A Appendix

On Lindley-Exponential Distribution: Properties and Application

Abstract

In this paper, we introduce a new distribution generated by Lindley random variable which offers a more flexible model for modelling lifetime data. Various statistical properties like distribution function, survival function, moments, entropy, and limiting distribution of extreme order statistics are established . Inference for a random sample from the proposed distribution is investigated and maximum likelihood estimation method is used for estimating parameters of this distribution. The applicability of the proposed distribution is shown through real data sets.

Keyword: Lindley Distribution, Entropy, Stress-Strength Reliability Model, Maximum Likelihood Estimator.

AMS 2001 Subject Classification: 60E05

1 Introduction

Lifetime distribution represents an attempt to describe, mathematically, the length of the life of a system or a device. Lifetime distributions are most frequently used in the fields like medicine, engineering etc. Many parametric models such as exponential, gamma, Weibull have been frequently used in statistical literature to analyze lifetime data. But there is no clear motivation for the gamma and Weibull distributions. They only have more general mathematical closed form than the exponential distribution with one additional parameter.
Recently, one parameter Lindley distribution has attracted the researchers for its use in modelling lifetime data, and it has been observed in several papers that this distribution has performed excellently. The Lindley distribution was originally proposed by Lindley [11] in the context of Bayesian statistics, as a counter example of fudicial statistics which can be seen that as a mixture of exp() and gamma(2, ). More details on the Lindley distribution can be found in Ghitany et al. [7].
A random variable X is said to have Lindley distribution with parameter if its probability density function is defined as:

(1)

with cumulative distribution function

Some of the advances in the literature of Lindley distribution are given by Ghitany et al. [5] who has introduced a two-parameter weighted Lindley distribution and has pointed that Lindley distribution is particularly useful in modelling biological data from mortality studies. Mahmoudi et. al. [12] have proposed generalized Poisson Lindley distribution. Bakouch et al. [3] have come up with extended Lindley (EL) distribution, Adamidis and Loukas [1] have introduced exponential geometric (EG) distribution. Shanker et. al. [14] have introduced a two-parameter Lindley distribution. Zakerzadeh et al.[15] have proposed a new two parameter lifetime distribution: model and properties. M.K. Hassan [7] has introduced convolution of Lindley distribution. Ghitany et al.[6] worked on the estimation of the reliability of a stress-strength system from power Lindley distribution. Elbatal et al.[4] has proposed a new generalized Lindley distribution.
Ristić [13] has introduced a new family of distributions with survival function given by

In this paper we introduce a new family of distribution generated by a random variable which follows one parameter Lindley distribution. The survival function of this new family is given as:

(2)

where and is a cumulative distribution function(cdf) which we use to generate a new distribution. The cdf is referred to as a transformer and the corresponding probability density function (pdf) is given by

(3)

We consider the transformer to follow exponential distribution with cdf . Hence the survival function of the new distribution is given by

(4)

with corresponding density given by

(5)

We refer the random variable with survival function (4) as Lindley-Exponential(L-E) distribution with parameters and which we denote by L-E().

The aim of this paper is to study the mathematical properties of the L-E distribution and to illustrate its applicability. The contents are organized as follows. The analytical shapes of the pdfin equations (5) are established in section 2. The quantile function presented in section 3. The expressions for the moment generating function and moments corresponding to Equation (5) are given in Section 4. Limiting distribution of sample statistics like maximum and minimum has been shown in section 5. In section 6, entropy of L-E distribution is presented. The maximum likelihood estimation procedure is considered in Section 7. The performance of the maximum likelihood estimators for small samples is assessed by simulation in Section 8. Section 9 gives estimation of stress-strength parameter R by using maximum likelihood estimation method. Finally we conclude the paper by showing applicability of the model to the real data sets.

2 Shape of the density

Here, the shape of pdf (5) follows from theorem 1.

Theorem 1: The probability density function of the L-E distribution is decreasing for and unimodel for . In the latter case, mode is a root of the following equation:

Proof: The first order derivative of is

(6)

where, . For , the function is negative. So for all . This implies that is decreasing for . Also note that, and . This implies that for , has a unique mode at such that for and for . So, is unimodal function with mode at . The pdf for various values of and are shown in Figure 1.

Figure 1: PDF plot for various values of and .

We, now, consider the hazard rate function (hrf) of the L-E distribution, which is given by

(7)

Proposition 1: For the hazard rate function follows relation .

Proof: The proof is straight forward and is omitted.

