THE FIVE PARAMETER LINDLEY DISTRIBUTION. Dept. of Statistics and Operations Research, King Saud University Saudi Arabia,

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1 Pak. J. Statist Vol. 31(4), THE FIVE PARAMETER LINDLEY DISTRIBUTION Abdulhakim Al-Babtain 1, Ahmed M. Eid 2, A-Hadi N. Ahmed 3 and Faton Merovci 4 1 Dept. of Statistics and Operations Research, King Saud University Saudi Arabia, hakim@ksu.edu.sa 2 Dept. of Statistics, Mathematics and Insurance, Benha University Egypt. ahmed_fuzzy@yahoo.com 3 Institute of Statistical Studies & Research, Cairo University, Egypt drhadi2010@gmail.com 4 Dept. of Mathematics, University of Prishtina Hasan Prishtina Prishtinë & University of Mitrovica Isa Boletini Mitrovicë, Kosovo Corresponding author fmerovci@yahoo.com ABSTRACT In this paper, a five parameter Lindley Distribution (FPLD) is proposed as a new generalization of the basic Lindley distribution. The structural properties of the new distribution are investigated. These include the compounding representation of the distribution, reliability analysis and statistical measures. Expressions for Lorenz and Bonferroni curves and Renyi entropy as a measure for uncertainty reduction are derived. Maximum likelihood estimation is used to evaluate the parameters. The new model contains twelve lifetime distributions as special cases such as the Lindley, Quasi Lindley, gamma, and exponential distributions, among others. This model has the advantage of being capable of modeling various shapes of aging and failure criteria. Finally, the usefulness of the new model for modeling reliability data is illustrated using a real data set. KEYWORDS Lindley distribution; mixture; reliability analysis; moment generating function; order statistics; maximum likelihood estimation. 1. INTRODUCTION In many applied sciences such as medicine, engineering and finance, amongst others, modeling and analyzing lifetime data are crucial. Several lifetime distributions have been used to model such kinds of data. The quality of the procedures used in a statistical analysis depends heavily on the assumed probability model or distribution along with relevant statistical methodologies. However, there still remain many important problems where the real data does not follow any of the classical or standard probability models. The one parameter family of distributions with density function (1) 2015 Pakistan Journal of Statistics 363

2 364 The Five Parameter Lindley Distribution is used by Lindley (1958) to illustrate the difference between fiducial distribution and posterior distribution. Sankaran (1970) introduced the discrete Poisson-Lindley distribution by combining the Poisson and Lindley distributions. Ghitany et al. (2008, a) have discussed various properties of (1). Another discrete version of this distribution has been suggested by Deniz and Ojeda (2011) with applications in count data related to insurance. Ghitany et al. (2008 b,c) obtained size-biased and zero-truncated version of Poisson-Lindley distribution with various properties and applications. The aim of this paper is to introduce a new generalization of Lindley (1958) distribution. This generalization is flexible enough to model different types of lifetime data having different forms of failure rate. The new distribution can accommodate both decreasing and increasing failure rates as its antecessors, as well as unimodal and bathtub shaped failure rates. The Lindley distribution is generalized by mixing. Several authors have considered versions from usual density functions by following this idea. For instance, Zakerzadeh and Dolati (2009) considered a generalized Lindley distribution as a generalization of the usual Lindley distribution, Rama and Mishra (2013) studied quasi Lindley distribution, Rama et al. (2013) introduced a new distribution for generalizing the Lindley model which called Janardan distribution, and Elbatal et al. (2013) have suggested a new generalized Lindley distribution. The introduced model will be named Five Parameter Lindley Distribution (FPLD). The basic idea behind this generalization is to use a unified approach that accommodates all the proceeding generalizations of Lindley distribution abovementioned, and to add new model that could offer a better fit to life time data. The proposed distribution includes nine models as special cases plus three new models (all special cases are shown in section 2). The procedure used here is based on certain mixtures of two gamma distributions with various weights. The research examines various properties of the new distribution. The rest of paper is organized as follows: Section 2 introduces the definition of the probability density function (pdf) of the FPLD including its cumulative distribution function (cdf) and the sub-models of the new suggested model. The reliability analysis including the survival function, the hazard (or failure) rate function, the reversed hazard rate function, the cumulative hazard rate function, and the mean residual lifetime is explored in Section 3. The statistical properties of the new distribution such as the moments, the moment generating function, and the distribution of order statistics are investigated in Section 4, with a proposed algorithm for generating random data from the new distribution in this section. Section 5 introduces Lorenz and Bonferroni curves and Renyi entropy as measures of inequality and uncertainty, respectively. Section 6 discusses the estimation of parameters by using maximum likelihood estimation. Finally, Section 7 provides an application for modeling real data sets to illustrate the performance of the new distribution. 2. GENERALIZATION AND RELATED SUB-MODELS In this section, we introduce the pdf and the cdf of the five parameter Lindley distribution and then the special cases of the FPLD are mentioned.

