A New Lifetime Distribution and Its Power Transformation

Newone-parameter and two-parameter distributions are introduced in this paper.The failure rate of the one-parameter distribution is unimodal (upside-down bathtub), while the failure rate of the two-parameter distribution can be decreasing, increasing, unimodal, increasing-decreasing-increasing, or decreasing-increasing-decreasing, depending on the values of its two parameters. The two-parameter distribution is derived from the one-parameter distribution by using a power transformation. We discuss some properties of these two distributions, such as the behavior of the failure rate function, the probability density function, themoments, skewness, and kurtosis, and limiting distributions of order statistics. Maximum likelihood estimation for the two-parameter model using complete samples is investigated. Different algorithms for generating random samples from the two new models are given. Applications to real data are discussed and compared with the fit attained by some oneand two-parameter distributions. Finally, a simulation study is carried out to investigate themean square error of themaximum likelihood estimators, the coverage probability, and the width of the confidence intervals of the unknown parameters.


Introduction
Lindley [1] proposed a one-parameter distribution, now known as the Lindley distribution, with the following probability density function (pdf): The failure rate function of the Lindley distribution is always increasing.The properties of the Lindley distribution are studied in detail by Ghitany et al. [2].There are situations in which the Lindley distribution may not be suitable from a theoretical or applied point of view, Ghitany et al. [3].For this reason, Ghitany et al. [3] used a power transformation,  =  1/ , to introduce the power Lindley distribution which is a more flexible distribution.The pdf of PL(, ) is (1 +   )  −  ,  > 0, ,  > 0. ( Ghitany et al. [3] showed that the hazard function of PL(, ) can be increasing, decreasing, and decreasing-increasingdecreasing depending on the values of the parameters.They also discussed some of the statistical properties of the distribution and used the maximum likelihood method to estimate its two unknown parameters and applied it to a real data set.
In spite of the flexibility of the PL(, ) to fit some real data sets, it fails to fit some other data sets.
The main aim of this paper is to introduce two new distributions.The first is a one-parameter distribution which is similar to the Lindley distribution and the second is the power transformation of the one-parameter distribution.We refer to these two distributions as () and TN(, ) respectively.The hazard function of () is only unimodal, while the hazard function of TN(, ) can be decreasing, increasing, unimodal, decreasing-increasingdecreasing, or increasing-decreasing-increasing depending on the values of its two parameters.The variety of shapes of the hazard function of the TN(, ) enables it to be a good model to fit different data sets.

Journal of Probability and Statistics
The rest of the paper is organized as follows.Section 2 introduces the new one-parameter distribution and some of its characteristics are discussed in Section 3. Section 4 presents the transformation of the new distribution, TN(, ).Different characteristics of TN(, ), such as the hazard function, quantiles, random sample generation, moments, and order statistics distributions, are discussed in Section 5. Section 6 discusses the maximum likelihood estimate of the two parameters of TN(, ).Applications of the two models are presented in Section 6. Monte Carlo Simulation study is carried out in Section 7 to examine the accuracy of the maximum likelihood estimators of the TN(, ) parameters as well as the coverage probability and average width of the confidence intervals for the parameters.Finally, Section 8 concludes this paper.

The New Distribution
Consider the random variable  whose pdf is given by The survival function (sf) of  is given by while its hazard rate function is given by For simplicity, from now on, we refer to this distribution as ().
Interpretation.There are two different interpretations of () as follows.

Characteristics of 𝑁(𝛽)
In this section, algorithms are described to obtain quantiles of () and to generate samples from ().Also, the moment generating function and the moments of this distribution are derived.

Random Sample Generation.
We provide below three equivalent algorithms to generate a random variate from ().

The Moments and the Moment Generating Function.
The moment generating function (mgf) of () may be written as Differentiating the above expression  times with respect to  and setting  to zero, we get th moments,   , as Based on the first four ordinary moments, the measures of skewness (sk) and kurtosis () of () can be obtained using Plots of the skewness and kurtosis of the distribution as a function of  are plotted in Figure 1.From the plots, sk and  are unimodal functions of .The skewness is always positive and the kurtosis is larger than 3; therefore, () is positively skewed and leptokurtic.

Power Transformation of the New Distribution
To get a more flexible distribution, we consider an extension of the new distribution () with the pdf (3) by using the power transformation  =  1/ ,  > 0. The pdf of  is given by The density of  is plotted in Figure 2 for three choices of  when  = 1.0, which shows that the density is symmetric when  = 3.535, left skewed when  < 3.535, and right skewed when  > 3.535.This implies that the power parameter  characterizes the shape of the density function.
More investigation of the density will be discussed, in the next section, based on the skewness and kurtosis measures.
From now on, we will use TN(, ) to refer to the power transformation of the new distribution ().
Interpretation.There are two different interpretations of  as follows.
Straightforward calculations yield the the survival function of TN(, ) as We derive some characteristics of TN(, ) in the next section.

