On modeling of lifetime data using one parameter akash, lindley and exponential distributions

doi:10.15406/bbij.2016.03.00061

eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 3 Issue 2

On modeling of lifetime data using one parameter akash, lindley and exponential distributions

Rama Shanker,¹

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Hagos Fesshaye,² Sujatha Selvaraj³

¹Department of Statistics, Eritrea Institute of Technology, Eritrea
²Department of Economics, College of Business and Economics, Eritrea
³Department of Banking and Finance, Jimma University, Ethiopia

Correspondence: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea

Received: December 02, 2015 | Published: January 28, 2016

Citation: Shanker R, Fesshaye H, Selvaraj S. On modeling of lifetime data using one parameter akash, lindley and exponential distributions. Biom Biostat Int J. 2016;3(2):56-62. DOI: 10.15406/bbij.2016.03.00061

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Abstract

The analysis and modeling of lifetime data are crucial in almost all applied sciences including medicine, insurance, engineering, and finance, amongst others. In the present paper an attempt has been made to discuss applications of Akash distribution introduced by Shanker,¹ Lindley distribution and exponential distributions for modeling lifetime data from various fields. Firstly a table for values of the various characteristics of Akash distribution and Lindley distribution has been presented for various values of their parameter which reflects their nature and behavior. The expressions for the index of dispersion of Akash, Lindley and exponential distributions have been obtained and the conditions under which Akash, Lindley and exponential distributions are over-dispersed, equi-dispersed, and under-dispersed has been given. Several lifetime data from medical science and engineering have been fitted using Akash distribution along with Lindley and exponential distributions to study the advantages and disadvantages of these distributions for modeling lifetime data.

Keywords: akash distribution, lindley distribution, exponential distribution, index of dispersion, estimation of parameter, goodness of fit

Introduction

The time to the occurrence of event of interest is known as lifetime or survival time or failure time in reliability analysis. The event may be failure of a piece of equipment, death of a person, development (or remission) of symptoms of disease, health code violation (or compliance). The modeling and statistical analysis of lifetime data are crucial for statisticians and research workers in almost all applied sciences including engineering, medical science/biological science, insurance and finance, amongst others.

Recently Shanker¹ has introduced a one parameter continuous distribution named, “Akash distribution” for modeling lifetime data from engineering and medical science and studied its various mathematical properties, estimation of its parameter, and its applications. A number of continuous distributions for modeling lifetime data have been introduced in statistical literature including exponential, Lindley, gamma, lognormal and Weibull, amongst others. 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. Though Akash, Lindley and exponential distributions are of one parameter, Akash and Lindley distributions have advantage over the exponential distribution that the exponential distribution has constant hazard rate and mean residual life function whereas the Akash and Lindley distributions have increasing hazard rate and decreasing mean residual life function. Further, Akash distribution of Shanker¹ has flexibility over both Lindley and exponential distributions.

Exponential, Lindley and Akash distributions

Exponential distribution

In statistical literature, exponential distribution was the first widely used lifetime distribution model in areas ranging from studies on the lifetimes of manufactured items Davis,² Epstein & Sobel,³ Epstein⁴ to research involving survival or remission times in chronic diseases Feigl & Zelen.⁵ The main reason for its wide usefulness and applicability as lifetime model is partly because of the availability of simple statistical methods for it Epstein & Sobel³ and partly because it appeared suitable for representing the lifetimes of many things such as various types of manufactured items Davis.²

Lindley distribution

The Lindley distribution is a two-component mixture of an exponential distribution having scale parameter $θ$ $θ$ and a gamma distribution having shape parameter 2 and scale parameter $θ$ $θ$ with mixing proportions $\frac{θ}{θ + 1}$ $\frac{θ}{θ + 1}$ and $\frac{1}{θ + 1}$ $\frac{1}{θ + 1}$ and is given by Lindley⁶ in the context of Bayesian Statistics as a counter example of fiducial Statistics. A detailed study about its various mathematical properties, estimation of parameter and application showing the superiority of Lindley distribution over exponential distribution for the waiting times before service of the bank customers has been done by Ghitany et al.⁷ The Lindley distribution has been generalized, extended, modified and its detailed applications in reliability and other fields of knowledge by different researchers including Hussain,⁸ Zakerzadeh & Dolati,⁹ Nadarajah et al.,¹⁰ Deniz & Ojeda,¹¹ Bakouch et al.,¹² Shanker & Mishra,^13,14 Shanker et al.,¹⁵ Elbatal et al.,¹⁶ Ghitany et al.,¹⁷ Merovci,¹⁸ Liyanage & Pararai,¹⁹ Ashour & Eltehiwy,²⁰ Oluyede & Yang,²¹ Singh et al.²² Sharma et al.²³ Shanker et al.,²⁴ Alkarni,²⁵ Pararai et al.,²⁶ Abouammoh et al.²⁷ are some among others.

