Abstract
In this paper, we introduce a new generalization of the power Lindley distribution referred to as the alpha power transformed power Lindley (APTPL). The APTPL model provides a better fit than the power Lindley distribution. It includes the alpha power transformed Lindley, power Lindley, Lindley, and gamma as special submodels. Various properties of the APTPL distribution including moments, incomplete moments, quantiles, entropy, and stochastic ordering are obtained. Maximum likelihood, maximum products of spacings, and ordinary and weighted least squares methods of estimation are utilized to obtain the estimators of the population parameters. Extensive numerical simulation is performed to examine and compare the performance of different estimates. Two important data sets are employed to show how the proposed model works in practice.
1. Introduction
The one-parameter Lindley distribution was primarily proposed by Lindley [1] as an alternative model for data with a nonmonotone hazard rate shape. Ghitany et al. [2] provided the properties of the Lindley distribution under a careful mathematical treatment. They also displayed in a numerical application that the Lindley distribution provides a better model than the exponential distribution. A random variable Y is said to follow the Lindley distribution with scale parameter θ > 0, if its probability density function (pdf) is given by which is two-component mixture of exponential () and gamma (). The Lindley distribution has only increasing failure rate which has been identified as a major difficulty in lifetime analysis. To overcome this situation, many generalizations of the Lindley distribution have been introduced in literature. Nadarajah et al. [3] proposed a generalized Lindley distribution. Bakouch et al. [4] introduced the extended Lindley (EL) distribution. Shanker et al. [5] introduced the two-parameter Lindley distribution. Ghitany et al. [6] proposed the power Lindley (PL) as an extension of Lindley distribution. Based on pdf (1) and by using the transformation X = , the pdf of the PL distribution is given byThe cumulative distribution function (cdf) corresponding to (2) is given by Many other generalizations of Lindley distribution have been proposed by several authors; see for example, a new generalized Lindley distribution [7], transmuted quasi Lindley distribution [8], exponentiated power Lindley [9], transmuted generalized Lindley distribution [10], transmuted generalized quasi Lindley distribution [11], and transmuted Kumaraswamy quasi Lindley distribution [12].
The alpha power transformation (APT) is one of the procedures that make the distributions richer and flexible to model the real life data. The APT has been proposed by Mahdavi and Kundu [13] with the parameter to incorporate skewness to the base distribution. The cdf of APT is specified byMahdavi and Kundu [13] applied the APT method to the exponential distribution and they studied various properties of the alpha power exponential distribution. Based on APT method, some new distributions are considered like APT Weibull [14], APT generalized exponential [15], APT Lindley [16], APT extended exponential [17], and APT inverse Lindley [18].
Following the similar idea of Mahdavi and Kundu [13], we introduce a new three-parameter distribution, the so-called APTPL distribution. The main purpose of the new model is that the additional parameter can give several desirable properties and more flexibility in the form of the hazard and density functions. The APTPL distribution contains a number of known lifetime submodels such as the APT Lindley (APTL) distribution, PL distribution, Lindley distribution, and gamma distribution. Properties, estimation of the parameters and applications with real data, are considered.
This paper is constructed as follows. We introduce and study the properties of APTPL in Sections 2 and 3, respectively. In Section 4, we obtain the maximum likelihood (ML) estimator, least square (LS) estimator, weighted least squares (WLS) estimator, and maximum product of spacing (MPS) estimator of APTPL distribution. In the same section, a simulation study is executed to investigate the effectiveness of the estimates. The analyses of two real data sets are employed in Section 5. The paper ends with concluding remarks.
2. The APTPL Model
Here, we now introduce the notion of APTPL distribution.
Definition 1. A random variable X is said to have a three-parameter APTPL distribution with the scale parameter θ>0 and shape parameters α, β >0, if its cdf is of the formwhere is a set of parameters. The pdf of the APTPL distribution is given byAlso, the reliability function, say and hazard rate function (hrf), say of X are given, respectively, as follows:andA random variable X with distribution (6) is denoted by X ~ APTPL ( Submodels of the APTPL distribution are as follows:(i)For β =1, the pdf (6) reduces to APTL [16].(ii)For =1, the pdf (6) reduces to PL [6].(iii)For β =1 and =1, the pdf (6) reduces to Lindley [1].(iv)For β =1 and =1, the pdf (6) reduces to gamma (2, θ). Some descriptive pdf and hrf plots of X are illustrated Figures 1 and 2 for specific parameter choices of (see Figures 1 and 2).
We conclude from Figure 1 that the pdf of the APTPL distribution can be reversed J-shaped, uni-model, and right and left skewed for some values of . Also, we conclude that, for some values of , the hrf of APTPL distribution can be increasing, decreasing, and reversed J-shaped.
