Location Based Generalized Akash Distribution: Properties and Applications

In recent years, several statistical finite mixture models have been proposed to model the lifetime data with heterogeneity. The Lindley distribution has been highlighted by many authors for these types of lifetime data analysis. This paper introduces a new Lindley family distribution called location-based generalized Akash distribution (NGAD) with monotonic increasing and bathtub failure rates. The density function of NGAD is flexible to cover the left-skewed, right-skewed and symmetrical shapes with different tail-weights. Its fundamen-tal structural properties and its ability to provide a suitable statistical model for various types of data sets are studied. The maximum likelihood (ML) method is used to estimate its unknown parameters and the performance of ML estimates are examined by a simulation study. Finally, several real-data sets with different characteristics are used to illustrate its flexibility. It is observed that NGAD provides a better fit than some other existing modified Lindley distributions.


Introduction
Several standard base-line distributions are available to model the lifetime data and the failure rate function is considered as the most crucial factor for these models. Examples of such distributions are exponential, gamma, Weibull, and log-normal distributions which have different capabilities to describe the shapes of the failure rate function. While the exponential distribution has a constant fail-R. Tharshan, P. Wijekoon ure rate, the gamma and Weibull distributions have both increasing and decreasing failure rates [1]. Researchers in the recent past have an increased interest in modifications of the above-mentioned standard distributions to handle a complicated task for the modeling of lifetime data. The finite mixture models that consist of a weighted sum of the standard or modifications of standard distributions are popular to handle the complexity by heterogeneity in a lifetime data analysis.
Each finite mixture failure rate may have various characteristics, such as monotonically increasing or decreasing, non-monotonic, constant, and bathtub shapes, and they may cover various tail-heaviness of a data set. We can measure the tailheaviness of a data set by the excess kurtosis (EK) that is defined as 3 γ − , where γ is the kurtosis of the data set. The  (1) where θ is the shape parameter, and λ is the respective random variable. Equation (1) presents two-component mixture of exponential ( θ ) and gamma ( 2,θ ) distributions with the mixing proportion 1 p θ θ = + . The statistical properties of LD have discussed by Ghitany, Atieh, and Nadarajah (2008) and showed that the Lindley distribution is more flexible and provides a better fit than the exponential distribution for lifetime data [8].
A considerable number of modifications of LD have been proposed by researchers with actual mixing components of LD based on the two-component mixture of exponential ( θ ) and gamma ( 2,θ ) with different mixing proportions. Some notables are mentioned as, Shanker and Sharma (2013) proposed a two-parameter Lindley distribution [9]; Shanker and Mishra (2013) obtained the Quasi Lindley distribution [10];  introduced the Shanker distribution [11]; and Shanker, Shukla, and Tekie (2017) introduced a three-parameter Lindley distribution [12], and abbreviations of these distributions are TwPLD, QLD, SD, and ThPLD, respectively.
To increase more flexibility in this line of developments, Monsef (2016) where θ and α are shape parameters and β is a location parameter. Equation (2) presents two-component mixture of an exponential ( , θ β ) and gamma ( 2, , θ β ) distributions with the mixing proportion p θ θ α = + .
On the other hand, Shanker (2015) [14] has obtained Akash distribution (AD) as an alternative to Lindley distribution with density function: where θ is a shape parameter. Equation (3) shows two-component mixture of exponential ( θ ) and gamma ( 3,θ ) with mixing proportion  In many lifetime distributions, the location parameter may be assumed to be zero. However, the location parameter is an important parameter that provides an estimate of the failure-free period, that is the earliest time at which a failure may be observed. When we include the location parameter in a distribution, it changes the starting point of the distribution. For example, time to achieve pain relief in a patient after who has applied with a treatment method does not start from the value of zero. Then, in this application, the location parameter of the relevant distribution cannot be taken as zero. In this paper, we introduce and study a new three-parameter Lindley family distribution with location parameter as a modification of AD. The new distribution will be called as location based generalized Akash distribution (NGAD).
The NGAD is a two-component mixture of exponential and gamma distributions which generalizes the AD. Further, a simulation study will be done to study the performance of the ML method in the parameter estimation of the NGAD.
The remaining part of this paper is organized as follows: In Section 2, we introduce the NGAD with its density and distribution functions. We present the statistical and reliability properties of NGAD in Section 3 and Section 4, respectively. The size-biased form of the NGAD is discussed in Section 5. Further, the estimation of the parameters of NGAD by using the ML method is discussed in Section 6. Finally, a simulation study is done to study the performance of the maximum likelihood estimators for NGAD, and real-world applications are used to illustrate its flexibility with the above-mentioned existing modified Lindley distributions.

Location Based Generalized Akash Distribution
In this section, we introduce the location based generalized Akash distribution with its probability density function (pdf) and cumulative distribution function (cdf). Note that the Akash distribution (3) contains only one shape parameter.
To develop the new distribution first, we define a new three-parameter Lindley family distribution by including the location parameter β as a finite mixture of exponential ( , θ β ) and gamma ( 3, , θ β ) with mixing proportion  The corresponding cumulative distribution function of NGAD is given by: Note that when 2, 0 η β = = , the NGAD reduces to the AD with parameter θ .

Statistical Properties
Here, some important statistical properties of the NGAD are derived such as r th moments and related measures, moment generating and characteristic functions, and quantile function.