3 Al-Babtain, Eid, Ahmed and Merovci Generalization Let and (2) (3) be two gammaand gamma densities, respectively. We define a new five parameter Lindley distribution as a mixture of (2) and (3) with probabilities and, respectively, as follows: Therefore, the pdf of the Five Parameter Lindley Distribution (FPLD), is defined as * + (4) We note that incorporates five parameters namely; an, subject to an are not allowed to be simultaneously zeros. The corresponding cumulative distribution function (cdf) of the FPLD is [ ] (5) where is known as the lower incomplete gamma function ratio. Also the upper incomplete gamma function ratio is given by Figures (1) and (2) illustrate some of the possible shapes of the pdf and the cdf, respectively, of the FPLD for different values of the parameters an chosen from the ranges specified in Equation (4).

4 366 The Five Parameter Lindley Distribution (a) (b) Figure 1: (a), (b) Different Shapes of the pdf for the FPLD (a)

5 Al-Babtain, Eid, Ahmed and Merovci 367 (b) Figure 2: (a), (b) The Distribution Function (cdf) of the FPLD 2.2 Sub-Models of the FPLD It is clear that the five parameter Lindley distribution is very flexible. Assigning particular numerical values of some subsets of the parameters yields several special generalizations of Lindley distribution. The special cases include nine distributions namely; the new generalized Lindley distribution (NGLD) introduced by Elbatal et al. (2013), generalized Lindley distribution (GLD) introduced by Zakerzadeh and Dolati (2009), quasi Lindley distribution (QLD) introduced by Rama and Mishra (2013), Lindley distribution (LD) by Lindley (1958), Erlang distribution, Janardan distribution introduced by Rama et al. (2013), gamma distribution, the exponential distribution (ED), and Chi-square distribution. In addition to yield all the previous distributions, our generalization model allowed us to create new three distributions namely; 4-parameter Lindley type I (4-p L type I) distribution, 4-parameter Lindley type II (4-p L type II) distribution and 2-parameter Lindley (2-p L) distribution. Table 1 shows the specific values of the parameters used to generate the abovementioned nine special cases, plus the three new models. Distribution Table 1: The Special Cases of the FPL Distribution Parameters Author Gamma Brow & Flood (1947) ED Steffensen (1930) LD Lindley (1958) Erlang A. K. Erlang (1917) QLD 1 2 Rama & Mishra (2013) GLD 1 Zakerzadeh & Dolati (2009) Janardan Rama et al. (2013) NGLD 1 1 Elbatal et al. (2013) Chi-square Fisher (1924) 4-p L type I 1 New 4-p L type II 1 New 2-p L New