3. Statistical Properties
Here, some statistical properties of the APTPL distribution including quantile, moments, moment generating function, incomplete moments, Bonferroni and Lorenz curves, probability weighed moments, Rényi entropy, and stochastic ordering are derived.
3.1. Quantile Function
The quantile function, say Q(u) =F−1(u), u∈ (0, 1), is obtained by inverting (5) as follows:which yields Multiply both sides by , then we have the Lambert equationHence, we have the negative Lambert W function of the real argumenti.e.,where u∈ (0, 1) and W−1(.) is the negative Lambert W function. The first quantile (25%), median (50%), and third quantile (75%) for some choices values of the parameters are computed numerically (see Table 1).
We detect form Table 1 that as the value of increases, for fixed values of and , the value of percentage points increases. As the value of increases, for fixed values of and , the percentage value points decrease. Also, as the value of increases, for fixed values of and , the percentage value points decrease.
3.2. Moments and Related Measures
The kth moment of a random variable X having the APTPL distribution is obtained as follows:Since the power series can be written as hence (14) can be expressed as follows:Using the binomial expansion, thenHence, the th moment of the APTPL distribution takes the formwhere Particularly, the mean is as follows:Similarly, the central moment of a given random variable can be defined as The coefficient of skewness (CS) and coefficient of kurtosis (CK) are defined by Thus, numerical values of the , variance (), coefficient of variation (CV), CS, and CK of the APTPL distribution for some certain values of parameters are obtained and recorded in Table 2.
Furthermore, the moment generating function of the APTPL distribution is given by
3.3. Incomplete Moments
The sth lower incomplete moment, say , of the APTPL distribution is given byUsing the power expansion (15) and binomial expansion, then (24) will be So, the sth lower incomplete moment of the APTPL distribution is where , is the lower incomplete moments. The first incomplete moment of the APTPL distribution can be obtained by setting s =1 in (26). The first incomplete moment is related to the Bonferroni and Lorenz curves, the mean residual, and mean waiting times. The Bonferroni and Lorenz curves are important in economics, reliability, demography, insurance, and medicine. The Lorenz curves, say , and Bonferroni curve, say , are defined by For the proposed model, they can be determined from (27) as and,
3.4. The Probability-Weighted Moments
The class of probability-weighted moments (PWMs), denoted by , for a random variable X, is defined as follows:Substituting (5) and (6) in (30), then the PWM of the APTPL distribution is Using the binomial expansion, then Using (15) and binomial expansions, then (32) will be Hence, the PWM of the APTPL distribution is given bywhere
3.5. Rényi Entropy
The Rényi entropy of a random variable represents a measure of variation of the uncertainty. The Rényi entropy is defined by Substituting (6) in (36), then we have Using (15), then is converted toUsing binomial expansions, then we haveThe above integral is complicated to obtain, so it will be solved numerically. Some of the numerical values of Rényi entropy for some certain values of parameters are given in Table 3.
3.6. Stochastic Ordering
Here, we compare the APTPL1 and the APTPL2 ( with respect to stochastic ordering information. Let and be two random variables with cdfs, reliability functions and pdfs and and ; and and , respectively, where and . A random variable is said to be smaller than in the following ordering (see [19]), if the following holds:(i)Stochastic order if for all .(ii)Likelihood ratio order if is decreasing in .(iii)Hazard rate order if is decreasing in .(iv)Mean residual life order if .
We have the following chain of implications among the various partial orderings discussed above:
Theorem 2. Let ~ APTPL1 ( and X2 ~ APTPL2. If , , and , then , , , and .
Proof. It is sufficient to show is a decreasing function of ; the likelihood ratio is therefore,Hence, if , , and , then , which implies that is stochastically greater than with respect to likelihood ratio order. Similarly, we can conclude , , and .
4. Estimation
The population parameters of the APTPL distribution can be estimated using ML, MPS, LS, and WLS methods of estimation.
4.1. Maximum Likelihood Estimators
To obtain the ML estimators of the APTPL distribution with set of parameters , let X1,…, Xn be observed values from this distribution. Hence, the log-likelihood function for the vector of parameters, say , can be written asThe ML equations of the APTPL distribution are obtained as follows:andEquating and with zeros and solving simultaneously, we obtain the ML estimators of α, , and θ.
4.2. Maximum Product of Spacings Estimators
The MPS method is a powerful alternative to ML method for estimating the population parameters of continuous distributions (see [20]). Let be the uniform spacings of a random sample from the APTPL distribution, where The MPS estimator is obtained by maximizing the geometric mean (GM) of the spacings with respect to , , and . Equivalently, the MPS estimator of , , and can be obtained by maximizing the logarithm of the GM of sample spacings (48). There is no closed solution, so the numerical technique is applied to find the required estimates.