Moments and Related Measures
Some basic important characteristics of the distribution such as central tendency, dispersion, skewness, kurtosis, and index of dispersion can be studied by using the moments. The following theorem gives the r th moment about the origin. Theorem 1. The r th moment about the origin of the NGAD is given by: µ , and 4 µ are obtained as:

Moment Generating and Characteristic Function
The moment generating function (mgf) and the characteristic function (cf) are directly associated with a probability distribution's characteristics. Further, these can be used to generate the moments of a distribution. The following theorem provides the moment generating function of the NGAD.
Theorem 2. The moment generating function say of the NGAD is given as follows: Proof.   ) .

Quantile Function
The quantile function is useful for quantile estimates and random number generation. The quantile function of NGAD can be derived by solving ( ) ,0 Then, the u th quantile function of NGAD is derived as: Equation (9) is not a closed-form. However, the u th quantile can be estimated and random varieties from NGAD can be generated by using the numerical methods.
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.

Reliability, Inequality and Entropy Measures
In this study, we derive important reliability measures of NGAD such as survival

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 NGAD is defined as: It is clear that, ( ) 1 The hazard rate function (hrf) is the instantaneous failure rate. It is widely used to describe the lifetime. The hrf of the NGAD is defined as: Note that, ( )   The corresponding reversed hazard rate function of NGAD is defined as:  (12) and the cumulative hazard rate function of NGAD is defined as:

Mean Residual Life Function
The residual life function is defined as the remaining lifetime of a unit of age Now, consider the integrals separately as follows: A η βθ βθ θλ βθ βθ θλ βθ θλ

Lorenz and Bonferroni Curves
The Lorenz and Bonferroni curves were formulated to measure the income inequality. They are widely used in economics, reliability, demography, medicine, and insurance. The following theorem derives the Lorenz curve for NGLD.

Renyi Entropy
The Renyi entropy is a measure of variation of uncertainty measure of a distribution say ( ) H γ λ [15]. It is an extension of Shannon entropy [16] and widley used in information theory. The following theorem derives the Renyi entropy associated with NGAD.
Theorem 5. The Renyi entropy of the NGAD is given by: Proof.

The Size-Biased of NGAD
When a recording of observations with an unequal chance, the weighted distributions are used significantly. This provides more flexibility to the standard distributions incorporating sampling probabilities which are proportional to a non-negative weighted function ( ) w λ . The applications of the weighted distributions in reliability, biomedical and ecological sciences have been studied by Patil and Rao (1978) [17]. The weighted random variable w Λ of NGAD is defined as: When ( ) , 0 where s Λ is the respective random variable.
The following theorem gives the density function for the sized-biased version of NGAD.
Theorem 6. The density function for r th order sized-biased form of NGAD is derived as: Note that ( )

Parameter Estimation and Inference
In this section, the method of ML is introduced to estimate the parameters of NGAD. Further, the confidence intervals for the unknown parameters are derived.
The second partial derivatives of the l are:

Applications
In this section, we examine the performance of the parameter estimates of NGAD by ML method and asymptotic theory respect to sample size n. Here, we perform a simulation study, and further, we illustrate the flexibility of the NGAD over the LD, QLD, TwPLD, LwLD, SD, AD, and ThPLD by using several real data sets.

Simulation Study
In this subsection, we present the simulation study results to examine the performance of the estimations by the ML method that proposed in Section 6. Equation (9) is used to carry out the simulation study by generating random samples from NGAD, where u is assumed to follow uniform distribution ( )

Real Data Applications
This subsection is considered to show the flexibility of NGAD over some other existing Lindley family distributions by fitting these to several real data sets. The Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and Kolmogorov-Smirnov Statistics (K-S Statistics) are utilized to compare the performance of LD, SD, AD, TwPLD, QLD, LwLD, and ThPLD. The ML method was used to estimate the unknown parameters. The first three real data sets that were considered for the goodness of fit of distributions are given below: Data set 1: The following data set represents the tensile strength, measured in GPa, of 69 carbon fibres tested under tension at gauge lengths of 20 mm and reported by Bader and Priest (1982) [ Some of the statistical measures for data set 1, 2 and 3 are given in Table 4.
The fitted density plot of each distribution is shown in Figure 3 for data set 1, 2, and 3 separately. Each fitted density compares with the empirical histogram of the real data sets. Here, we may observe that the fitted density for the NGAD shows a closer fit with the empirical distributions. The AIC, BIC, and K-S statistics with critical values for NGAD, LwLD, TwPLD, QLD, ThPLD, AD, LD, and SD are shown in Table S1 (Appendix). Based on minimum AIC, BIC values and significant results by K-S statistics, the NGAD provides a better fit than all other distributions for light-tailed distributions ( 0 EK < ) as well as heavy-tailed distributions ( 0 EK > ). Further, it is noted that the LwLD performs better than NGAD for the data set with a considerably high positive EK value.
Then, similarly we have fitted NGAD and LwLD to several real-data sets (Appendix) having different EK values and compare their performances. Table 5 summarizes the results of the goodness of fit test for these data sets. It is clear that NGAD may perform better than LwLD for considerably small EK values.

Conclusion
In this study, a new generalized Akash distribution has been introduced by incorporating the location parameter to improve the flexibility of the failure rate function. Then, its structural properties including parameter estimation have been discussed. This new distribution yields a more flexible density and failure rate shapes. Further, it has the capability to model the bathtub and monotonic increasing failure rate shapes. The simulation study indicates that the ML method performs well in the estimation of the unknown parameters for NGAD. To illustrate the theoretical findings, the real-world applications were used, and the results reveal that the NGAD is superior over AD, Lindley distribution (LD) and some other existing modified LDs that have been developed based on exponential and gamma mixtures with different mixing proportions. Based on the special characteristics, the proposed model may attract wider applications in reliability, mortality, actuarial, ecological sciences, among others.