Review Article Volume 2 Issue 5
^{1}Department of Statistics, Eritrea Institute of Technology, Eritrea
^{2}Department of Economics, College of Business and Economics, Eritrea
^{3}Department of Banking and Finance, Jimma University, Ethiopia
Correspondence: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea
Received: June 15, 2015  Published: June 25, 2015
Citation: Shanker R, Fesshaye H, Selvaraj S. On modeling of lifetimes data using exponential and lindley distributions. Biom Biostat Int J. 2015;2(5):140147. DOI: 10.15406/bbij.2015.02.00042
In this paper, firstly the nature of exponential and Lindley distributions have been studied using different graphs of their probability density functions and cumulative distribution functions. The expressions for the index of dispersion for both exponential and Lindley distributions have been obtained and the conditions under which the exponential and Lindley distributions are overdispersed, equidispersed, and underdispersed has been given. Several real lifetimes datasets has been fitted using exponential and Lindley distributions for comparative study and it has been shown that in some cases exponential distribution provides better fit than the Lindley distribution whereas in other cases Lindley distribution provides better fit than the exponential distribution.
Keywords: exponential distribution, Lindley distribution; index of dispersion, estimation of parameter, goodness of fit
The time to the occurrence of some event is of interest for some populations of individuals in every field of knowledge. The event may be death of a person, failure of a piece of equipment, development of (or remission) of symptoms, health code violation (or compliance). The times to the occurrences of events are known as “lifetimes” or “survival times” or “failure times” according to the event of interest in the fields of study. The statistical analysis of lifetime data has been a topic of considerable interest to statisticians and research workers in areas such as engineering, medical and biological sciences. Applications of lifetime distributions range from investigations into the endurance of manufactured items in engineering to research involving human diseases in biomedical sciences.
There are a number of continuous distributions for modeling lifetime data such as exponential, Lindley, gamma, lognormal and Weibull. The exponential, Lindley and the Weibull distributions are more popular in practice than the gamma and the lognormal distributions because the survival functions of the gamma and the lognormal distributions cannot be expressed in closed forms and both require numerical integration. Both exponential and Lindley distributions are of one parameter and the Lindley distribution has advantage over the exponential distribution that the exponential distribution has constant hazard rate and mean residual life function whereas the Lindley distribution has increasing hazard rate and decreasing mean residual life function.
In this paper, firstly the nature of exponential and Lindley distribution has been studied by drawing different graphs for probability densities and cumulative distribution functions for the same values of parameter. Several examples of lifetimes datasets from different fields of knowledge has been considered and an attempt has been made to study the goodnessof fit for both exponential and Lindley distributions to see the superiority of one over the other.
Exponential Distribution
The exponential distribution was the first widely used lifetime distribution model in areas ranging from studies on the lifetimes of manufactured items^{13} to research involving survival or remission times in chronic diseases.^{4} The main reason for its wide applicability as lifetime model is partly because of the availability of simple statistical methods for it^{2} and partly because it appeared suitable for representing the lifetimes of many things such as various types of manufactured items.^{1}
Let $T$
be a continuous random variable representing the lifetimes of individuals in some population and following exponential distribution. The probability density function (p.d.f.), cumulative distribution function (c.d.f.), survival function, hazard function, and mean residual life function of $T$
, respectively, are given by
$f\left(t\right)=\theta {e}^{\theta x};\theta >0,t>0$
$F\left(t\right)=1{e}^{\theta t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};\theta >0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0$
$S\left(t\right)=1F\left(t\right)={e}^{\theta t}$
$h\left(t\right)=\frac{f\left(t\right)}{1F\left(t\right)}=\frac{f\left(t\right)}{S\left(t\right)}=\theta $
<$m\left(t\right)=\frac{1}{\theta}$
Lindley distribution
Lindley distribution is a mixture of exponential $\left(\theta \right)$
and gamma $\left(2,\theta \right)$
distributions with mixing proportion $\frac{\theta}{\theta +1}$
and is given by Lindley (1958) in the context of Bayesian Statistics as a counter example of fiducial Statistics. Let $T$
be a continuous random variable representing the lifetimes of individuals in some population and following Lindley distribution. The probability density function (p.d.f.), cumulative distribution function (c.d.f.), survival function, hazard function, and mean residual life function of $T$
, respectively, are given by
$f\left(t\right)=\frac{{\theta}^{2}}{\theta +1}\left(1+t\right){e}^{\theta t};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0$
$F\left(t\right)=1\frac{\theta +1+\theta t}{\theta +1}{e}^{\theta t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};\theta >0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0$
$S\left(t\right)=1F\left(t\right)=\frac{\theta +1+\theta t}{\theta +1}{e}^{\theta t}$
$h\left(t\right)=\frac{f\left(t\right)}{1F\left(t\right)}=\frac{f\left(t\right)}{S\left(t\right)}=\frac{{\theta}^{2}\left(1+t\right)}{\theta +1+\theta t}$
$m\left(t\right)=\frac{\theta +2+\theta t}{\theta \left(\theta +1+\theta t\right)}$
