MODIFIED-LINDLEY DISTRIBUTION AND ITS APPLICATIONS TO THE REAL DATA

In this paper, a new three-parameter lifetime distribution is proposed by mixing modified Weibull and generalized gamma distributions. The point estimation on the distribution parameters are discussed through several estimators. The interval estimation is also studied with two methods based on asymptotic normality and likelihood ratio. A Monte Carlo simulation study is performed to evaluate the biases and mean square errors behaviors of point estimates for a different sample of size. A simulation study is also conducted to investigate the coverage probabilities of confidence intervals. The distribution modeling analyses are provided based on several real data sets to demonstrate the fitting ability of the introduced distribution.

©2022 Ankara University Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics respectively, and θ > 0 is a parameter.
L distribution can be represented as a mixture of two distributions with pdf where f E (x; θ) = θe −θx and f G (x; θ) = θ 2 xe −θx are the pdfs of exponential and gamma distributions respectively and p = θ 1+θ is the mixing proportion of distributions. Since the L distribution is IFR, it is unsuitable for modelling the data that obeys the non-linear hazard rate structure. [12] introduced the power Lindley (PL) distribution, which generalizes the Lindley distribution with the following pdf where f W (x; α, θ) = αθx α−1 e −θx α and f GG (x; α, θ) = αθ 2 x 2α−1 e −θx α are the pdfs of Weibull and generalized gamma (GG) distributions respectively and p = θ 1+θ is the mixing proportion of distributions. [12] investigated properties of the PL distribution with an application and outlined that the PL distribution is a better model than the other L and exponential based distribution. Moreover, several generalizations have been proposed in the literature in order to increase the flexibility and usefulness of the L model. Some of them are: generalized Lindley (GL) [32], exponentiated Lindley (EL) [22], discrete Lindley [11], extended Lindley [6], beta Lindley [20,21], exponentiated power Lindley (EPL) [31], odd log logistic power Lindley [1], odd log-logistic Lindley Poisson [24], odd Burr Lindley [3], binomial discrete Lindley [16], Weibull-Lindley [4] and generalized power Lindley [15] among others.
This paper aims to introduce a new flexible distribution that generalizes the L and PL distributions with the same structure of (1). Furthermore, we are also motivated to propose a new L distribution because introduced model has various pdf shapes as well as non-monotone hazard rate function (hrf) shapes unlike L and PL models.
The paper is organized as follows: In Section 2, a new lifetime distribution is proposed and several distributional properties are discussed. Several point estimation methods are discussed for the distribution parameters in Section 3. In Section 4, the interval estimation is considered with two well-known methods. The Section 5 close the paper with three distribution modeling analyses based on real data.

Modified Lindley Distribution and Some Properties
A random variable X has a Modified Lindley (MoL) distribution if its pdf is given by where α, β, θ > 0 are parameters and Ξ = (α, β, θ). Indeed MoL distribution is a mixture of two distribution with the following representation: where p = θ θ+1 is the weighting parameter of the distributions, g 1 (x; α, β, θ) is the pdf of Modified Weibull (MW) distribution introduced in [17], with the following pdf and g 2 (x; α, θ) is the pdf of a GG distribution introduced in [28], with the following pdf From (2), we see that the MoL distribution is a two-component mixture of MW and GG distributions with weighting parameter p. We denote the MoL distribution with parameter Ξ by M oL(Ξ). While β → 0, MoL distribution reduces to the PL distribution. While β → 0 and α → 1, it is reduced to L distribution.
The cdf and hrf of the MoL distribution are and respectively. The plots of the pdf and hrf are given in Figure 1 to identify their possible shapes. These figures show that the MoL distribution can be unimodal, bimodal, decreasing and firstly decreasing then unimodal shaped. On the other hand, the hrf of MoL can be both monotone and non-monotone structures. In distribution theory, stochastic ordering is an essential measure for evaluating the comparative behavior of random variables. It is known that X < lr Y ⇒ X < hr Y ⇒ X < st Y , see [25]. For more information about stochastic ordering with different applications, one can see [27]. Likelihood ratio ordering is shortly defined as follow: X is less than Y in the likelihood ratio order (denoted by X < lr Y ) if f X (x) / f Y (x) increases in x over the union of the supports of X and Y . Theorem 1. If X ∼MoL(α, β, θ 1 ) and Y ∼MoL(α, β, θ 2 ) and θ 1 < θ 2 , then X < lr Y .
Proof. See Appendix.
Corollary 2. The mean and rth central moment of the MoL(Ξ) are given, respectively, by and Using (6), the skewness and kurtosis coefficients can be obtained by respectively. The mean, variance, skewness and kurtosis are computed for some choices of parameters and given in Table 1. From Table 1, it is seen that the coefficient of kurtosis can take negative and positive values. This shows that the distribution has a flexible structure in data modeling. In addition, it is seen that the new distribution is flatter than the normal distribution. When θ increases, the kurtosis coefficient increases and the variance decreases. E(X) decreases when the parameter β increases.