6 368 The Five Parameter Lindley Distribution 3. RELIABILITY ANALYSIS In this section, we present the survival function, the hazard rate function, the reversed hazard rate function, the cumulative hazard rate function and the mean residual lifetime for the five parameter Lindley distribution. 3.1 The Survival Function The survival function which is the probability of an item not failing prior to some time is defined by. Therefore, the survival function of the FPLD is given by [ ] (6) Figure (3) illustrates the survival behavior of a FPLD as the values of the parameters are changed. Figure 3: The Reliability (Survival) Function of the FPLD 3.2 The Hazard Rate Function Let be the life time of a device (or a component in a system). Suppose a component follow that has a pdf as in (4). One of the most important characteristics of is its hazard rate function defined by ( ) which provides information about a small interval after time. The hazard rate function for a FPLD can be shown to be [ ] [ ] (7)

7 Al-Babtain, Eid, Ahmed and Merovci 369 We note that might be constant, increasing, decreasing, or bathtub shaped depending on the values of the parameters involved. For example, if and then, a constant, while for and it will be increasing, and it is going to be decreasing if and, and the bathtub-type curve appears for, and. The next result describes some particular cases for the hazard rate function arising from the five parameter Lindley distribution by assigning relevant values of the parameters. Theorem 1: The hazard rate function of the particular cases from the five parameter Lindley distribution are given by (i) If α=k=η=1 and β=2 the failure rate is same as the LD(θ). (ii) If α=1, β=2, and η=θ the failure rate is same as the QLD(k,θ). (iii) If α=1, β=2, and η=0 the failure rate is same as the ED(θ). (iv) If α=k=1, θ=(θ/η), and β=2 the failure rate is same as the JD(θ,η). (v) If k=η=1 the failure rate is same as the NGLD(θ,α,β). (vi) If η=0 the failure rate is same as the Gamma(θ,α). Proof: (i) If and the failure rate is same as the LD (ii) If, and the failure rate is same as the QLD (iii) If, and the failure rate is same as the ED (iv) If, and the failure rate is same as the JD (v) If the failure rate is same as the NGLD [ ] [ ] (vi) If the failure rate is same as the Gamma

8 370 The Five Parameter Lindley Distribution Figure 4 illustrates the behavior of the hazard rate function of the new model to illustrate through its different shapes. (a) (b) Figure 4: (a), (b) Increasing, Decreasing, Constant, Bathtub and Upside-Down Shapes for the Hazard Rate Function of the FPLD 3.3 The Reversed Hazard Rate Function The reversed hazard rate function is the probability of observing an outcome in a neighborhood of, conditional on the outcome being no more than. This is defined as ( ) Therefore, the reversed hazard rate function of a random variable distributed according to the FPLD( ) is given by

9 Al-Babtain, Eid, Ahmed and Merovci 371 * + (8) 3.4 The Cumulative Hazard Rate Function Many generalized models have been proposed in reliability literature through the relationship between the reliability function and its cumulative hazard rate function given by ln. The cumulative hazard rate function of the FPLD is given by ln { [ ]} (9) where is the total number of failure or deaths over an interval of time, and is a non-decreasing function of satisfying; (a), (b). Figure 5 illustrates the behavior of the cumulative hazard rate function of the five parameter Lindley distribution at different values of the parameters. Figure 5: The Cumulative Hazard Rate Function of the FPLD 3.5 The Mean Residual Lifetime The additional lifetime given that the component has survived up to time is called the residual life function of the component, then the expectation of the random variable that represent the remaining lifetime is called the mean residual lifetime (MRL) and is given by or equivalently ( )

10 372 The Five Parameter Lindley Distribution While the hazard rate function provides information about a small interval after time (just after ), the MRL considers information about the whole interval after (all after ). The MRL as well as the hazard rate function or the reliability function is very important as each of them can be used to characterize a unique corresponding life time distribution. where The MRL function m(x) for FPL random variable is given by { [ ]} The MRL function given in Equation (10) satisfies the following properties; (i), (ii) (iv),, (iii), -, and is the first non-central moment of the FPLD (the mean of the distribution). 4. STATISTICAL PROPERTIES This section investigates the statistical properties of the FPLD as the moments (noncentral and central), the moment generating function and an algorithm for random number generating. 4.1 The Moment Generating Function The following theorem gives the moment generating function (mgf) of FPLD. Theorem 2: If X has the FPLD, then the mgf of X say is given as follows (10) { [. / ] } (11) Proof : [ ] * + * +