The Lindley distribution has been extensively studied and generalized by many researchers such as^{512} are among others. A discrete version of the Lindley distribution has been obtained by^{13,14} obtained the Lindley mixture of Poisson distribution.
The graphs of the probability densities functions of exponential and Lindley distributions are presented for different values of parameter and shown in Figure 1. The graphs of the cumulative distribution functions of exponential and Lindley distributions are presented for different values of parameter and are shown in Figure 2.
The expressions for coefficient of variation (C.V.), coefficient of Skewness $\left(\sqrt{{\beta}_{1}}\right)$ , coefficient of Kurtosis, and index of dispersion of exponential and Lindley distributions are summarized in the following Table 1. It can be easily verified that the Lindley distribution is over dispersed$\left(\mu <{\sigma}^{2}\right)$ , equidispersed$\left(\mu ={\sigma}^{2}\right)$ and underdispersed$\left(\mu >{\sigma}^{2}\right)$ for $\theta <(=)>{\theta}^{\ast}=1.170086487$ respectively, whereas as exponential distribution is over dispersed$\left(\mu <{\sigma}^{2}\right)$ , equidispersed$\left(\mu ={\sigma}^{2}\right)$ and under dispersed$\left(\mu >{\sigma}^{2}\right)$ for $\theta <(=)>{\theta}^{\ast}=1$ respectively
The exponential and Lindley distribution has been fitted to a number of real lifetime data  sets to tests their goodness of fit. Goodness of fit tests for fifteen real lifetime data sets has been presented here.
In order to compare exponential and Lindley distributions,$2\mathrm{ln}L$
, AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion), KS Statistics ( KolmogorovSmirnov Statistics) for all fifteen real lifetime data sets have been computed. The formulae for computing AIC, AICC, BIC, and KS Statistics are as follows:
$AIC=2\mathrm{ln}L+2k$
,$AICC=AIC+\frac{2k\left(k+1\right)}{\left(nk1\right)}$
, $BIC=2\mathrm{ln}L+k\mathrm{ln}L$
and
$D=\underset{x}{\text{Sup}}\left{F}_{n}\left(x\right){F}_{0}\left(x\right)\right$
, where $k$
the number of parameters, $n$
is the sample size and ${F}_{n}\left(x\right)$
is the empirical distribution function. The best distribution corresponds to lower$2\mathrm{ln}L$
, AIC, AICC, BIC, and KS statistics.
The fittings of exponential and Lindley distributions are based on maximum likelihood estimates (MLE). Let ${t}_{1},{t}_{2},\mathrm{....},{t}_{n}$ be a random sample of size n from exponential distribution. The likelihood function, $L$ and the log likelihood function, $\mathrm{ln}L$ of exponential distribution are given by $L={\theta}^{n}{e}^{n\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\overline{t}}$ and $\mathrm{ln}L=n\mathrm{ln}\theta n\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\overline{t}$ . The MLE $\widehat{\theta}$ of the parameter $\theta $ of exponential distribution is the solution of the equation $\frac{d\mathrm{ln}L}{d\theta}=0$ and is given by $\widehat{\theta}=\frac{1}{\overline{t}}$ , where $\overline{t}$ is the sample mean. Let ${t}_{1},{t}_{2},\mathrm{....},{t}_{n}$ be a random sample of size n from Lindley distribution. The likelihood function, $L$ and the log likelihood function,$\mathrm{ln}L$ of Lindley distribution are given by $L={\left(\frac{{\theta}^{2}}{\theta +1}\right)}^{n}{\displaystyle \prod _{i=1}^{n}\left(1+{t}_{i}\right)}\text{\hspace{0.17em}}{e}^{n\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\overline{t}}$ and $\mathrm{ln}L=n\mathrm{ln}\left(\frac{{\theta}^{2}}{\theta +1}\right)+{\displaystyle \sum _{i=1}^{n}\mathrm{ln}\left(1+{t}_{i}\right)}n\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\overline{t}$ . The MLE $\widehat{\theta}$ of the parameter $\widehat{\theta}$ of Lindley distribution is the solution of the equation $\frac{d\mathrm{ln}L}{d\theta}=0$ and is given by$\widehat{\theta}=\frac{\left(\overline{t}1\right)+\sqrt{{\left(\overline{t}1\right)}^{2}+8\text{\hspace{0.17em}}\overline{t}}}{2\text{\hspace{0.17em}}\overline{t}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{t}>0$ , where $\overline{t}$ is the sample mean. It was shown by [5] showed that the estimator $\widehat{\theta}$ of Lindley distribution is positively biased, consistent and asymptotically normal.
From above table it is obvious that the fittings of Lindley distribution is better than the exponential distribution in Datasets 16,12,14,15. Whereas the fittings of exponential distribution is better than the Lindley distribution in Datasets 711,13 (Table 2).
In this paper we have tried to find the suitability of exponential and Lindley distributions for modeling real lifetimes data. It has been observed that neither exponential distribution nor Lindley distribution is appropriate for modeling real lifetime data in all cases. As per the nature of the data related with overdispersion, equidispersion, and underdispersion, in some cases exponential is better than Lindley while in other cases Lindley is better than exponential. Further, the decision about the suitability of exponential and Lindley for modeling real lifetime data depends on the nature of the data. Of course, Lindley is more flexible than exponential but exponential has some advantage over Lindley due to its simplicity.
Exponential Distribution 
Lindley Distribution 
$C.V.=\frac{\sigma}{{{\mu}^{\prime}}_{1}}=1$ 
$C.V.=\frac{\sigma}{{{\mu}^{\prime}}_{1}}=\frac{\sqrt{{\theta}^{2}+4\theta +2}}{\theta +2}$ 
$\sqrt{{\beta}_{1}}=2$ 
$\sqrt{{\beta}_{1}}=\frac{2\left({\theta}^{3}+6{\theta}^{2}+6\theta +2\right)}{{\left({\theta}^{2}+4\theta +2\right)}^{3/2}}$ 
${\beta}_{2}=9$ 
${\beta}_{2}=\frac{3\left(3{\theta}^{4}+24{\theta}^{3}+44{\theta}^{2}+32\theta +8\right)}{{\left({\theta}^{2}+4\theta +2\right)}^{2}}$ 
Index of dispersion 
Index of dispersion 
Table 1 Index of dispersion of exponential and Lindley distributions
0.55 
0.93 
1.25 
1.36 
1.49 
1.52 
1.58 
1.61 
1.64 
1.68 
1.73 
1.81 
2.00 
0.74 
1.04 
1.27 
1.39 
1.49 
1.53 
1.59 
1.61 
1.66 
1.68 
1.76 
1.82 
2.01 
0.77 
1.11 
1.28 
1.42 
1.50 
1.54 
1.60 
1.62 
1.66 
1.69 
1.76 
1.84 
2.24 
0.81 
1.13 
1.29 
1.48 
1.50 
1.55 
1.61 
1.62 
1.66 
1.70 
1.77 
1.84 
0.84 
1.24 
1.30 
1.48 
1.51 
1.55 
1.61 
1.63 
1.67 
1.70 
1.78 
1.89 