Point Estimation
In this section, the maximum likelihood, least square, weighted least square, Anderson-Darling, Cramer-von Mises, and maximum product spacing methods are discussed to estimate the MoL distribution parameters. It is noticed that these estimates are also used in [2], [13], [14], [29], [30] among others. Let X 1 , X 2 , . . . , X n be a random sample from the MoL(Ξ) distribution with realizations x 1 , x 2 , . . . , x n . Furthermore, X (1) , X (2) , . . . , X (n) be the corresponding ordered statistics with realizations x (1) , x (2) , . . . , x (n) . Then the log likelihood function can be written by Hence, the maximum likelihood estimate (MLE) Ξ of Ξ is written by The maximum product spacing estimate (MPSE) was proposed by [9]. The MPSE Ξ M P S of parameter Ξ are achieved by maximizing where, F is MoL cdf given in (3) and F (x (0) ; Ξ) = 0 and F (x (n+1) ; Ξ) = 1. Note that the MPSE can be written by The least square estimate (LSE) Ξ LSE of parameter Ξ are obtained by minimizing the function where F is MoL cdf given in (3). Hence, LSE of Ξ is given by The weighted least square estimate (WLSE) Ξ W LSE of Ξ are obtained by minimizing The Anderson-Darling (ADE) type estimate Ξ AD of parameters Ξ are obtained by minimizing The ADE of Ξ is written by The Cramer-von Mises (CVME) type estimate, Ξ CV M of parameter Ξ are obtained by minimizing The CVME of Ξ is given by In order to achieve the values of estimates, the R functions such as constrOptim, optim or maxLik can be used.
The simulation study is performed for the bias and mean square errors (MSEs) of estimates and the results are presented by graphically. We consider N = 1000 trials of size n = 20, 25, . . . , 1000 from the MoL distribution with true parameter Ξ = (5, 5, 2). All estimates are achieved by using constrOptim routine in the R. The simulation results are presented in Figs. [2][3][4] show that all estimates are consistent since the MSEs decrease to zero for large sample size. The CVME and MPSE have the maximum amount of the biases for all parameters while CVME and WLSE have the maximum MSEs for all parameters. On the other hand, MPSE is the best estimator according to MSEs for small sample size. It is noticed that the MPSE and MLE has almost same MSEs for moderate and large sample size cases. The ADE and LSE have the lowest bias for all parameters. As a final comment on the simulation study, we recommend that the MLE or MPSE should be used to estimate the parameters.

Interval Estimation of MoL Distribution Parameters
In this section, the confidence intervals (CIs) are discussed for the parameters a, β and θ. In general, CIs are constructed by using MLE based on pivotal quantities through the asymptotic normality(AN) property of MLE. These CIs are most where z a , is the a th quantile of the standard normal distribution, se ( α), se β and se θ are the roots of the diagonal member of I −1 Ξ which is a consistent estimate of I −1 (Ξ) and the se (·) stands for standard error. There is another method called uncorrected likelihood ratio (ULR). It is noticed that AN and ULR CIs are asymptotically equivalent [10].

