distributions. The KL discrimination of Recently, Alizadeh Noughabi (2019) proposed a new estimate of (1) and then constructed a test statistic for testing the validity of a model. His test statistic is Here,  is a model parameter which is usually unknown, and  is a reasonable equivariant estimate of  .
We reject the null hypothesis for large values of (3). According to Alizadeh Noughabi (2019), the test statistic is nonnegative, i.e., 0 T  , and also the test based on T is consistent.

Critical points and power comparison
Because deriving the exact distribution of the test statistic is complicated, we obtain the critical values of the test statistic by Monte Carlo simulations. These values for different sample sizes are presented in Table 1.
It should be mentioned that the empirical percentiles given in Table 1 provides an excellent type I error control.   In the above test statistics, 0 (x) F is the cumulative distribution function of the Lindley distribution and  is the maximum likelihood estimate of the parameter  , which are presented in equations (4) and (5), respectively. It is obvious that for large values of the above test statistics the null hypothesis 0 H will be rejected.     A have a good performance and therefore can be used in practice. Finally, we summarized the results in Table 4.
This table presents the best test in terms of power against different alternatives.

Conclusions
In this paper, we have proposed a powerful goodness-of-fit tests for the Lindley distribution and have shown that the suggested test has a good performance. Through Monte Carlo simulations, we have carried out an extensive power study on the suggested test with the competing tests. It is shown that our test outperforms in most cases all other tests. Finally, we have used two real data sets and have illustrated how the proposed test can be applied to test the goodnessof-fit for the Lindley distribution when a random sample is available.