Data Set 1 The data set represents the strength of 1.5cm glass fibers measured at the National Physical Laboratory, England. Unfortunately, the units of measurements are not given in the paper, and they are taken from Smith and Naylor.^{15}
5 
25 
31 
32 
34 
35 
38 
39 
39 
40 
42 
43 
43 
43 
44 
44 
47 
47 
48 
49 
49 
49 
51 
54 
55 
55 
55 
56 
56 
56 
58 
59 
59 
59 
59 
59 
63 
63 
64 
64 
65 
65 
65 
66 
66 
66 
66 
66 
67 
67 
67 
68 
69 
69 
69 
69 
71 
71 
72 
73 
73 
73 
74 
74 
76 
76 
77 
77 
77 
77 
77 
77 
79 
79 
80 
81 
83 
83 
84 
86 
86 
87 
90 
91 
92 
92 
92 
92 
93 
94 
97 
98 
98 
99 
101 
103 
105 
109 
136 
147 




Data Set 2 The data is given by Birnbaum and SaundersM^{16} on the fatigue life of 6061 – T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle 31,000 psi. The data ( $\times $ ${10}^{3}$ ) are presented below (after subtracting 65).
5 
25 
31 
32 
34 
35 
38 
39 
39 
40 
42 
43 
43 
43 
44 
44 
47 
47 
48 
49 
49 
49 
51 
54 
55 
55 
55 
56 
56 
56 
58 
59 
59 
59 
59 
59 
63 
63 
64 
64 
65 
65 
65 
66 
66 
66 
66 
66 
67 
67 
67 
68 
69 
69 
69 
69 
71 
71 
72 
73 
73 
73 
74 
74 
76 
76 
77 
77 
77 
77 
77 
77 
79 
79 
80 
81 
83 
83 
84 
86 
86 
87 
90 
91 
92 
92 
92 
92 
93 
94 
97 
98 
98 
99 
101 
103 
105 
109 
136 
147 