MODIFIED-LINDLEY DISTRIBUTION AND ITS APPLICATIONS TO THE REAL DATA261
with α L < α and α U > α, where χ 2 (1) (a) is the a th quantile of the χ 2 distribution with 1 degrees of freedom. The 100 × (1 − γ) % ULR CIs can be produced in the same manner for the other parameters β and θ.
In the simulation study, 5000 trials are used to predict the coverage probabilities (CPs) of the AN and ULR CIs. The nominal level is fixed at 0.95. In order to get CPs of ULR CIs, there is no need to obtain the CIs limits. It is possible that the CPs of ULR CIs can be simulated by a likelihood ratio test on the true parameter. The simulated CPs of these intervals are given in Table 2. Let us discuss the true parameter cases Ξ = (1, 1, 0.5) , (1, 1, 2.5) , (5, 5, 2), (1, 2, 3) , (3, 0.5, 1.5) and (2, 1, 0.25) . From Table 2, it is observed that the CPs of ULR reach to the desired level when the all sample of size discussed here (say n ≥ 50) for all parameters. However, the CPs of AN can not reach the desired level for small sample of size case especially for parameter β. The CPs reach the nominal level when the sample of size increases (say n ≥ 250 or n ≥ 500 according to selected true parameters). Under discussion given here, it is indicated that ULR CIs powerful tool to construct the CIs for the MoL parameters.

Real-life Data Analysis
In this section, we provide three applications to the real data sets to demonstrate empirically the potentiality of the proposed model. All data sets, we compare the MoL model with MW, PL, GL, EPL, EL and L models. In order to reveal the best model, the estimated log-likelihood values ℓ( Ξ), Akaike information criteria (AIC), consistent Akaike information criteria (CAIC), Kolmogorov-Smirnov (KS), Cramer von Mises (W * ) and Anderson-Darling (A * ) goodness of-fit statistics are computed for all models.
The first data set represents the times between successive failures (in thousands of hours) in events of secondary reactor pumps studied by [5], [19] and [26].  We give the summary statistics of the data sets in Table 3. The first and third data sets have right skewness as well as the second data set has the left skewness. Tables 4-6 list the MLEs, standard errors, ℓ( Ξ) and goodness-of-fits statistics from the fitted models.  show that the MoL model can be chosen as the best model based on all criteria. In addition, we give the parameter estimation results and goodness-of-fit statistics of the MoL distribution based on other estimation methods in Table 7. Figures 5-7 show the fitted densities, cdfs and probability-probability (P-P) plots of the MoL model. We also sketch the P-P plots of others models in Figures 8-10. From Figures 8-10, we clearly show that the MoL model fits this data set better than the other models. In Table 8, 95% AN and ULR confidence limits of the parameters are presented for the all data sets. In general the limits of AN and ULR intervals are close to each other. Figure 11 demonstrate the ULR intervals for the third real data.     Taking the derivative with respect to x, for θ 1 < θ 2 , − ((θ 1 − θ 2 )) is greater than zero. So W ′ 1 (x) > 0 when θ 1 < θ 2 is taken. W 1 (x) is an increasing function in x. Secondly, the same steps are applied  Figure 9. The PP plots for the second data set for GG density ratio. The GG density ratio is given by W 2 (x) = g 2 (x; α, β, θ 1 ) g 2 (x; α, β, θ 2 ) = θ 2 1 exp (−θ 1 x α ) θ 2 2 exp (−θ 2 x α ) Taking the derivative with respect to x,  Figure 10. The PP plots for the third data set for θ 1 < θ 2 , − ((θ 1 − θ 2 )) is greater than zero. So W ′ 2 (x) > 0 when θ 1 < θ 2 . W 2 (x) is an increasing function in x. Since both W 1 (x) and W 2 (x) are increasing functions in x, W (x) = W 1 (x) + W 2 (x) is also an increasing function in x. The proof is completed.