Parameter Estimation and Joint Confidence Regions for the Parameters of the Generalized Lindley Distribution

We deal with the problem of estimating the parameters of the generalized Lindley distribution. Besides the classical estimator, inverse moment and modified inverse estimators are proposed and their properties are investigated. A condition for the existence and uniqueness of the inverse moment and modified inverse estimators of the parameters is established. Monte Carlo simulations are conducted to compare the estimators’ performances. Two methods for constructing joint confidence regions for the two parameters are also proposed and their performances are discussed. A real example is presented to illustrate the proposedmethods.


Introduction
Lindley [1] originally introduced the Lindley distribution to illustrate a difference between fiducial distribution and posterior distribution.This distribution is becoming increasingly popular for modeling lifetime data and has a wide applicability in survival and reliability as closed forms for the survival and hazard functions and good flexibility of fit.Its density function is given by We denoted this by writing LD().The Lindley distribution is a mixture of an exponential distribution with scale  and a gamma distribution with shape 2 and scale , where the mixing proportion is  = /(1 + ).Ghitany et al. [2] provided a comprehensive treatment of the statistical properties of the Lindley distribution.Mazucheli and Achcar [3] used the Lindley distribution as a good alternative to analyze lifetime data within the competing risks approach as compared with the use of standard exponential or even the Weibull distribution commonly used in this area.Krishna and Kumar [4] considered the reliability estimation in Lindley distribution with progressively type II right censored sample.Al-Mutairi et al. [5] dealt with the estimation of the stress-strength parameter when the variables are independent Lindley random variables with different shape parameters.
Some researchers have proposed and studied new classes of distributions based on the Lindley distribution.See, for example, Sankaran [6], Ghitany et al. [7], Bakouch et al. [8], Shanker et al. [9], and Ghitany et al. [10].In this paper, we focus on the generalized Lindley distribution (GLD) introduced by Nadarajah et al. [11].It has the attractive feature of allowing for monotonically decreasing, monotonically increasing, and bath tub shaped hazard rate functions while not allowing for constant hazard rate functions.It has better hazard rate properties than the gamma, lognormal, and the Weibull distributions.
The cumulative distribution function and the probability density function are, respectively, given by (; , ) =  2 ( + 1)  −  + 1 (1 −  − ( +  + 1)  + 1 ) where  > 0 and  > 0 are two parameters.We denote this distribution as GLD(, ).When  = 1, the generalized Lindley distribution reduces to the one parameter Lindley distribution.Singh et al. [12] developed the Bayesian estimation for the generalized Lindley distribution under squared error and general entropy loss functions in case of complete sample of observations.Singh et al. [13] considered the generalized Lindley distribution and proposed the progressive type II censoring scheme which allows the removal of the live units from a life-test with beta-binomial probability law during the execution of the experiment.
Nadarajah et al. [11] considered the classical maximumlikelihood estimation of the parameters of a generalized Lindley distribution.The results showed that the bias is not satisfied especially for a small or even moderate sample size.As for the moment estimates, two nonlinear equations need to be solved simultaneously and the existence and uniqueness of the roots are not clear and guaranteed.
In this paper, we consider the problem of estimating the two parameters of the generalized Lindley distribution.We propose inverse moment and modified inverse moment estimators and study their properties.The conditions of the existence and uniqueness of the estimators are established.Monte Carlo simulations are used to compare the performances of the estimators.We also investigate the methods for constructing joint confidence regions for the two parameters and study their performances.
The rest of this paper is organized as follows.In Section 2, we briefly review the classical maximum-likelihood estimation of the parameters of the generalized Lindley distribution.In Section 3, the moment estimator is discussed.In Section 4, we propose two new methods of estimating the parameters and study their properties.Joint confidence regions for the two parameters are proposed in Section 5. Section 6 conducts simulations to assess the methods.Finally, in Section 7, a real example is presented to illustrate the proposed methods.