Data Set 3 The data set is from Lawless.^{17} The data given arose in tests on endurance of deep groove ball bearings. The data are the number of million revolutions before failure for each of the 23 ball bearings in the life tests and they are:
17.88 
28.92 
33.00 
41.52 
42.12 
45.60 
48.80 
51.84 
51.96 
54.12 
55.56 
67.80 
68.44 
68.64 
68.88 
84.12 
93.12 
98.64 
105.12 
105.84 
127.92 
128.04 
173.40 

Data Set 4 The data is from Picciotto^{18} and arose in test on the cycle at which the Yarn failed. The data are the number of cycles until failure of the yarn and they are:
86 
146 
251 
653 
98 
249 
400 
292 
131 
169 
175 
176 
76 
264 
15 
364 
195 
262 
88 
264 
157 
220 
42 
321 
180 
198 
38 
20 
61 
121 
282 
224 
149 
180 
325 
250 
196 
90 
229 
166 
38 
337 
65 
151 
341 
40 
40 
135 
597 
246 
211 
180 
93 
315 
353 
571 
124 
279 
81 
186 
497 
182 
423 
185 
229 
400 
338 
290 
398 
71 
246 
185 
188 
568 
55 
55 
61 
244 
20 
284 
393 
396 
203 
829 
239 
236 
286 
194 
277 
143 
198 
264 
105 
203 
124 
137 
135 
350 
193 
188 




Data Set 5 This data represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal.^{19}
12 
15 
22 
24 
24 
32 
32 
33 
34 
38 
38 
43 
44 
48 
52 
53 
54 
54 
55 
56 
57 
58 
58 
59 
60 
60 
60 
60 
61 
62 
63 
65 
65 
67 
68 
70 
70 
72 
73 
75 
76 
76 
81 
83 
84 
85 
87 
91 
95 
96 
98 
99 
109 
110 
121 
127 
129 
131 
143 
146 
146 
175 
175 
211 
233 
258 
258 
263 
297 
341 
341 
376 






Data Set 6 This data is related with behavioral sciences, collected by N Balakrishnan et al.^{20} The scale “General Rating of Affective Symptoms for Preschoolers (GRASP)” measures behavioral and emotional problems of children, which can be classified with depressive condition or not according to this scale. A study conducted by the authors in a city located at the south part of Chile has allowed collecting real data corresponding to the scores of the GRASP scale of children with frequency in parenthesis, which are:
19(16) 
20(15) 
21(14) 
22(9) 
23(12) 
24(10) 
25(6) 

26(9) 
27(8) 
28(5) 
29(6) 
30(4) 
31(3) 
32(4) 

33 
34 
35(4) 
36(2) 
37(2) 
39 
42 
44 
Data Set 7 The data set reported by Efron^{21} represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using radiotherapy (RT).
6.53 
7 
10.42 
14.48 
16.10 
22.70 
34 
41.55 
42 
45.28 
49.40 
53.62 
63 
64 
83 
84 
91 
108 
112 
129 
133 
133 
139 
140 
140 
146 
149 
154 
157 
160 
160 
165 
146 
149 
154 
157 
160 
160 
165 
173 
176 
218 
225 
241 
248 
273 
277 
297 
405 
417 
420 
440 
523 
583 
594 
1101 
1146 
1417 