In Figure 2, hazard function for different values of parameters and .

Figure 2: Hazard function for various values of and .

3 The Quantile Function of L-E distribution

The cdf, , can be obtained by using eq.(4). Further, it can be noted that is continuous and strictly increasing so the quantile function of is , . In the following theorem, we give an explicit expression for in terms of the Lambert function. For more details on Lambert function we refer the reader to Jodrá [9].

Theorem 2: For any , the quantile function of the L-E distribution is

(8)

where denotes the negative branch of the Lambert W function.

Proof: By assuming , the cdf can be written as

for fixed and , the quantile function is obtained by solving . By re-arranging the above, we obtain

taking exponential and multiplying on both sides, we get

(9)

By using definition of Lambert-W function (, where is a complex number), we see that is the Lambert function of the real argument . Thus, we have

(10)

Moreover, for any it is immediate that , and it can also be checked that since . Therefore, by taking into account the properties of the negative branch of the Lambert W function, we have

Also by substituting in cdf and solving it for , we get

(11)

Further the first three quantiles we obtained by substituting in equation (11).

(12)

4 Moments

The moment generating function of the random variable follow L-E distribution is given as

(13)

where, and known as digamma function.
Hence the first and second raw moments can be obtained by and respectively.

(14)
(15)

where is Eulergamma constant =0.577216.

Table 1 displays the mode, mean and median for L-E distribution for different choices of parameter and . It can be observed from the table that all the three measures of central tendency decrease with increase in and increase with an increase in . Also for any choice of and it is observed that Mean Median Mode , which is an indication of positive skewness.

0.1 0.5 1 1.5 2 2.5
1.1 Mode 0.001219 0.000244 0.000122 0.000081 0.000061 0.000049
Mean 7.446760 1.489350 0.744676 0.496451 0.372330 0.297871
Median 4.478034 0.895607 0.447803 0.298536 0.223902 0.179121
1.5 Mode 1.508590 0.301719 0.150850 0.100573 0.075429 0.060344
Mean 9.861580 1.972300 0.986158 0.657438 0.493079 0.394463
Median 6.920488 1.384098 0.692048 0.461366 0.346024 0.276820
2 Mode 4.174510 0.834902 0.417451 0.278300 0.208725 0.166980
Mean 12.367100 2.473421 1.236710 0.824470 0.618350 0.494680
Median 9.528006 1.905601 0.952801 0.635200 0.476400 0.381120
2.5 Mode 6.540190 1.308040 0.654019 0.436013 0.327010 0.261608
Mean 14.444000 2.888800 1.444400 0.962930 0.722200 0.577761
Median 11.704978 2.340996 1.170498 0.780332 0.585249 0.468199
3 Mode 8.569080 1.713820 0.856908 0.571272 0.428454 0.342763
Mean 16.204700 3.240930 1.620470 1.080310 0.810233 0.648186
Median 13.549240 2.709848 1.354924 0.903283 0.677462 0.541970
3.5 Mode 10.317200 2.063400 1.031720 0.687811 0.515858 0.412687
Mean 17.726300 3.545200 1.772630 1.181760 0.886310 0.709053
Median 15.138649 3.027730 1.513865 1.009243 0.756932 0.605546
4 Mode 11.840500 2.368100 1.184050 0.789366 0.592025 0.473620
Mean 19.062700 3.812500 1.906270 1.270850 0.953130 0.762510
Median 16.529903 3.305981 1.652990 1.101994 0.826495 0.661196
Table 1: Mode, Mean and Median for various value of parameter

5 Limiting Distribution of Sample Minima and Maxima

We can derive the asymptotic distribution of the sample minimum by using theorem 8.3.6 of Arnold t. al.[2], it follows that the asymptotic distribution of is Weibull type with shape parameter if

for all . Then, by using L ’Hópital’s rule, it follows that

Since

and

Hence, we obtain that the asymptotic distribution of the sample minima is of the Weibull type with shape parameter .
Further, it can be seen that

by using L-Hópital’s rule,

Since

and

Hence, it follows from Theorem 1.6.2 in Leadbetter et al. (1983) that there must be norming constants , and such that

(16)

and

(17)

as . By following Corollary 1.6.3 in Leadbetter et al. (1983), we can determine the form of the norming constants. As an illustration, one can see that and , where denotes the inverse function of .