Maximum-Likelihood Estimation
In this section, we briefly review the MLEs of the parameters of GLD distribution.Let  1 ,  2 , . . .,   be a random sample from GLD(, ) with pdf and cdf as (3) and (2), respectively.The log-likelihood function is given by The score equations are thus as follows: From ( 6) we obtain the MLE of  as a function of : The MLE of  is the root of the following equation: Such nonlinear equation does not have closed form solution.We can apply numerical method such as Newton-Raphson method to compute .
Let  denote a GLD random variable.It follows that By equating the population moments with the sample moments, we obtain The method of moments estimators is the roots of the two equations.Similar to the MLEs, such nonlinear equations do not have closed form solutions.We can apply numerical method such as Newton-Raphson method to determine the roots.

Inverse Moment Estimation of Parameters
Unlike the regular method of moments, the idea of the inverse moment estimation (IME) is as follows: for a given random sample  1 , . . .,   from a distribution with unknown parameters, first transform the original sample to a quasisample  1 , . . .,   , where   contains the unknown parameters but its distribution does not depend on the unknown parameters; that is,   is a pivot variable,  = 1, . . ., .The population moments of the new sample do not depend on the unknown parameters while the sample moments do.Let the population moments of the quasisample equal the sample moments and solve for the unknown parameters.
Let  1 , . . .,   form a sample from GLD(, ) with pdf given in (3); it is known that (  ),  = 1, . . ., , follow the uniform distribution (0, 1), and thus − log (  ),  = 1, . . ., , follow standard exponential distribution Exp (1).By the method of inverse moment estimation, we let that is, Thus, the IME of  is obtained as a function of , which is identical to the MLE of .In the following, we determine the IME of .
Proof.The proof can be found by Arnold et al. [14].
Proof.The proof can be found by Wang [15].
In the following, we prove the existence and uniqueness of the root in (20) and ( 21).

Joint Confidence Regions for 𝜆 and 𝛼
Let  1 ,  2 , . . .,   form a sample from GLD(, ), and It is obvious that  and  are independent.Define We obtain that  1 and  2 are independent using the known bank-post office story in statistics.
Let   (V 1 , V 2 ) denote the percentile of  distribution with left-tail probability  and V 1 and V 2 degrees of freedom.Let  2  (V) denote the percentile of  2 distribution with left-tail probability  and V degrees of freedom.

Mathematical Problems in Engineering
By using the pivotal variables  1 and  2 , a joint confidence region for the two parameters  and  can be constructed as follows.
Theorem 7 (method 1).Let  1 ,  2 , . . .,   form a sample from (, ); then, based on the pivotal variables  1 and  2 , a 100(1 − )% joint confidence region for the two parameters (, ) is determined by the following inequalities: where   is the root of  for the equation ) and   is the root of  for the equation ] is a function of  and does not depend on .From Theorem 6, we have lim and   1 = (1/( − 1))(  /  )  > 0. Therefore, for any  > 0, equation  1 =  has a unique positive root of : ) . (33) On the other hand, by Lemma 3, we have 2 and  3 are also independent.By using the pivotal variables  2 and  3 , a joint confidence region for the two parameters  and  can be constructed as follows.
Theorem 8 (method 2).Let  1 ,  2 , . . .,   form a sample from (, ); then, based on the pivotal variables  2 and  3 , a 100(1 − )% joint confidence region for the two parameters (, ) is determined by the following inequalities: where  *  is the root of  for the equation is the root of  for the equation =1 log(  /  ) is a function of  and does not depend on .From Theorem 6, for any  > 0, equation  3 =  has a unique positive root of : ) . (36)

