A Modification of the Quasi Lindley Distribution

In this paper, we introduce a modification of the Quasi Lindley distribution which has various advantageous properties for the lifetime data. Several fundamental structural properties of the distribution are explored. Its density function can be left-skewed, symmetrical, and right-skewed shapes with various rages of tail-weights and dispersions. The failure rate function of the new distribution has the flexibility to be increasing, decreasing, constant, and bathtub shapes. A simulation study is done to examine the performance of maximum likelihood and moment estimation methods in its unknown parameter estimations based on the asymptotic theory. The potentiality of the new distribution is illustrated by means of applications to the simulated and three real-world data sets.


Introduction
The modeling of the lifetime data is a crucial one in many applied sciences, especially engineering, actuarial science, medicine, and others. Several lifetime distributions, for instance, the exponential, gamma, Weibull, log-normal distributions, and their modifications, have been used to model the lifetime data [1].
These distributions and their modifications have their own characteristics in terms of the shapes of the failure rate function, covering the tail-heaviness, horizontal symmetry, and dispersion. The tail-heaviness for a data set can be measured by the excess kurtosis (EK) which is defined as 3 τ − , where τ is the kurtosis of the data set. The 0 EK > is called a fatter tail (Leptokurtic) and 0 EK < is called a thinner tail (Platykurtic) distribution. Further, the symmetry and dispersion for a data set can be measured by skewness (SK), and Fano factor (FF) values, respectively, where the Fano factor value is the variance-to-mean ratio.
The modification of a lifetime distribution may be done by using the finite mixture model to handle the complexity by heterogeneity. The Lindley distribution (LD) is one of the finite mixture models under the Bayesian framework, and it was introduced by Lindley (1958) [2] having the density function:  (1) where θ is the shape parameter that controls the shape of the distribution, and y is the respective random variable. The density function of this distribution is based on a two-component mixture of two different continuous distributions namely exponential ( θ ) and gamma ( 2,θ ) distributions with the mixing pro- , where the p is defined by using the shape parameter (s) of the latent variable distribution. The LD has the increasing failure rate function while the exponential distribution has the constant failure rate function. In statistical literature, Ghitany et al. (2008) [3] showed that the Lindley distribution is more flexible and provides a better fit than the exponential distribution for lifetime data, especially its flexible mathematical format and failure rate criteria.   (2) where α is the shape parameter introduced from the latent variable distribution and θ is the scale parameter introduced from the mixing components. Equation (2) where , α δ , and η are the shape parameters introduced from the latent variable distribution, and θ and β are scale and location parameters, respectively, introduced from the mixing components. Equation (3)  ) and its sub-models.
Here, FPGLD ( , 0, , , θ β α δ η = ) means FPGLD by setting its location parameter 0 β = . This comparison study will be helpful to define the mixing proportion of MQLD that provides a better fit without having additional shape parameter(s) in the new distribution.
The paper is outlined as follows: in Section 2, we introduce the MQLD with its density and distribution functions. We present the statistical properties of MQLD including moments and moment generating functions, and quantile function in Section 3. In Section 4, we derive the reliability properties of MQLD.
The size-biased form of the MQLD is discussed in Section 5. Section 6 covers the unknown parameter estimation of MQLD. Finally, a simulation study is performed to verify the asymptotic property of unknown parameter estimation methods, and simulated and real-world data sets are used to illustrate its applicability over some other existing Lindley family distributions.

Formulation of the New Distribution
In this section, we introduce a finite mixture of two non-identical distributions called modified Quasi Lindley distribution with its probability density function (pdf) and cumulative distribution function (cdf).

Defining the Mixing Proportion p
For the comparison study, it is simulated 50 random samples of size, ) and its sub-models for given η and δ values in Table 2 are fitted to the simulated random samples. Table 2  The detailed study results could be provided upon request of reviewers.

Defining the pdf and cdf
Suppose Y be a non-negative random variable that is derived as a finite mixture of two non-identical distributions, exponential ( θ ), and gamma ( , δ θ ) with the mixing proportion, The first derivative of ( ) log f y for y is given by:  ) and its sub-models.  The corresponding cdf of MQLD is given by: where ( )

Statistical Properties
In this section, we provide some important statistical properties of MQLD such as r th moments about the origin and about the mean, moment related measures, moment generating and characteristic functions, and quantile function.

Moments and Related Measures
We may utilize the moments to study the characteristics of a distribution such as horizontal symmetry, dispersion, and tail-heaviness. The following proposition gives the r th moment about the origin: Proposition 1. The r th moment about the origin of the MQLD is given by: and 4 in Equation (6), the first four moments about the origin are derived as: respectively. Then, the r th -order moments about the mean can be obtained by using the relationship between moments about the mean and moments about the origin, i.e.

Moment Generating and Characteristic Function
Own characteristics of a probability distribution are directly associated with the moment generating function (mgf) and the characteristic function (cf). The Proof.