11 Al-Babtain, Eid, Ahmed and Merovci 373 using the expansion. / one has { [. / ] } This completes the proof. Depending on the previous theorem, we can conclude the basic statistical properties as follows; (i) The non-central moments are the coefficients of In Equation (11), for. Therefore, the mean and the variance of the FPL random variable are, respectively, given by and where (12) (13) is the second non-central moment which is given by (ii) The central moments can be obtained easily from the moments through the relation. / (14) where. Then the central moments of the FPLD are given by. / ( ) (15) (iii) Finally, the coefficient of variation, the coefficient of skewness, and the coefficient of kurtosis of FPLD are, respectively, obtained according the following relations (16), - [ ] (17)

12 374 The Five Parameter Lindley Distribution, - [ ] (18) 4.2 Distribution of Order Statistics Let denote independent random variables from a distribution function with pdf, and then the pdf of (the order sample arrangement) is given by, -, -. (19) Using Equations (4) and (5) into Equation (19), then the pdf of according to the FPLD is given by, * { [ ]} + - { [ ]} (20) Hence, the pdf of the largest order statistic and the smallest order statistic are, respectively, given by, * + - and, * { [ ]} (21) + - { [ ]} (22) 4.3 Random Variates Generation The probability density function of the FPLD can be expressed in terms of the gamma density function as follows To generate random variates, for, from FPL, we can use the following algorithm; 1. Generate,, from Uniform distribution. 2. Generate,, from Gamma.

13 Al-Babtain, Eid, Ahmed and Merovci Generate,, from Gamma. 4. If, then the set of random variates ; otherwise set, 5. MEASURES OF INEQUALITY AND UNCERTAINTY In this section Lorenz and Bonferroni curves are introduced as measures of inequality. Also, Renyi entropy will be mentioned as an important measure of uncertainty. 5.1 Lorenz and Bonferroni Curves Lorenz and Bonferroni curves are the most widely used inequality measures in income and wealth distribution (Dagum, 2004). In fact, Lorenz and Bonferroni curves are depending on the length-biased distribution with pdf defined by where is the pdf of the base distribution with mean E. Accordingly, Lorenz and Bonferroni curves denoted by and defined by (23) respectively, where is the cdf of the length-biased distribution. Now, we shall derive the expressions of and based on and for FPLD. It is easily shown that the pdf of the length-biased distribution can be obtained as follows [ (24) ] (25) with cdf defined by (26) It follows from (12), (24), and (26) that and are and [ ] (27) [ ] [ ] (28)

14 376 The Five Parameter Lindley Distribution 5.2 Renyi Entropy If is a random variable having an absolutely continuous cdf and pdf, then the basic uncertainty measure for distribution (called the entropy of ) is defined as, ln( )-. Statistical entropy is a probabilistic measure of uncertainty or ignorance about the outcome of a random experiment, and is a measure of a reduction in that uncertainty. Abundant entropy and information indices, among them the Renyi entropy, have been developed and used in various disciplines and contexts. Information theoretic principles and methods have become integral parts of probability and statistics and have been applied in various branches of statistics and related fields. Renyi entropy is an extension of Shannon entropy. Renyi entropy of the FPLD is defined to be * + ln. / (29) where an. Renyi entropy tends to Shannon entropy as. Now, ( ) (30) Using. / Then one has ( ) ( ) (31) sin the e pansion one an ha e ( ) ( ) (32) Substitute (32) into (31) and make the necessary simplifications, gives ( ) ( ) (33) Using the gamma function to evaluate the integral in(33) and collecting the entire above evaluations then substitute into (29), the Renyi entropy of the FPLD can be written as