Characteristics of TN(𝛼, 𝛽)
5.1.The Hazard Function.The hazard rate function of TN(, ) is For  = 1, the hazard function is unimodal.Its limiting values at zero and infinity are , and it reaches a maximum value of where  −1 (⋅) denotes the Lambert  function which is the inverse of the function   .
For  ̸ = 1, the shape of the hazard function is difficult to ascertain analytically.The shape was determined numerically by examining the derivative of the hazard out to the 99.99th percentile of the distribution, and the results are shown in Figure 3.For  < 1, the hazard is decreasing except for a small region with  close to 1 and  < 0.5 where the hazard is initially decreasing, then increasing, and finally decreasing (DID).For  > 1, the hazard is strictly increasing for large  ( > 2.6 in the figure).For smaller  with  close to 1, the hazard can be unimodal (for very small ) or initially increasing, then decreasing, and finally increasing (IDI) (for slightly larger ). Figure 4 shows the hazard for five choices of  and  which demonstrate the five possible shapes.

Quantiles and Random Sample
Generations.The 100th quantile of TN(, ),   , can be derived from that of (),   , as follows: Figure 5 depicts the three quartiles  1 ,  2 , and  3 , which can be obtained from the th quantile by setting  = 0.25, 0.50, and 0.75, respectively.From Figure 5, the Interquartile range (IQR =  3 −  1 ) decreases dramatically when  increases.
The following algorithm generates a random variate from TN(, ).

The Moments and Shape Measures.
Let  follow TN(, ).After some algebra, the th ordinary moment of  is derived as Therefore, the mean and variance of  are Figure 6 depicts the mean and variance of TN(, ) as functions of  when  = 1 which shows that the mean decreases dramatically in  and takes its minimum of 0.8298 at  = 1.61 then it increases steadily to take its maximum of 0.9687, while the variance is decreasing.
Based on the first four ordinary moments, the measures of skewness (sk) and kurtosis () of TN(, ) can be obtained by substituting (17) into ( 9) and (10), respectively.Plots of the skewness and kurtosis of TN(, ) distribution as functions of , when  = 1.0, are given in Figure 7. From these plots, (1) the skewness is positive when  < 3.535 and negative when  > 3.535 and the kurtosis is (i) equal to 3 when either  = 2.4977 or  = 4.9151 which means that the distribution is mesokurtic; (ii) greater than 3 when either  < 2.4977 or  > 4.9151 which means that the distribution is leptokurtic; (iii) smaller than 3 when 2.4977 <  < 4.9151 which means that the distribution is platykurtic.This analysis shows how the power parameter  improves (), because the power transformation model can be used for data with a wide variety of distributional shapes.
For the power transformation, we have lim Therefore, (20) follows by Theorem 8.3.3 of Arnold et al. [4].
The following Theorem gives the limiting distribution of the th order statistic of the  lifetimes  1 ,  2 , . . .,   .

Theorem 6. The limiting distributions of 𝑋
where Proof.It follows from Theorem 5 and (8.4.2) of Arnold et al. [4].

Maximum Likelihood Estimation
Maximum likelihood estimation (MLE) is one of the most common methods for estimating the parameters of a statistical model.Assume that  independent and identical items, whose lifetimes follow TN(, ), are put on a life test simultaneously.Let  1 ,  2 , . . .,   be the failure times of  the items and let x = ( 1 , . . .,   ).The likelihood function for (, ) is The log-likelihood function is L (, ; x) =  ln  +  ln  −  ln (1 + ) where The first partial derivatives of L, with respect to  and , are , (, )   (, ) , where where The information matrix is The MLE of  and , say α and β, are the solution of the system of nonlinear equations obtained by setting L  = 0 and L  = 0 such that the F(α, β) is positive definite.This system has no analytic solution, so numerical methods, such as the Newton-Raphson method, Burden and Faires [5], should be used.
Large-Sample Intervals.The MLE of the parameters  and  are asymptotically normally distributed with means equal to the true values of  and  and variances given by the inverse of the information matrix.In particular, where F−1 is the inverse of F(α, β), with main diagonal elements F11 and F22 given by Using (32), large-sample (1 − )100% confidence intervals for  and  are where  /2 is the upper 100/2 quantile of the standard normal distribution.