Although the Lindley distribution has been used to model lifetime data by many researchers and Hussain⁸ has shown that the Lindley distribution is important for studying stress-strength reliability modeling, it has been observed that there are many situations in the modeling of lifetime data where the Lindley distribution may not be suitable from a theoretical or applied point of view. In fact, Shanker et al.²⁴ has detailed comparative study about the applicability of Lindley and exponential distributions for modeling various types of lifetime data and observed that none is a suitable model in all cases.

Akash distribution

Shanker¹ introduced a new distribution named, ‘Akash distribution’ which is flexible than the Lindley distribution for modeling lifetime data in reliability and in terms of its hazard rate shapes. Akash distribution is a two- component mixture of an exponential distribution having scale parameter $θ$ $θ$ and a gamma distribution having shape parameter 3 and scale parameter $θ$ $θ$ with mixing proportions $\frac{θ^{2}}{θ^{2} + 2}$ $\frac{θ^{2}}{θ^{2} + 2}$ and $\frac{1}{θ^{2} + 2}$ $\frac{1}{θ^{2} + 2}$ and has been shown by Shanker¹ that Akash distribution gives better fit than Lindley and exponential distributions in modeling lifetime data.

Let $T$ $T$ be a continuous random variable representing the lifetimes of individuals in some population. The expressions for probability density function, $f (t)$ $f (t)$ , cumulative distribution function, $F (t)$ $F (t)$ , survival function, $S (t)$ $S (t)$ , hazard rate function, $h (t)$ $h (t)$ , mean residual life function, $m (t)$ $m (t)$ , mean $μ_{1}^{'}$ $μ_{1}^{'}$ , variance $μ_{2}$ $μ_{2}$ , third moment about mean $μ_{3}$ $μ_{3}$ , fourth moment about mean $μ_{4}$ $μ_{4}$ , coefficient of variation (C.V.), coefficient of Skewness $(\sqrt{β_{1}})$ $(\sqrt{β_{1}})$ , coefficient of Kurtosis $(β_{2})$ $(β_{2})$ , and index of dispersion $(γ)$ $(γ)$ of exponential, Lindley and Akash distributions are summarized in the following .