Data Set 8 The data set reported by Efron^{21} represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using a combination of radiotherapy and chemotherapy (RT+CT).
12.20 
23.56 
23.74 
25.87 
31.98 
37 
41.35 
47.38 
55.46 
58.36 
63.47 
68.46 
78.26 
74.47 
81.43 
84 
92 
94 
110 
112 
119 
127 
130 
133 
140 
146 
155 
159 
173 
179 
194 
195 
209 
249 
281 
319 
339 
432 
469 
519 
633 
725 
817 
1776 








Data Set 9 This data set represents remission times (in months) of a random sample of 128 bladder cancer patients reported in Lee and Wang.^{22}
0.08 
2.09 
3.48 
4.87 
6.94 
8.66 
13.11 
23.63 
0.20 
2.23 
3.52 
4.98 
6.97 
9.02 
13.29 
0.40 
2.26 
3.57 
5.06 
7.09 
9.22 
13.80 
25.74 
0.50 
2.46 
3.64 
5.09 
7.26 
9.47 
14.24 
25.82 
0.51 
2.54 
3.70 
5.17 
7.28 
9.74 
14.76 
6.31 
0.81 
2.62 
3.82 
5.32 
7.32 
10.06 
14.77 
32.15 
2.64 
3.88 
5.32 
7.39 
10.34 
14.83 
34.26 
0.90 
2.69 
4.18 
5.34 
7.59 
10.66 
15.96 
36.66 
1.05 
2.69 
4.23 
5.41 
7.62 
10.75 
16.62 
43.01 
1.19 
2.75 
4.26 
5.41 
7.63 
17.12 
46.12 
1.26 
2.83 
4.33 
5.49 
7.66 
11.25 
17.14 
79.05 
1.35 
2.87 
5.62 
7.87 
11.64 
17.36 
1.40 
3.02 
4.34 
5.71 
7.93 
11.79 
18.10 
1.46 
4.40 
5.85 
8.26 
11.98 
19.13 
1.76 
3.25 
4.50 
6.25 
8.37 
12.02 
2.02 
3.31 
4.51 
6.54 
8.53 
12.03 

20.28 
2.02 
3.36 
6.76 
12.07 
21.73 
2.07 
3.36 
6.93 
8.65 
12.63 
22.69 

Data Set 10 This data set is given by Linhart and Zucchini,^{23} which represents the failure times of the air conditioning system of an airplane:
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 









Data Set 11 This data set used by Bhaumik et al.,^{24} is vinyl chloride data obtained from clean upgradient monitoring wells in mg/l:
0.8 
0.8 
1.3 
1.5 
1.8 
1.9 
1.9 
2.1 
2.6 
2.7 
2.9 
3.1 
3.2 
3.3 
3.5 
3.6 
4.0 
4.1 
4.2 
4.2 
4.3 
4.3 
4.4 
4.4 
4.6 
4.7 
4.7 
4.8 
4.9 
4.9 
5 
5.3 
5.5 
5.7 
5.7 
6.1 
6.2 
6.2 
6.2 
6.3 
6.7 
6.9 
7.1 
7.1 
7.1 
7.1 
7.4 
7.6 
7.7 
8 
8.2 
8.6 
8.6 
8.6 
8.8 
8.8 
8.9 
8.9 
9.5 
9.6 
9.7 
9.8 
10.7 
10.9 
11 
11 
11.1 
11.2 
11.2 
11.5 
11.9 
12.4 
12.5 
12.9 
13 
13.1 
13.3 
13.6 
13.7 
13.9 
14.1 
15.4 
15.4 
17.3 
17.3 
18.1 
18.2 
18.4 
18.9 
19 
19.9 
20.6 
21.3 
21.4 
21.9 
23.0 
27 
31.6 
33.1 
38.5 




Data Set 12 This data set represents the waiting times (in minutes) before service of 100 Bank customers and examined and analyzed by Ghitany et al.^{5} for fitting the Lindley^{25} distribution.
74 
57 
48 
29 
502 
12 
70 
21 
29 
386 
59 
27 
153 
26 
326 