6 Entropy

In many field of science such as communication, physics and probability, entropy is an important concept to measure the amount of uncertainty associated with a random variable . Several entropy measures and information indices are available but among them the most popular entropy measure called Rényi entropy is defined as

(18)

In our case

substituting and using power series expansion , the above expression reduces to

(19)

where known as exponential integral function. For more details
see http://functions.wolfram.com/06.34.02.0001.01.
Thus according to (18) the Rényi entropy of L-E distribution is given by

(20)

Moreover, the Shannon entropy is defined by . This is a special case derived from

7 Maximum Likelihood function

In this section we shall discuss the point and interval estimation on the parameters that index the L-E. Let the log-likelihood function of single observation(say ) for the vector of parameter can be written as

The associated score function is given by , where

(21)
(22)

As we know th xpctd value of score function equals zero, i.e. , which implies

The total log-likelihood of the random sample of size from is given by and th total score function is given by , where is the log-likelihood of observation. The maximum likelihood estimator of is obtained by solving equation(21) and (22) numerically or this can also be obtained easily by using nlm() function in R. Moreover the Fisher information matrix is given by

(23)

where

(24)

The above expressions depend on some expectations which easily computed using numerical integration. Under the usual regularity conditions, the asymptotic distribution of

(25)

where . The asymptotic multivariate normal distribution of can be usd to construct approximate confidence intervals. An asymptotic confidence interval with significance level for each parameter and is

(26)

where denotes quantile of standard normal random variable.

8 Simulation

In this section, we investigate the behavior of the ML estimators for a finite sample size (). Simulation study based on different L-E distribution is carried out. The random variable are generated by using cdf technique presented in section 4 from L-E are generated. A simulation study consisting of following steps is being carried out for each triplet , where and .

  1. Choose the initial values of for the corresponding elements of the parameter vector , to specify L-E distribution;

  2. choose sample size ;

  3. generate independent samples of size from L-E;

  4. compute the ML estimate of for each of the samples;

  5. compute the mean of the obtained estimators over all samples, the average bias and the average mean square error , of simulated estimates.

bais bais bais bais bais bais bais bais
20 0.0448 0.1836 0.0476 0.4173 0.0519 0.8777 0.0534 1.1682
50 0.0187 0.0673 0.0196 0.1596 0.0164 0.2694 0.0148 0.3427
75 0.0092 0.0403 0.0131 0.0792 0.0136 0.2166 0.0120 0.2608
100 0.0067 0.0267 0.0091 0.0728 0.0074 0.1318 0.0095 0.1988
20 0.1200 0.1012 0.1073 0.1873 0.1158 0.3884 0.1085 0.5350
50 0.0478 0.0405 0.0412 0.0655 0.0507 0.1723 0.0522 0.2444
75 0.0364 0.0268 0.0265 0.0370 0.0286 0.1091 0.0285 0.1415
100 0.0150 0.0136 0.0198 0.0315 0.0183 0.0548 0.0244 0.1266
20 0.3599 0.0650 0.3270 0.1268 0.3628 0.2334 0.3401 0.3451
50 0.1003 0.0204 0.1100 0.0460 0.1200 0.1021 0.1209 0.1324
75 0.0686 0.0136 0.0654 0.0312 0.0803 0.0653 0.0955 0.1173
100 0.0562 0.0136 0.0457 0.0234 0.0511 0.0417 0.0588 0.0922
Table 2: Average bias of the simulated estimates.
MSE MSE MSE MSE MSE MSE MSE MSE
20 0.0164 0.1988 0.0177 1.0549 0.0204 5.3140 0.0200 8.3846
50 0.0051 0.0459 0.0055 0.2089 0.0053 0.7757 0.0051 1.5790
75 0.0032 0.0270 0.0031 0.1047 0.0033 0.4656 0.0032 0.9631
100 0.0024 0.0161 0.0022 0.0763 0.0022 0.2870 0.0024 0.7081
20 0.1313 0.0802 0.0998 0.2457 0.1012 1.1482 0.1067 2.2693
50 0.0314 0.0200 0.0283 0.0716 0.0302 0.3081 0.0316 0.7259
75 0.0195 0.0119 0.0163 0.0431 0.0178 0.1875 0.0171 0.3556
100 0.0112 0.0078 0.0128 0.0331 0.0111 0.1131 0.0125 0.3082
20 0.8890 0.0344 0.7999 0.1393 0.9890 0.5046 0.9142 1.1439
50 0.1733 0.0097 0.1815 0.0425 0.1633 0.1597 0.1857 0.3790
75 0.1131 0.0063 0.1008 0.0244 0.1075 0.1069 0.1028 0.2132
100 0.0730 0.0046 0.0712 0.0185 0.0730 0.0785 0.0625 0.1586
Table 3: Average MSE of the simulated estimates.