Exponential distribution	Lindley distribution	Akash distribution
$f (t) = θ e^{- θ t}$ $f (t) = θ e^{- θ t}$	$f (t) = \frac{θ^{2}}{θ + 1} (1 + t) e^{- θ t}$ $f (t) = \frac{θ^{2}}{θ + 1} (1 + t) e^{- θ t}$	$f (t) = \frac{θ^{3}}{θ^{2} + 2} (1 + t^{2}) e^{- θ t}$ $f (t) = \frac{θ^{3}}{θ^{2} + 2} (1 + t^{2}) e^{- θ t}$
$F (t) = 1 - e^{- θ t}$ $F (t) = 1 - e^{- θ t}$	$F (t) = 1 - \frac{θ + 1 + θ t}{θ + 1} e^{- θ t}$ $F (t) = 1 - \frac{θ + 1 + θ t}{θ + 1} e^{- θ t}$	$F (t) = 1 - [1 + \frac{θ t (θ t + 2)}{θ^{2} + 2}] e^{- θ t}$ $F (t) = 1 - [1 + \frac{θ t (θ t + 2)}{θ^{2} + 2}] e^{- θ t}$
$S (t) = e^{- θ t}$ $S (t) = e^{- θ t}$	$S (t) = \frac{θ + 1 + θ t}{θ + 1} e^{- θ t}$ $S (t) = \frac{θ + 1 + θ t}{θ + 1} e^{- θ t}$	$S (t) = [1 + \frac{θ t (θ t + 2)}{θ^{2} + 2}] e^{- θ t}$ $S (t) = [1 + \frac{θ t (θ t + 2)}{θ^{2} + 2}] e^{- θ t}$
$h (t) = θ$ $h (t) = θ$	$h (t) = \frac{θ^{2} (1 + t)}{θ + 1 + θ t}$ $h (t) = \frac{θ^{2} (1 + t)}{θ + 1 + θ t}$	$h (t) = \frac{θ^{3} (1 + t^{2})}{θ t (θ t + 2) + (θ^{2} + 2)}$ $h (t) = \frac{θ^{3} (1 + t^{2})}{θ t (θ t + 2) + (θ^{2} + 2)}$
$m (t) = \frac{1}{θ}$ $m (t) = \frac{1}{θ}$	$m (t) = \frac{θ + 2 + θ t}{θ (θ + 1 + θ t)}$ $m (t) = \frac{θ + 2 + θ t}{θ (θ + 1 + θ t)}$	$m (t) = \frac{θ^{2} t^{2} + 4 θ t + (θ^{2} + 6)}{θ [θ t (θ t + 2) + (θ^{2} + 2)]}$ $m (t) = \frac{θ^{2} t^{2} + 4 θ t + (θ^{2} + 6)}{θ [θ t (θ t + 2) + (θ^{2} + 2)]}$
$μ_{1}^{'} = \frac{1}{θ}$ $μ_{1}^{'} = \frac{1}{θ}$	$μ_{1}^{'} = \frac{θ + 2}{θ (θ + 1)}$ $μ_{1}^{'} = \frac{θ + 2}{θ (θ + 1)}$	$μ_{1}^{'} = \frac{θ^{2} + 6}{θ (θ^{2} + 2)}$ $μ_{1}^{'} = \frac{θ^{2} + 6}{θ (θ^{2} + 2)}$
$μ_{2} = \frac{1}{θ^{2}}$ $μ_{2} = \frac{1}{θ^{2}}$	$μ_{2} = \frac{θ^{2} + 4 θ + 2}{θ^{2} {(θ + 1)}^{2}}$ $μ_{2} = \frac{θ^{2} + 4 θ + 2}{θ^{2} {(θ + 1)}^{2}}$	$μ_{2} = \frac{θ^{4} + 16 θ^{2} + 12}{θ^{2} {(θ^{2} + 2)}^{2}}$ $μ_{2} = \frac{θ^{4} + 16 θ^{2} + 12}{θ^{2} {(θ^{2} + 2)}^{2}}$
$μ_{3} = \frac{2}{θ^{3}}$ $μ_{3} = \frac{2}{θ^{3}}$	$μ_{3} = \frac{2 (θ^{3} + 6 θ^{2} + 6 θ + 2)}{θ^{3} {(θ + 1)}^{3}}$ $μ_{3} = \frac{2 (θ^{3} + 6 θ^{2} + 6 θ + 2)}{θ^{3} {(θ + 1)}^{3}}$	$μ_{3} = \frac{2 (θ^{6} + 30 θ^{4} + 36 θ^{2} + 24)}{θ^{3} {(θ^{2} + 2)}^{3}}$ $μ_{3} = \frac{2 (θ^{6} + 30 θ^{4} + 36 θ^{2} + 24)}{θ^{3} {(θ^{2} + 2)}^{3}}$
$μ_{4} = \frac{9}{θ^{4}}$ $μ_{4} = \frac{9}{θ^{4}}$	$μ_{4} = \frac{3 (\begin{matrix} 3 θ^{4} + 24 θ^{3} + 44 θ^{2} + 32 θ + 8 \end{matrix})}{θ^{4} {(θ + 1)}^{4}}$ $μ_{4} = \frac{3 (\begin{array}{l} 3 θ^{4} + 24 θ^{3} + 44 θ^{2} \\ + 32 θ + 8 \end{array})}{θ^{4} {(θ + 1)}^{4}}$	$μ_{4} = \frac{3 (\begin{matrix} 3 θ^{8} + 128 θ^{6} + 408 θ^{4} + 576 θ^{2} + 240 \end{matrix})}{θ^{4} {(θ^{2} + 2)}^{4}}$ $μ_{4} = \frac{3 (\begin{array}{l} 3 θ^{8} + 128 θ^{6} + 408 θ^{4} \\ + 576 θ^{2} + 240 \end{array})}{θ^{4} {(θ^{2} + 2)}^{4}}$
$C . V = \frac{σ}{{μ^{'}}_{1}^{'}} = 1$ $C . V = \frac{σ}{{μ^{'}}_{1}} = 1$	$C . V = \frac{σ}{{μ^{'}}_{1}^{'}} = \frac{\sqrt{θ^{2} + 4 θ + 2}}{θ + 2}$ $C . V = \frac{σ}{{μ^{'}}_{1}} = \frac{\sqrt{θ^{2} + 4 θ + 2}}{θ + 2}$	$C . V = \frac{σ}{μ_{1}^{'}} = \frac{\sqrt{θ^{4} + 16 θ^{2} + 12}}{θ^{2} + 6}$ $C . V = \frac{σ}{μ_{1}^{'}} = \frac{\sqrt{θ^{4} + 16 θ^{2} + 12}}{θ^{2} + 6}$
$\sqrt{β_{1}} = 2$ $\sqrt{β_{1}} = 2$	$\sqrt{β_{1}} = \frac{2 (θ^{3} + 6 θ^{2} + 6 θ + 2)}{{(θ^{2} + 4 θ + 2)}^{3 / 2}}$ $\sqrt{β_{1}} = \frac{2 (θ^{3} + 6 θ^{2} + 6 θ + 2)}{{(θ^{2} + 4 θ + 2)}^{3 / 2}}$	$\sqrt{β_{1}} = \frac{2 (θ^{6} + 30 θ^{4} + 36 θ^{2} + 24)}{{(θ^{4} + 16 θ^{2} + 12)}^{3 / 2}}$ $\sqrt{β_{1}} = \frac{2 (θ^{6} + 30 θ^{4} + 36 θ^{2} + 24)}{{(θ^{4} + 16 θ^{2} + 12)}^{3 / 2}}$
$β_{2} = 9$ $β_{2} = 9$	$β_{2} = \frac{3 (\begin{matrix} 3 θ^{4} + 24 θ^{3} + 44 θ^{2} + 32 θ + 8 \end{matrix})}{{(θ^{2} + 4 θ + 2)}^{2}}$ $β_{2} = \frac{3 (\begin{array}{l} 3 θ^{4} + 24 θ^{3} + 44 θ^{2} \\ + 32 θ + 8 \end{array})}{{(θ^{2} + 4 θ + 2)}^{2}}$	$β_{2} = \frac{3 (\begin{matrix} 3 θ^{8} + 128 θ^{6} + 408 θ^{4} + 576 θ^{2} + 240 \end{matrix})}{{(θ^{4} + 16 θ^{2} + 12)}^{2}}$ $β_{2} = \frac{3 (\begin{array}{l} 3 θ^{8} + 128 θ^{6} + 408 θ^{4} \\ + 576 θ^{2} + 240 \end{array})}{{(θ^{4} + 16 θ^{2} + 12)}^{2}}$
$γ = \frac{σ^{2}}{μ_{1}^{'}} = \frac{1}{θ}$ $γ = \frac{σ^{2}}{μ_{1}^{'}} = \frac{1}{θ}$	$γ = \frac{σ^{2}}{μ_{1}^{'}} = \frac{θ^{2} + 4 θ + 2}{θ (θ + 1) (θ + 2)}$ $γ = \frac{σ^{2}}{μ_{1}^{'}} = \frac{θ^{2} + 4 θ + 2}{θ (θ + 1) (θ + 2)}$	$γ = \frac{σ^{2}}{μ_{1}^{'}} = \frac{θ^{4} + 16 θ^{2} + 12}{θ (θ^{2} + 2) (θ^{2} + 6)}$ $γ = \frac{σ^{2}}{μ_{1}^{'}} = \frac{θ^{4} + 16 θ^{2} + 12}{θ (θ^{2} + 2) (θ^{2} + 6)}$