15 Al-Babtain, Eid, Ahmed and Merovci 377 0ln ln 2 ( ) ( ) 31 (34) where is a constant as ESTIMATION OF THE PARAMETERS In this section, we use the method of likelihood to estimate the parameters involved and use them to create confidence intervals for the unknown parameters. Let be a sample size from FPL distribution. Then the likelihood function is given by [ ] then, e p [ ], - [ ] Hence, the log-likelihood function ln becomes (35) ln ln ln ln [ ] (36) Therefore, the maximum likelihood estimators (MLEs) of and are derived from the derivatives of. They should satisfy the following equations [ ] (37)

16 378 The Five Parameter Lindley Distribution [ ] (38) [ ] (39) [ ] (40) [ where is the diagamma function, and it is defined as ] (41) To solve the equations (37) through (41), it is usually more convenient to use nonlinear optimization algorithms such as quasi-newton algorithm to numerically maximize the log-likelihood function. In order to compute the standard errors and asymptotic confidence intervals we use the usual large sample approximation, in which the MLEs can be treated as being approximately trivariate normal. Hence as et al. (2009): ( ) [ ( ) ( where ( ) and, the asymptotic distribution of the MLE is given by, see Zaindin )] is the approximate variance-covariance matrix with its elements obtained from

17 Al-Babtain, Eid, Ahmed and Merovci 379 By solving this inverse dispersion matrix, these solutions will yield the asymptotic variances and covariances of these MLEs for and. Approximate confidence intervals for and can be determined as where is the upper percentile of the standard normal distribution. 7. APPLICATION In this section, we use a real data set to compare the fits of the FPL distribution with three sub-models. In each case, the parameters are estimated by maximum likelihood as described in Section 6, using the R software. The data set consist of uncensored data set from Nichols and Padgett on the breaking stress of carbon fibers (in Gba). The data are given below: 3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 3.56, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 1.57, 2.67, 2.93, 3.22, 3.39, 2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.70, 2.03, 1.89, 2.88, 2.82, 2.05, 3.65, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.35, 2.55, 2.59, 2.03, 1.61, 2.12, 3.15, 1.08, 2.56, 1.80, The summary of the above data is given by Units Minimum 1st Qu. Median Mean 3rd Qu. Maximum In order to compare the two distribution models, we consider criteria like KS (Kolmogorov Smirnov),, AIC (Akaike information criterion), AICC (corrected

18 380 The Five Parameter Lindley Distribution Akaike information criterion), and BIC (Bayesian information criterion) for the data set. The better distribution corresponds to smaller KS,, AIC and AIC C values: I I I BI ( ) where denotes the log-likelihood function evaluated at the maximum likelihood estimates, is the number of parameters, and is the sample size. Also, for calculating the values of KS we use the sample estimates of and. Table 2 shows the parameter estimation based on the maximum likelihood and least square estimation, and gives the values of the criteria AIC, AICC, BIC, and KS test. The values in Table 2 indicate that the FPL distribution leads to a better fit over all the other models. Model Table 2 MLEs, the Measures AIC, AIC C and BIC, and KS Test under Considered Models based on Real Data Parameter Estimates Log L AIC AIC C BIC KS =5.224 FPLD NGLD =6.183 = =0.425*10-3 =0.021 =2.788 =7.412 = Lindley = Exponential = A density plot compares the fitted densities of the models with the empirical histogram of the observed data (Figures 6 and 7). The fitted density for the FPL model is closer to the empirical histogram than the fits of the other models.

19 Al-Babtain, Eid, Ahmed and Merovci 381 (a) (b) Figure 6: (a) Estimated Densities of the FPL, NGL, Lindley and Exponential Distributions for the Data. (b) Estimated cdf Function from the Fitted the FPL, NGL, Lindley and Exponential Distributions and the Empirical cdf for the Data.