9 Application to Real Datasets

In this section, we illustrate, the applicability of L-E Distribution by considering two different datasets used by different researchers. We also fit L-E distribution, Power-Lindley distribution [6] , New Generalized Lindley Distribution [4], Lindley Distribution, Weibull distribution and Exponential distribution. Namely
(i) Power-Lindley distribution (PL):

(ii) New Generalized Lindley distribution (NGLD()):

(iii) Lindley Distribution (L)

In each of these distributions, the parameters are estimated by using the maximum likelihood method, and for comparison we use negative log-likelihood values (), the Akaike information criterion (AIC) and Bayesian information criterion (BIC) which are defined by and , respectively, where is the number of parameters estimated and is the sample size. Further K-S(Kolmogorov-Smirnov) test statistic defined as , where is empirical distribution function and is cumulative distribution function is calculated and shown for all the datasets.

9.1 Illustration 1

We consider an uncensored data set corresponding to remission times (in months) of a random sample of 128 bladder cancer patients(Lee and Wang[10]) as presented in Appendix A.1.The data sets are presented in appendix A.1 in Table (6). The results for these data are presented in Table 4. We observe that the L-E distribution is a competitive distribution as compared with other distributions. In fact, based on the values of the AIC, BIC and as well as the value of the K-S test statistic, we observe that the L-E distribution provides the best fit for these data among all the models considered. In Figure 2, we have plotted probability density function and empirical distribution function for all considered distributions for these data.

Model Parameter -LL AIC BIC K-S statistic
L-E = 0.0962, =1.229 401.78 807.564 807.780 0.0454
PL =0.385, =0.744 402.24 808.474 808.688 0.0446
L =0.196 419.52 841.040 843.892 0.0740
NGLD =0.180, =4.679, =1.324 412.75 831.501 840.057 0.1160
Table 4: The estimates of parameters and goodness-of-fit statistics for Illustration 1.
Figure 3: PDF plot for various values of and .

9.2 Illustration 2

As second example, we consider 100 observations on waiting time (in minutes) before the customer service in a bank (see Ghitany et al.[7]). The data sets are presented in appendix A.2 in Table (7). The results for these data are presented in Table 5. From these results we can observe that L-E distribution provide smallest AIC and BIC values as compare to Power lindley, new generalized Lindley distribution, Lindley and exponential and hence best fits the data among all the models considered. The results are presented in Table 5 and probability density function and empirical distribution function are shown in Figure 3.

Model Parameter -LL AIC BIC K-S
L-E =2.650; =0.1520 317.005 638.01 638.1337 0.0360
PL =0.1530;=1.0832 318.319 640.64 640.64 0.0520
L =0.187 319.00 640.00 640.00 0.0680
E =0.101 329.00 660.00 660.00 0.1624
NGLD = 0.2033; =2.008; =2.008 317.3 640.60 640.60 0.0425
Table 5: The estimates of parameters and goodness-of-fit statistics for Illustration 2.
Figure 4: PDF plot for various values of and .

10 Estimation of the Stress-Strength Parameter

The stress-strength parameter plays an important role in the reliability analysis as it measures the system performance. Moreover, provides the probability of a system failure, the system fails whenever the applied stress is greater than its strength i.e. . Here L-E denotes the strength of a system subject to stress , and L-E, X and Y are independent of each other. In our case, the stress-strength parameter R is given by

(27)

Remarks:
(i) R is independent of
(ii) When , R=0.5. This is intuitive that X and Y are i.i.d. and there is an equal chance that X is bigger than Y.

Since R in equation (27) is a function of stress-strength parameters and we need to obtain the maximum likelihood estimators (MLEs) of and to compute the MLE of R under invariance property of the MLE. Suppose that and are independent random samples from L-E and L-E respectively. Thus, the likelihood function based on the observed sample is given by

The log - likelihood function is given by

where and .
The MLE of and , say and respectively, can be obtained as the solutions of the following equations

(28)

from above equations

(29)

Hence, using the invariance property of the MLE, the maximum likelihood estimator of can be obtained by substituting for =1,2 in equation (27).

(30)

10.1 Asymptotic Confidence

For an estimator to be asymptotically efficient for estimating for large samples, we should have