Table 1 Characteristics of Exponential, Lindley and Akash Distributions

It can be easily verified that the Akash distribution is over- dispersed $(μ < σ^{2})$ $(μ < σ^{2})$ , equi-dispersed $(μ = σ^{2})$ $(μ = σ^{2})$ and under-dispersed $(μ > σ^{2})$ $(μ > σ^{2})$ for $θ < (=) > θ^{*} = 1.515400063$ $θ < (=) > θ^{*} = 1.515400063$ respectively. Further, Lindley distribution is over- dispersed $(μ < σ^{2})$ $(μ < σ^{2})$ , equi-dispersed $(μ = σ^{2})$ $(μ = σ^{2})$ and under-dispersed $(μ > σ^{2})$ $(μ > σ^{2})$ for $θ < (=) > θ^{*} = 1.170086487$ $θ < (=) > θ^{*} = 1.170086487$ respectively, whereas as exponential distribution is over- dispersed $(μ < σ^{2})$ $(μ < σ^{2})$ , equi-dispersed $(μ = σ^{2})$ $(μ = σ^{2})$ and under- dispersed $(μ > σ^{2})$ $(μ > σ^{2})$ for $θ < (=) > θ^{*} = 1$ $θ < (=) > θ^{*} = 1$ respectively.

A table of values for coefficient of variation (C.V.), coefficient of Skewness $(\sqrt{β_{1}})$ $(\sqrt{β_{1}})$ , coefficient of Kurtosis $(β_{2})$ $(β_{2})$ , and index of dispersion $(γ)$ $(γ)$ for Akash and Lindley distributions for various values of their parameter for comparative study are summarized in the Table 2.