4.3. Least Squares and Weighted Least Squares Estimators
Suppose is a random sample of size n from APTPL distribution and suppose is the corresponding ordered sample. Then, the LS estimators can be obtained by minimizing the sum of squares errors, with respect to the unknown parameters. So, the LS estimators of the population parameters of APTPL distribution are obtained by minimizing the following quantity with respect to , and Also, the WLS estimators of APTPL distribution can be obtained by minimizing the following with respect to the population parameters , and where
4.4. Numerical Illustration
As seen from the three previous subsections, that the ML, MPS, LS, and WLS estimators of the APTPL are very hard to obtain. Therefore, a simulation study is implemented to study the behavior of estimates, based on certain measures, which are mean square errors (MSEs) and absolute biases (ABs). 1000 random sample X1,…Xn of sizes n = 10, 20, 30, and 100 are generated from the APTPL distribution. Exact values of parameters are considered as
and . The ML MPS, LS, and WLS estimates of and their measures are calculated. Numerical outcomes are listed in Table 4 and we conclude the following.(i)As the sample sizes increase, the MSEs and ABs of the ML, MPS, LS, and WLS estimates of parameters decrease.(ii)Based on ML and MPS methods, for fixed values of , and as the value of increases, the MSEs and ABs increase for and estimates and decrease for estimates.(iii)Based on the ML method, as the values of and increase while the value of decreases, the MSEs and ABs for and estimates increase, whereas the values of increase for approximately sample size and different sets.(iv)The MSEs and ABs of the MPS estimates are smaller than the corresponding for the other estimators.(v)The MSEs and ABs of the WLS estimates for and estimates are smaller than the corresponding for LS estimates in set 1 at most sample sizes.(vi)Based on the LS method, as the values of and increase, while the value of decreases, the MSEs for and estimates decrease for all sample sizes and different set of parameters.(vii)Generally, the MPS estimates of all parameters have the smallest MSEs followed, respectively, by ML, LS, and WLS estimates. Also, the MSEs of LS estimates are better than the MSEs of WLS estimates for most sample sizes.
5. Real Data Analysis
In this section, we analyze two real data sets to demonstrate the performance of the APTPL distribution in practice. One of the data sets is taken from Linhart and Zucchini [21, page 69]. The other data are taken from Aarset [22] representing the failure times of 50 devices. For the two data sets, we compare the results of the fits with the APTL, PL, EL, Lindley (L), and exponential (E) distributions. The pdfs of APTL and EL are given byandTo test the superiority of the APTPL distribution, measures of goodness-of-fit can be applied in comparison to some other models. Mainly we use minus log-likelihood (-log L) Kolmogorov-Smirnov (KS) test statistic, Akaike information criterion (AIC), corrected Akaike information criterion (CAIC), Bayesian information criterion (BIC), and Hannan-Quinn information criterion (HQIC).
5.1. Linhart and Zucchini Data
The first data consists of a sample of 30 failure times of air-conditioned system of an airplane. The data are 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95. The ML estimates, SEs of the model parameters, and analytical measures are provided in Table 5.
According to the criteria mentioned in Table 5, we found that the APTPL model is the best fitted model compared to the other competitive models. The histogram of data and the estimated pdfs and cdfs for the fitted models are displayed in Figure 3. It is clear from Figure 3 that the APTPL model provides a better fit to this data.
5.2. Aarset Data
The second data consists of 50 failure times of devices. The data are 0.1, 0.2, 1, 1, 1, 1, 1, 2, 3, 6, 7, 11, 12, 18, 18, 18, 18, 18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 60, 63, 63, 67, 67, 67, 67, 72, 75, 79, 82, 82, 83, 84, 84, 84, 85, 85, 85, 85,85, 86, 86. The ML estimates, SEs of the model parameters, and analytical measures are provided in Table 6.
As we see from Table 6 that the APTPL distribution gives better fit than the other models; therefore, it could be more adequate model for explaining the used data set. More information can be provided in Figure 4.
Also, from Figure 4, we conclude that the APTPL distribution provides better fits then we expect that the proposed model may be an interesting alternative model for a wider range of statistical research.
6. Concluding Remarks
In this paper a new three-parameter power Lindley distribution, called the APTPL distribution, based on alpha power transformation is proposed. The main objective behind this generalization is to give more flexibility for more data that can be analyzed using the proposed distribution. Various statistical properties are discussed such as moments, moment generating function, incomplete moments, and quantile function. The model parameters are estimated via the maximum likelihood, maximum product of spacing, and ordinary and weighted least squares methods. Simulation study is employed to examine the behavior of different estimates. The usefulness of the proposed model is illustrated motivated by two real data.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.