Data Set 13 This data is for the times between successive failures of air conditioning equipment in a Boeing 720 airplane, Proschan:^{26}
1.1 
1.4 
1.3 
1.7 
1.9 
1.8 
1.6 
2.2 
1.7 
2.7 
4.1 
1.8 
1.5 
1.2 
1.4 
3 
1.7 
2.3 
1.6 
2 






Data set 14 This data set represents the lifetime’s data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross and Clark.^{27}
18.83 
20.8 
21.657 
23.03 
23.23 
24.05 
24.321 
25.5 
25.52 
25.8 
26.69 
26.77 
26.78 
27.05 
27.67 
29.9 
31.11 
33.2 
33.73 
33.76 
33.89 
34.76 
35.75 
35.91 
36.98 
37.08 
37.09 
39.58 
44.045 
45.29 
45.381 





Data Set 15 This data set is the strength data of glass of the aircraft window reported by Fuller et al.:^{28}

Model 
Parameter Estimate 
2ln L 
AIC 
AICC 
BIC 
KS Statistic 








Data 1 
Lindley 
0.996116 
162.56 
164.56 
164.62 
166.7 
0.371 
Exponential 
0.663647 
177.66 
179.66 
179.73 
181.8 
0.402 

Data 2 
Lindley 
0.028859 
983.11 
985.11 
985.15 
987.71 
0.242 
Exponential 
0.014635 
1044.87 
1046.87 
1046.91 
1049.48 
0.357 

Data 3 
Lindley 
0.027321 
231.47 
233.47 
233.66 
234.61 
0.149 
Exponential 
0.013845 
242.87 
244.87 
245.06 
246.01 
0.263 

Data 4 
Lindley 
0.00897 
1251.34 
1253.34 
1253.38 
1255.95 
0.098 
Exponential 
0.004505 
1280.52 
1282.52 
1282.56 
1285.12 
0.19 

Data 5 
Lindley 
0.019841 
789.04 
791.04 
791.1 
793.32 
0.133 
Exponential 
0.010018 
806.88 
808.88 
808.94 
811.16 
0.198 

Data 6 
Lindley 
0.077247 
1041.64 
1043.64 
1043.68 
1046.54 
0.448 
Exponential 
0.04006 
1130.26 
1132.26 
1132.29 
1135.16 
0.525 

Data 7 
Lindley 
0.008804 
763.75 
765.75 
765.82 
767.81 
0.245 
Exponential 
0.004421 
744.87 
746.87 
746.94 
748.93 
0.166 

Data 8 
Lindley 
0.00891 
579.16 
581.16 
581.26 
582.95 
0.219 
Exponential 
0.004475 
564.02 
566.02 
566.11 
567.8 
0.145 

Data 9 
Lindley 
0.196045 
839.06 
841.06 
841.09 
843.91 
0.116 
Exponential 
0.106773 
828.68 
830.68 
830.72 
833.54 
0.077 

Data 10 
Lindley 
0.033021 
323.27 
325.27 
325.42 
326.67 
0.345 
Exponential 
0.016779 
305.26 
307.26 
307.4 
308.66 
0.213 

Data 11 
Lindley 
0.823821 
112.61 
114.61 
114.73 
116.13 
0.133 
Exponential 
0.532081 
110.91 
112.91 
113.03 
114.43 
0.089 

Data 12 
Lindley 
0.186571 
638.07 
640.07 
640.12 
642.68 
0.058 
Exponential 
0.101245 
658.04 
660.04 
660.08 
662.65 
0.163 

Data 13 
Lindley 
0.01636 
181.34 
183.34 
183.65 
184.05 
0.386 
Exponential 
0.008246 
173.94 
175.94 
176.25 
176.65 
0.277 

Data 14 
Lindley 
0.816118 
60.5 
62.5 
62.72 
63.49 
0.341 
Exponential 
0.526316 
65.67 
67.67 
67.9 
68.67 
0.389 

Data 15 
Lindley 
0.062988 
253.99 
255.99 
256.13 
257.42 
0.333 
Exponential 
0.032455 
274.53 
276.53 
276.67 
277.96 
0.426 
Table 2 MLE’s, 2ln L, AIC, AICC, BIC, KS Statistics, and pvalues of the fitted distributions of data sets 115
None.
Author declares that there are no conflicts of interest.
©2015 Shanker, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work noncommercially.
2 7