Values of $θ$ $θ$ for Akash Distribution
	0.01	0.05	0.09	0.5	0.8	1.5	2
C.V	0.577379	0.578071	0.579679	0.641249	0.716741	0.882958	0.959166
$\sqrt{β_{1}}$ $\sqrt{β_{1}}$	1.154643	1.153268	1.150133	1.083974	1.10564	1.388077	1.61372
$β_{2}$ $β_{2}$	4.999867	4.996681	4.989352	4.784948	4.735717	5.472724	6.391304
$γ$ $γ$	100.0067	20.03328	11.17079	2.284444	1.615097	1.008913	0.766667
Values of $θ$ $θ$ for Lindley Distribution
	0.01	0.05	0.09	0.5	0.8	1.5	2
C.V	0.710607	0.723943	0.736298	0.824621	0.863075	0.914732	0.935414
$\sqrt{β_{1}}$ $\sqrt{β_{1}}$	1.414317	1.416546	1.421076	1.512281	1.580387	1.698866	1.756288
$β_{2}$ $β_{2}$	6.000294	6.006807	6.020488	6.342561	6.621505	7.172516	7.469388
$γ$ $γ$	100.4926	20.46458	11.55007	2.266667	1.448413	0.780952	0.583333

Table 2 Values of C.V $\sqrt{β_{1}}$ $\sqrt{β_{1}}$ , $β_{2}$ $β_{2}$ and $γ$ $γ$ of Akash and Lindley Distributions for varying values of the parameter $θ$ $θ$

Applications

The Akash, Lindley and exponential distributions have been fitted to a number of real lifetime data - sets to tests their goodness of fit. Goodness of fit tests for sixteen real lifetime data- sets have been presented here. In order to compare Akash, Lindley and exponential distributions, $- 2 ln L$ $- 2 \ln L$ , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion), K-S Statistics ( Kolmogorov-Smirnov Statistics) for all sixteen real lifetime data- sets have been computed Table 3. The formulae for computing AIC, AICC, BIC, and K-S Statistics are as follows:

	Model	Parameter Estimate	-2ln L	AIC	AICC	BIC	K-S Statistic
Data 1	Akash Lindley	1.355445 0.996116	163.73 162.56	165.73 164.56	165.79 164.62	169.93 166.70	0.355 0.371
	Exponential	0.663647	177.66	179.66	179.73	181.80	0.402
Data 2	Akash Lindley	0.043876 0.028859	950.97 983.11	952.97 985.11	953.01 985.15	955.58 987.71	0.184 0.242
	Exponential	0.014635	1044.87	1046.87	1046.91	1049.48	0.357
Data 3	Akash Lindley	0.04151 0.027321	227.06 231.47	229.06 233.47	229.25 233.66	230.20 234.61	0.107 0.149
	Exponential	0.013845	242.87	244.87	245.06	246.01	0.263
Data 4	Akash Lindley	0.013514 0.00897	1255.83 1251.34	1257.83 1253.34	1257.87 1253.38	1260.43 1255.95	0.071 0.098
	Exponential	0.004505	1280.52	1282.52	1282.56	1285.12	0.190
Data 5	Akash Lindley	0.030045 0.019841	794.70 789.04	796.70 791.04	796.76 791.10	798.98 793.32	0.184 0.133
	Exponential	0.010018	806.88	808.88	808.94	811.16	0.198
Data 6	Akash Lindley	0.11961 0.077247	981.28 1041.64	983.28 1043.64	983.31 1043.68	986.18 1046.54	0.393 0.448
	Exponential	0.04006	1130.26	1132.26	1132.29	1135.16	0.525
Data 7	Akash Lindley	0.013263 0.008804	803.96 763.75	805.96 765.75	806.02 765.82	810.01 767.81	0.298 0.245
	Exponential	0.004421	744.87	746.87	746.94	748.93	0.166
Data 8	Akash Lindley	0.013423 0.00891	609.93 579.16	611.93 581.16	612.02 581.26	613.71 582.95	0.280 0.219
	Exponential	0.004475	564.02	566.02	566.11	567.80	0.145
Data 9	Akash Lindley	0.3105 0.196045	887.89 839.06	889.89 841.06	889.92 841.09	892.74 843.91	0.198 0.116
	Exponential	0.106773	828.68	830.68	830.72	833.54	0.077
Data 10	Akash Lindley	0.050293 0.033021	354.88 323.27	356.88 325.27	357.02 325.42	358.28 326.67	0.421 0.345
	Exponential	0.016779	305.26	307.26	307.40	308.66	0.213
Data 11	Akash Lindley	1.165719 0.823821	115.15 112.61	117.15 114.61	117.28 114.73	118.68 116.13	0.156 0.133
	Exponential	0.532081	110.91	112.91	113.03	114.43	0.089
Data 12	Akash Lindley	0.295277 0.186571	641.93 638.07	643.93 640.07	643.95 640.12	646.51 642.68	0.100 0.058
	Exponential	0.101245	658.04	660.04	660.08	662.65	0.163
Data 13	Akash Lindley	0.024734 0.01636	194.30 181.34	196.30 183.34	196.61 183.65	197.01 184.05	0.456 0.386
	Exponential	0.008246	173.94	175.94	176.25	176.65	0.277
Data 14	Akash Lindley	1.156923 0.816118	59.52 60.50	61.52 62.50	61.74 62.72	62.51 63.49	0.320 0.341
	Exponential	0.526316	65.67	67.67	67.90	68.67	0.389
Data 15	Akash Lindley Exponential	0.097062 0.062988 0.032455	240.68 253.99 274.53	242.68 255.99 276.53	242.82 256.13 276.67	244.11 257.42 277.96	0.266 0.333 0.426
Data 16	Akash Lindley Exponential	0.964726 0.659000 0.407941	224.28 238.38 261.74	226.28 240.38 263.74	226.34 240.44 263.80	228.51 242.61 265.97	0.348 0.390 0.434