Comparison of the Four Estimation Methods.
In this section, we conduct simulations to compare the performances of the MIMEs, IMEs, MLEs, and MOMs mainly with respect to their biases and mean squared errors (MSEs), for various sample sizes and for various true parametric values.The random data  from the GLD(, ) distribution can be generated as follows: where  follows uniform distribution over [0, 1] and () giving the principal solution for  in  =   is pronounced as Lambert  function; see Jodrá [16].
We consider sample sizes  = 30, 40, 50, 60, 80, 100 and  = 2.0, 2.5, 3.0, 3.5, 4.0.We take  = 2 in all our computations.For each combination of sample size  and parameter , we generate a sample of size  from GLD( = 2, ) and estimate the parameters  and  by the MLE, MOM, IME, and MIME methods.The average values of α/ and λ/2 as well as the corresponding MSEs over 1000 replications are computed and reported.
Table 1 reports the average values of α/ and the corresponding MSE is reported within parenthesis.Figures 1(a Table 2 reports the average values of λ/ = λ/2 and the corresponding MSE is reported within parenthesis.Figures 2(a From Tables 1 and 2, it is observed that for the four methods the average relative biases and the average relative MSEs decrease as sample size  increases as expected.The asymptotic unbiasedness of all the estimators is verified.The average MSEs of α/ and λ/ = λ/2 depend on the parameter .For the four methods, the average relative MSEs of λ/2 decrease as  goes up.The average relative MSEs of α/ increase as  goes up.Considering only MSEs, we can observe that the estimation of 's is more accurate for smaller values while the estimation of 's is more accurate for larger values of .MOM, MLE, and IME overestimate both of the two parameters  and .MIME overestimates only .
As far as the biases and MSEs are concerned, it is clear that MIME works the best in all the cases considered for estimating the two parameters.Its performance is followed by IME, MLE, and MOM, especially for small sample sizes.The four methods are close for larger sample sizes.
Considering all the points, MIME is recommended for estimating both parameters of the GLD(, ) distribution.MOM is not suggested.

Comparison of the Two Joint Confidence
Regions.In Section 5, two methods to construct the confidence regions of the two parameters  and  are proposed.In this section, we conduct simulations to compare the two methods.First, we assess the precisions of the two methods of interval estimators for the parameter .We take sample sizes  = 30, 40, 50, 60, 80, 100 and  = 2.0, 2.5, 3.0, 3.5, 4.0.We take  = 2 in all our computations.For each combination of sample size  and parameter , we generate a sample of size  from GLD( = 2, ) and estimate the parameter  by the two proposed methods (32) and (35).
The mean widths as well as the coverage rates over 1000 replications are computed and reported.Here the coverage rate is defined as the rate of the confidence intervals that contain the true value  = 2 among these 1,000 confidence intervals.The results are reported in Table 3.
It is observed that the mean widths of the intervals decrease as sample sizes  increase as expected.The mean widths of the intervals decrease as the parameter  increases.The coverage rates of the two methods are close to the nominal level 0.95.Considering the mean widths, the interval estimate of  obtained in method 2 performs better than that obtained in method 1. Method 2 for constructing the interval estimate of  is recommended.
Next we consider the two joint confidence regions and the empirical coverage rates and expected areas.The results of the methods for constructing joint confidence regions for (, ) with confidence level  = 0.95 are reported in Table 4.
We can find that the mean areas of the joint regions decrease as sample sizes  increase as expected.The mean areas of the joint regions increase as the parameter  increases.The coverage rates of the two methods are close to the nominal level 0.95.Considering the mean areas, the joint region of (, ) obtained in method 2 performs better than that obtained in method 1. Method 2 is recommended.
(38) Nadarajah et al. [11] fit the data with generalized Lindley distribution and showed that it can be a better model than those based on the gamma, lognormal, and the Weibull distributions.The MLEs of the parameters are λMLE = 2.5395 and αMLE = 27.8766 with log-likelihood value −16.4044.The Kolmogorov-Smirnov distance and its corresponding  value In addition, based on method 1, the 95% joint confidence region for the parameters (, ) is given by the following inequalities: Based on method 2, the 95% joint confidence region for the parameters (, ) is given by the following inequalities: Considering the widths of , method 2 is suggested.

Conclusion
In this paper, we study the problem of estimating the two parameters of the generalized Lindley distribution introduced by Nadarajah et al. [11].We propose the inverse moment estimator and modified inverse moment estimator and study their statistical properties.The existence and uniqueness of inverse moment and modified inverse moment estimates of the parameters are proved.Monte Carlo simulations are used to compare their performances.We also investigate the methods for constructing joint confidence regions for the two parameters and study their performances.
), 2(b), 2(c), and 2(d) show the relative biases and the MSEs of the four estimators of  for sample sizes  = 40 and  = 80.Figures 2(e) and 2(f) show the relative biases and the MSEs of the four estimators of  for  = 3.0.The other cases are similar.

Figures 3
Figures 3(a) and 3(b) show the 95% joint confidence regions of (, ).Considering the widths of , method 2 is suggested.

Table 1 :
Average relative estimates and MSEs of .

Table 2 :
Average relative estimates and MSEs of .