20 382 The Five Parameter Lindley Distribution Figure 7: PP plots for the fitted FPLD, NGLD, Lindley distribution and Exponential distribution for the data set. CONCLUDING REMARKS There has been a great interest among statisticians and applied researchers in constructing flexible lifetime models to facilitate better modelling of survival data. Consequently, a significant progress has been made towards the generalization of some well-known lifetime models and their successful application to problems in several areas. In this paper, we introduce a new five-parameter distribution obtained using the idea of mixture of distributions. We refer to the new model as the Five Parameter Lindley Distribution (FPLD) and study some of its mathematical and statistical properties. We provide the pdf, the cdf and the hazard rate function of the new model and explicit expressions for the moments. The model parameters are estimated by the method of maximum likelihood. The new model is compared with three lifetime models and provides consistently better fit than them. We hope that the proposed distribution will

21 Al-Babtain, Eid, Ahmed and Merovci 383 serve as an alternative model to other models available in the literature for modelling positive real data in many areas such as engineering, survival analysis, hydrology and economics. ACKNOWLEDGMENT The authors are grateful to two anonymous referees for careful reading and the Editor for very useful comments and suggestions. This project was supported by King Saud University, Deanship of Scientific Research, Collage of Sciences Research Center. REFERENCES 1. Balakrishnan, K. (1996). Exponential distribution: theory, methods and applications. CRC press. 2. Bozdogan, H. (1987). Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52(3), Brown, G.W. and Flood, M.M. (1947). Tumbler mortality. Journal of the American Statistical Association, 42(240), Dagum, C. (2004). Specification and analysis of wealth distribution models with applications. American Statistical Association, Business and Economic Statistics Section, Toronto, Gómez-Deniz, E. and Calderon-Ojeda, E. (2011). The discrete Lindley distribution: properties and applications. Journal of Statistical Computation and Simulation, 81(11), Elbatal, I., Merovci, F. and Elgarhy, M. (2013). A new generalized Lindley distribution. Mathematical Theory and Modeling, 3(13), Erlang, A.K. (1917). Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges. The Post Office Electrical Engineer s Journal, 10, Ghitany, M.E., Al-Mutairi, D.K. and Nadarajah, S. (2008). Zero-truncated Poisson Lindley distribution and its application. Mathematics and Computers in Simulation, 79(3), Ghitany, M.E., Atieh, B. and Nadarajah, S. (2008). Lindley distribution and its application. Mathematics and computers in simulation, 78(4), Guess, F. and Proschan, F. (1985). Mean residual life: theory and applications (No. FSU-STATISTICS-M702). Dept. of Statistics, Florida State University Tallahassee. 11. Hogg, R.V., McKean, J. and Craig, A.T. (2005). Introduction to mathematical statistics. Pearson Education. 12. Jeffrey, A. and Zwillinger, D. (Eds.). (2007). Table of integrals, series and products. Academic Press. 13. Klein, J.P. and Zhang, M.J. (2005). Survival analysis, software. John Wiley & Sons, Ltd. 14. Lindley, D.V. (1958). Fiducial distributions and Bayes' theorem. Journal of the Royal Statistical Society. Series B (Methodological), Nichols, M.D. and Padgett, W.J. (2006). A bootstrap control chart for Weibull percentiles. Quality and Reliability Engineering International, 22(2),

22 384 The Five Parameter Lindley Distribution 16. Renyi, A. (1961). On measures of entropy and information. In Fourth Berkeley Symposium on Mathematical Statistics and Probability (pp ). 17. Sankaran, M. (1970). The Discrete Poisson-Lindley Distribution. Biometrics, 26, Shanker, R. and Mishra, A. (2013). A quasi Lindley distribution. African Journal of Mathematics and Computer Science Research, 6(4), Shanker, R., Sharma, S., Shanker, U. and Shanker, R. (2013). Janardan distribution and its application to waiting times data. Indian Journal of Applied Research, 3(8), Zakerzadeh, H. and Dolati, A. (2009). Generalized Lindley Distribution. Journal of Mathematical Extension, 3(2),

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