Table 3 MLE’s, -2ln L, AIC, AICC, BIC, K-S Statistics of the fitted distributions of data sets 1-16

$A I C = - 2 ln L + 2 k$ $A I C = - 2 \ln L + 2 k$ , $A I C C = A I C + \frac{2 k (k + 1)}{(n - k - 1)}$ $A I C C = A I C + \frac{2 k (k + 1)}{(n - k - 1)}$ , $B I C = - 2 ln L + k ln n$ $B I C = - 2 \ln L + k \ln n$ and $D = Sup x | F_{n} (x) - F_{0} (x) |$ $D = \underset{x}{Sup} | F_{n} (x) - F_{0} (x) |$ , where $k$ $k$ the number of parameters, $n$ $n$ is the sample size and $F_{n} (x)$ $F_{n} (x)$ is the empirical distribution function. The best distribution corresponds to lower values of $- 2 ln L$ $- 2 \ln L$ , AIC, AICC, BIC, and K-S statistics. The fittings of Akash, Lindley and exponential distributions are based on maximum likelihood estimates (MLE).

Let $t_{1}, t_{2}, ...., t_{n}$ $t_{1}, t_{2}, ...., t_{n}$ be a random sample of size n from exponential distribution. The likelihood function, $L$ $L$ and the log likelihood function, $ln L$ $\ln L$ of exponential distribution are given by $L = θ^{n} e^{- n θ ¯ t}$ $L = θ^{n} e^{- n θ \bar{t}}$ and $ln L = n ln θ - n θ ¯ t$ $\ln L = n \ln θ - n θ \bar{t}$ . The MLE $ˆ θ$ $\hat{θ}$ of the parameter $θ$ $θ$ of exponential distribution is the solution of the equation $\frac{d ln L}{d θ} = 0$ $\frac{d \ln L}{d θ} = 0$ and is given by $ˆ θ = \frac{1}{¯ t}$ $\hat{θ} = \frac{1}{\bar{t}}$ , where $¯ t$ $\bar{t}$ is the sample mean.

Let $t_{1}, t_{2}, ...., t_{n}$ $t_{1}, t_{2}, ...., t_{n}$ be a random sample of size n from Lindley distribution. The likelihood function, $L$ $L$ and the log likelihood function, $ln L$ $\ln L$ of Lindley distribution are given by $L = {(\frac{θ^{2}}{θ + 1})}^{n} n \prod i = 1 (1 + t_{i}) e^{- n θ ¯ t}$ $L = {(\frac{θ^{2}}{θ + 1})}^{n} \prod_{i = 1}^{n} (1 + t_{i}) e^{- n θ \bar{t}}$ and $ln L = n ln (\frac{θ^{2}}{θ + 1}) + n \sum i = 1 ln (1 + t_{i}) - n θ ¯ t$ $\ln L = n \ln (\frac{θ^{2}}{θ + 1}) + \sum_{i = 1}^{n} \ln (1 + t_{i}) - n θ \bar{t}$ . The MLE $\hat{θ}$ of the parameter $θ$ of Lindley distribution is the solution of the equation $\frac{d \ln L}{d θ} = 0$ and is given by $\hat{θ} = \frac{- (\bar{t} - 1) + \sqrt{{(\bar{t} - 1)}^{2} + 8 \bar{t}}}{2 \bar{t}}; \bar{t} > 0$ , where $\bar{t}$ is the sample mean.