Quantile Function
We may use the quantile function to estimate the quantiles and simulate the random samples for a probability distribution. The quantile function can be derived by solv- The quantile function of MQLD is obtained as: Since Equation (9) is not a closed-form, we cannot estimate the quantiles and simulate the random variables from MQLD directly. However, these can be done by using numerical methods. Further, By substituting 0.25, 0.5 u = and 0.75 in Equation (9), the first three quartiles can be derived by solving the following equations, respectively.

Distribution of Order Statistics
The linear combinations of order statistics are used to estimate the unknown parameters for a distribution. Let 1 2 , , , n Y Y Y  be n independent random variables from MQLD and 1: respectively. By substituting (10) and (11), the pdf and cdf of : k n Y for MQLD are obtained as:

Reliability, Inequality and Entropy Measures
In this section, we derive and study some important reliability measures of MQLD, namely survival function/reliability function ( )

Survival and Hazard Rate Functions
The survival function and hazard rate function are crucial functions to specify a survival distribution. The survival function is the probability of surviving up to a point λ . Then, the survival function of MQLD is defined as: Note that, ( ) The instantaneous failure rate is described by the hazard rate function (hrf).
The hrf of MQLD is given by:  The reversed hazard rate function of MQLD is defined as: and the corresponding cumulative hazard rate function that represents the total number of failures over an interval of time is defined for MQLD as:

Mean Residual Life Function
The

Lorenz and Bonferroni Curves
The Lorenz and Bonferroni curves are used to measure the income inequality.
They are widely used in reliability, insurance, economises, and medicine. The

Renyi Entropy
The entropy measure is a measure of the variation of uncertainty for a distribu-

The Size-Biased of MQLD
The weighted distributions are used to record the observations with an unequal chance. The application of the weighted distributions in reliability, medical, and ecological sciences have studied by Patil et al. (1978) [14]. The weighted random variable w Y of MQLD is defined as:

Parameter Estimation
This section introduces the parameter estimation methods of MQLD by using the method of moment estimation, maximum likelihood estimation method, and weighted least square estimation method.

Method of Moment Estimation (MME)
The method of moment estimators of , θ α , and δ , abbreviated as MME MMÊ, θ α , and MMÊ δ can be derived by equating the raw-moments, say r µ′ , to the sample moments, say

Maximum Likelihood Estimation (MLE)
The MLE method is the most commonly employed due to its better asymptotic properties. Suppose The asymptotic confidence intervals for the parameters , θ α , and δ are derived by the asymptotic theory. The estimators are asymptotic three-variate normal with mean ( ) , , θ α δ and the observed information matrix: Therefore, the ( ) θ α , and MLÊ δ , respectively, and can be derived by diagonal elements of ( ) 1 , , I θ α δ − and 2 a z is the critical value at a level of significance.

Simulation Study
In this section, we examine the performance of the MME and MLE method in the unknown parameter estimation of MQLD with respect to the sample size n.

Performance of MME and MLE Methods
The simulation study is designed to examine the performance of  Observations from Table 3 and Table 4, the biases and MSEs decrease as n increases in both methods. Then, both methods verify the asymptotic property.
However, comparing between MME and MLE method for given combination of parameter values and different sample sizes, it is clear that the MLE method is better than the MME since its' ability to converge to the actual parameter value is stronger than the method of moment estimation. Further, we have noted that this ability is very strong for a large sample. Among the MLEs of unknown parameters, θ and δ are overestimated and α is underestimated for both combinations of parameters. Further, MLÊ θ has low biases and MSEs while MLÊ δ has high biases and MSEs.

Comparison Study among MQLD, QLD and LD
This comparison study is performed to show how the MQLD provides a better fit than QLD and LD for the various data sets that are simulated from MQLD.
Since the ranges of skewness, and kurtosis of QLD are, 2) Fit the MQLD, QLD, and LD to the 24 generated random samples.
3) Make the comparisons based on minimum 2 log L − values.
Here, the estimates of the unknown parameters for the distributions are derived by the MLE method. Tables 5-7 summarize 2 log L − values of MQLD, QLD, and LD for the generated random samples. Based on minimum 2 log L − value, the MQLD performs better than QLD, and LD in all given ranges of SK, EK, and FF.   Table 8.
The empirical histogram of the data sets and the fitted densities of MQLD, QLD, and LD are displayed in Figure 5. One can observe that the fitted density of MQLD gives a closer fit with the empirical distributions of the data sets. Table   9 lists the MLEs, SDs, 2 log L − , AICs, BICs, and K-S statistics with critical values for the fitted models to the data set 1, 2, and 3. It is noted that from Table 9, the MQLD provides the lowest values for the 2 log L − , AIC, and BIC among all fitted models. Then, it is clear from Table 9 and Figure 5 results that the MQLD provides a better fit than the QLD and LD.  Figure 5. Empirical histograms of the data sets with fitted densities of MQLD, QLD, TwPLD, and LD.

Conclusion
In this paper, we have introduced a new three-parameter Lindley family distri-