Applications
In this section, we analyze four data sets to illustrate the applicability of the two new distributions proposed in this paper.The first data consists of 61 observed recidivism failure times (in days) of individuals released directly from correctional institutions to parole in the District of Columbia, Columbia, USA [6].The second data set consists of 43 active repair times (in hours) for an airborne communication transceiver [7].The third data set consists of 57 times (in thousands of operating hours) of unscheduled maintenance actions for the number 4 diesel engine of the U.S.S.Grampus, up to 16 thousand hours of operation [8].The forth data set consists of the tensile strength (measured in GPa) of 69 carbon fibers tested under tension at gauge lengths of 20 mm [9].We will refer to these data sets as failure times, repair times, maintenance actions, and tensile strength data, respectively.For each data set, we fit the proposed two distributions as well as Lindley and power Lindley distributions.For the sake of comparison, we apply goodness-of-fit tests to verify which distribution better fits these data sets.We consider the well-known Kolmogorov-Smirnov (K-S) statistic, the Cramér-von Mises (C-M), and Anderson-Darling (A-D) statistics [10].Furthermore, we consider the Akaike information criterion AIC = −2 L + 2, where L is the loglikelihood function at the MLE of the parameters and  is the number of model parameters.Table 1 shows the MLE of the parameters of each model, the corresponding maximum loglikelihood value, and the AIC for the four data sets.Table 2 presents the results of the goodness of fit tests for the four data sets using each model.
For every data set, we plotted (1) the scaled total time on test transform (TTT-transform) plot which gives qualitative information about the hazard rate shape [11]; (2) the hazard functions for the four fitted models; (3) the empirical and fitted density and distribution functions.Figures 8, 9, 10, and 11 show the four plots for the four data sets 1-4, respectively.The scaled TTT-transform plots show that the repair data set has a unimodal hazard, while the rest of data sets have increasing hazards.
The inverse of information matrix at the MLE using the four data sets are listed below.
Failure times: Active repair: Maintenance actions: Tensile strength:      For the first three data sets, TN(, ) model has the smallest value of the Kolmogorov-Smirnov (largest  value), the Cramér-von Mises, and Anderson-Darling goodness-offit tests statistics which indicate that the best fit is provided by the TN model for these data sets.For the forth data set, the power Lindley model provides the best fit in the sense of having the smallest test statistics.For all data sets, TN(, ) is a better fit than ().For the first two data sets, () is a better fit than both () and PL(, ) while it is the worst fit for the last two data sets.The AIC statistic is the lowest for TN(, ) for all data sets except for tensile strength where it is slightly higher.
Further, for testing () as a submodel of the TN(, ), we use the likelihood ratio test statistic (LRT) to check if the fit using the TN(, ) is statistically superior to a fit using the () for each data set.The LRT for testing  0 :  = 1 against  1 :  ̸ = 1 is Λ = 2( L 1 − L 0 ), where L 1 and L 0 are the maximum log-likelihood values under  1 and  0 , respectively.Under  0 , Λ   →  2  1 .The LRT rejects  0 if Λ >  2  1, , where  2 1, denotes the upper 100% point of chi-square distribution with 1 degree of freedom.Table 3 lists the values of the LRT and the corresponding  value for the four data sets.Based on the  values, the () is not rejected against the TN(, ) to fit the repair times data set, while it is rejected, at any level of where θ is the MLE of  using the th sample,  = 1, 2, . . ., , and  = , .Coverage probability is the proportion of the  simulated confidence intervals which include the true parameter .

Conclusion
In this paper, we have proposed new one-parameter and two-parameter distributions, called the () and TN(, ), respectively.The TN(, ) was obtained by using a power transformation of the () distributed variable.The TN(, ) provides more flexibility than the () in terms of the shape of the density and hazard rate functions as well as its skewness and kurtosis.We derived the maximum likelihood estimates of the parameters and their variancecovariance matrix.We proposed different algorithms to generate samples from the two proposed distributions.Applications of the two proposed distributions to real data sets show better fits than Lindley and power Lindley distributions.Finally, we examined the accuracy of the maximum likelihood estimators of the TN(, ) parameters as well as the coverage probability and average width of the confidence intervals for the parameters using simulation.

Figure 2 :
Figure 2: The TN(, ) density function for some values of  when  = 1.

1 Figure 4 :
Figure 4: The hazard function of TN(, ) for some parameter values.

Figure 8 :
Figure 8: The TTT-transform, fitted hazard, pdf, and cdf of the failure times data.

Figure 9 :
Figure 9: The TTT-transform, hazard, pdf, and cdf of the active repair times data.

Figure 10 :
Figure 10: The TTT-transform, hazard, pdf, and cdf of the maintenance actions data.

Figure 11 :
Figure 11: The TTT-transform, hazard, pdf, and cdf of the tensile strength data.

Table 1 :
Parameter estimates, maximum log-likelihood, and AIC for the four data sets.

Table 2 :
Statistics K-S ( value), C-M, and A-D for the four data sets.

Table 3 :
The LRT and  value for the four data sets.