Let $t_{1}, t_{2}, ...., t_{n}$ be a random sample of size n from Akash distribution. The likelihood function, $L$ and the log likelihood function, $\ln L$ of Akash distribution are given by $L = {(\frac{θ^{3}}{θ^{2} + 2})}^{n} \prod_{i = 1}^{n} (1 + t_{i}^{2}) e^{- n θ \bar{t}}$ and $\ln L = n \ln (\frac{θ^{3}}{θ^{2} + 2}) + \sum_{i = 1}^{n} \ln (1 + t_{i}^{2}) - n θ \bar{t}$ . The MLE $\hat{θ}$ of the parameter $θ$ of Akash distribution is the solution of the equation $\frac{d \ln L}{d θ} = 0$ is the solution of following non-linear equation $\bar{t} θ^{3} - θ^{2} + 2 \bar{t} θ - 6 = 0$ , where $\bar{t}$ is the sample mean.

It is obvious from the goodness of fit of Akash, Lindley and exponential distributions that the Akash distribution provides better fit than the Lindley and exponential distributions in data-sets 2, 3, 6, 14, 15, and 16; the Lindley distribution gives better fit than the exponential and Akash distributions in data-sets 1, 4, 5 and 12; the exponential distribution gives better fit than the Lindley and the Akash distributions in data sets 7, 8, 9, 10, 11, and 13.

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
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	1.5	1.54	1.6	1.62	1.66	1.69	1.76	1.84	2.24
0.81	1.13	1.29	1.48	1.5	1.55	1.61	1.62	1.66	1.7	1.77	1.84	0.84
1.24	1.3	1.48	1.51	1.55	1.61	1.63	1.67	1.7	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 & Naylor.²⁸

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 & Saunders²⁹ 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).

17.88	28.92	33	41.5	42.12	45.6	48.8	51.8	52	54.12	55.56	67.8
68.44	68.64	68.9	84.1	93.12	98.6	105	106	128	128	173.4

Data Set 3: The data set is from Lawless.³⁰ 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.

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 4: The data is from Picciotto ³¹ 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.

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 5: This data represents the survival times (in days) of 72 guinna pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal.³²

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 6: This data is related with behavioral sciences, collected by Balakrishnan N, Victor Leiva & Antonio Sanhueza:³³ 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.

6.53	7	10.42	14.48	16.1	22.7	34	41.55	42	45.28	49.4	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 7: The data set reported by Efron³⁴ represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using radiotherapy (RT).

12.2	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 8: The data set reported by Efron³⁴ 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).

0.08	2.09	3.48	4.87	6.94	8.66	13.11	23.63	0.2	2.23	3.52	4.98	6.97
9.02	13.29	0.4	2.26	3.57	5.06	7.09	9.22	13.8	25.74	0.5	2.46	3.64
5.09	7.26	9.47	14.24	25.82	0.51	2.54	3.7	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	0.9	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
1.4	3.02	4.34	5.71	7.93	11.79	18.1	1.46	4.4	5.85	8.26	11.98	19.13
1.76	3.25	4.5	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 9: This data set represents remission times (in months) of a random sample of 128 bladder cancer patients reported in Lee & Wang.³⁵

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 10: This data set is given by Linhart & Zucchini [36], which represents the failure times of the air conditioning system of an airplane.

5.1	1.2	1.3	0.6	0.5	2.4	0.5	1.1	8	0.8	0.4	0.6	0.9
0.4	2	0.5	5.3	3.2	2.7	2.9	2.5	2.3	1	0.2	0.1	0.1
1.8	0.9	2	4	6.8	1.2	0.4	0.2

Data Set 11: This data set used by Bhaumik et al.,³⁷ is vinyl chloride data obtained from clean up gradient 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,	5.0,	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,	8.0,	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,	11.0,
11.0,	11.1,	11.2,	11.2,	11.5,	11.9,	12.4,	12.5,	12.9,	13.0,	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,	19.0,	19.9,
20.6,	21.3,	21.4,	21.9,	23.0,	27.0,	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.⁷ for fitting the Lindley⁶ 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.³⁸

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 & Clark.³⁹

18.83	20.8	21.66	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	44.05	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 [40].

1.312	1.314	1.479	1.552	1.7	1.803	1.861	1.865	1.944	1.958	1.966	1.997
	2.006	2.021	2.027	2.055	2.063	2.098	2.14	2.179	2.224	2.24	2.253
	2.27	2.272	2.274	2.301	2.301	2.359	2.382	2.382	2.426	2.434	2.435
	2.478	2.49	2.511	2.514	2.535	2.554	2.566	2.57	2.586	2.629	2.633
	2.642	2.648	2.684	2.697	2.726	2.77	2.773	2.8	2.809	2.818	2.821
	2.848	2.88	2.954	3.012	3.067	3.084	3.09	3.096	3.128	3.233	3.433
	3.585	3.858

Data Set 16: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm.^41-44

Acknowledgments

None.

Conflicts of interest

None.

References

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Davis DJ. An analysis of some failure data. Journal of American Statistical Association. 1952;47(258):113−150.
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Hussain E. The non-linear functions of Order Statistics and Their Properties in selected probability models. Pakistan; 2006.
Zakerzadeh H, Dolati A. Generalized Lindley distribution. Journal of Mathematical extension. 2009;3(2):13−25.
Nadarajah S, Bakouch HS, Tahmasbi R. A generalized Lindley distribution. Sankhya Series. 2011;73(2):331−359.
Deniz EG, Ojeda EC. The discrete Lindley distribution-Properties and Applications. Journal of Statistical Computation and Simulation. 2011;81(11):1405−1416.
Bakouch SH, Al-Zahrani BM, Al-Shomrani AA, et al. An extended Lindley distribution. Journal of Korean Statistical Society. 2012;41(1):75−85.
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Ghitany M, Al-Mutairi D, Balakrishnan N, et al. Power Lindley distribution and associated inference. Computational Statistics & Data Analysis. 2013;64:20−33.
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Singh SK, Singh U, Sharma VK. The Truncated Lindley distribution-inference and Application. Journal of Statistics Applications & Probability. 2014;3(2):219−228.
Sharma V, Singh S, Singh U, et al. The inverse Lindley distribution-A stress-strength reliability model with applications to head and neck cancer data. Journal of Industrial & Production Engineering. 2015;32(3):162−173.
Shanker R, Hagos F, Sujatha S. On modeling of Lifetimes data using exponential and Lindley distributions. Biometrics & Biostatistics International Journal. 2015;2(5):1−9.
Alkarni S. Extended Power Lindley distribution-A new Statistical model for non-monotone survival data. European journal of statistics and probability. 2015;3(3):19−34.
Pararai M, Liyanage GW, Oluyede BO. A new class of generalized Power Lindley distribution with applications to lifetime data. Theoretical Mathematics & Applications. 2015;5(1):53−96.
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Smith RL, Naylor JC. A comparison of Maximum likelihood and Bayesian estimators for the three parameter Weibull distribution. Applied Statistics. 1987;36(3):358−369.
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Lawless JF. Statistical models and methods for lifetime data. John Wiley and Sons, New York, USA; 1982.
Picciotto R. Tensile fatigue characteristics of a sized polyester/viscose yarn and their effect on weaving performance. North Carolina State, University at Raleigh, USA; 1970.
Bjerkedal T. Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. Am J Hyg. 1960;72(1):130−148.
Balakrishnan N, Victor L, Antonio S. A mixture model based on Birnhaum-Saunders Distributions, A study conducted by Authors regarding the Scores of the GRASP (General Rating of Affective Symptoms for Preschoolers), in a city located at South Part of the Chile. 2010.
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Lee ET, Wang JW. Statistical methods for survival data analysis. 3rd ed, USA: John Wiley and Sons, New York; 2003.
Linhart H, Zucchini W. Model Selection. USA: John Wiley, New York; 1986.
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Proschan F. Theoretical explanation of observed decreasing failure rate. Technometrics. 1963;5(3):375−383.
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Fuller EJ, Frieman S, Quinn J, et al. Fracture mechanics approach to the design of glass aircraft windows: A case study. SPIE Proc. 1994;2286:419−430.
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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

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

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

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

5.1	1.2	1.3	0.6	0.5	2.4	0.5	1.1	8	0.8	0.4	0.6	0.9
0.4	2	0.5	5.3	3.2	2.7	2.9	2.5	2.3	1	0.2	0.1	0.1
1.8	0.9	2	4	6.8	1.2	0.4	0.2

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

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

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

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

5.1	1.2	1.3	0.6	0.5	2.4	0.5	1.1	8	0.8	0.4	0.6	0.9
0.4	2	0.5	5.3	3.2	2.7	2.9	2.5	2.3	1	0.2	0.1	0.1
1.8	0.9	2	4	6.8	1.2	0.4	0.2

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Biometrics & Biostatistics International Journal

On modeling of lifetime data using one parameter akash, lindley and exponential distributions

Abstract

Introduction

Exponential, Lindley and Akash distributions

Applications

Acknowledgments

Conflicts of interest

References

Citations

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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

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

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

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

5.1	1.2	1.3	0.6	0.5	2.4	0.5	1.1	8	0.8	0.4	0.6	0.9
0.4	2	0.5	5.3	3.2	2.7	2.9	2.5	2.3	1	0.2	0.1	0.1
1.8	0.9	2	4	6.8	1.2	0.4	0.2