A GENERALIZATION OF SUJATHA DISTRIBUTION AND ITS APPLICATIONS WITH REAL LIFETIME DATA

A two-parameter generalization of Sujatha distribution (AGSD), which includes Lindley distribution and Sujatha distribution as particular cases, has been proposed. It's important mathematical and statistical properties including its shape for varying values of parameters, moments, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, and stress-strength reliability have been discussed. Maximum likelihood estimation method has been discussed for estimating its parameters. AGSD provides better fit than Sujatha, Aradhana, Lindley and exponential distributions for modeling real lifetime data.


INTRODUCTION
The probability density function (PDF) and the cumulative distribution function (CDF) of Lindley (1958)  respectively.Detailed study on 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. (2008).Shanker et al. (2015) had critically studied the modeling of lifetime data using exponential and Lindley distributions and concluded that there are several lifetime data where these distributions are not suitable from theoretical or applied point of view.In recent years much work have been done on Lindley distribution, its mixture with other distributions, extensions, and generalizations by many researchers including Zakerzadeh and Dolati (2009), Nadarajah et al. (2011), Deniz and Ojeda (2011), Bakouch et al. (2012), Shanker andMishra (2013 a, 2013 b), Shanker and Amanuel (2013), Shanker et al. (2015), Ghitany et al. (2013), Shanker et al. (2016Shanker et al. ( a, 2016Shanker et al. ( b, 2016 Shanker ( 2016 Shanker (2016 c) has discussed its important properties including its shape for varying values of parameters, moments, coefficient of variation, skewness , kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, and stress-strength reliability.Shanker (2016 c) has discussed the maximum likelihood estimation of parameter and showed the applications of Sujatha distribution to model lifetime data from biomedical science and engineer.Shanker (2016 d) has also obtained a Poisson mixture of Sujatha distribution namely Poisson-Sujatha distribution (PSD) and studied its properties, estimation of parameter and applications for count data.Shanker andHagos (2016, 2015) have obtained and discussed the size-biased and zero-truncated Poisson-Sujatha distribution and their various statistical and mathematical properties, estimation of parameter and applications to model count data which structurally exclude zero-counts.Recently, Shanker (2016 e) has introduced a quasi Sujatha distribution (QSD), for modeling lifetime data from biomedical science and engineering.In this paper, a generalization of Sujatha distribution (AGSD), of which one parameter Lindley (1958) distribution and Sujatha distribution introduced by Shanker (2016 c) are particular cases, has been proposed.It's important properties including hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, stress-strength reliability have also been discussed.The estimation of the parameters has been discussed using maximum likelihood estimation.Some numerical examples have been given to test the goodness of fit of AGSD and the fit has been compared with one parameter Sujatha, Aradhana, Lindley and exponential distributions.

A GENERALIZATION OF SUJATHA DISTRIBUTION
The probability density function of a generalization of Sujatha distribution (AGSD) can be introduced as: where  is a scale parameter and  is a shape parameter.It can be easily verified that (2.1) reduces to the one parameter Lindley (1958) distribution (1.1) and Sujatha distribution (1.3), introduced by Shanker (2016 c) for 0   and 1   .It can be easily shown that AGSD is a three-component mixture of exponential    , gamma   2, and gamma   3, distributions.We have Since AGSD includes Lindley and Sujatha distributions as particular cases, it is expected to give better fit than both Lindley and Sujatha distributions for modeling lifetime data from biological sciences and engineering.
Further, for  , the PDF of AGSD (2.1) reduces to the PDF of gamma distribution with shape parameter 3 and scale parameter  .
The corresponding cumulative distribution function of (2.1) can be obtained as:

MOMENTS
The r th moment about origin, r   of AGSD (2.1) can be obtained as: The first four moments about origin of AGSD (2.1) are thus obtained as: Using the relationship between moments about mean and the moments about origin, the moments about mean of AGSD (2.1) are obtained as To study the nature of C.V, 1  , 2  and  , their values for varying values of the parameters  and  have been computed and presented in tables 1,2,3 and 4.
The corresponding hazard rate function,   hxand the mean residual life function,   mxof AGSD (2.1) are thus obtained as:

STOCHASTIC ORDERINGS
Stochastic ordering of positive continuous random variables is an important tool for judging the comparative behaviour of continuous distributions.
A random variable X is said to be smaller than a random variable Y in the (i) The following results due to Shaked and Shanthikumar (1994) are well known for establishing stochastic ordering of distributions The AGSD (2.1) is ordered with respect to the strongest 'likelihood ratio' ordering as shown in the following theorem: .

MEAN DEVIATIONS
The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and the median.These are known as the mean deviation about the mean and the mean deviation about the median and are defined as: , respectively, where,  can be calculated using the following relationships Using PDF (2.1) and expression for the mean of AGSD (2.1), we get Using expressions from (6.1), (6.2), (6.3), and (6.4) and after some mathematical simplifications, the mean deviation about the mean,  

BONFERRONI AND LORENZ CURVES AND INDICES
The Bonferroni and Lorenz curves (Bonferroni, 1930) and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medical science.The Bonferroni and Lorenz curves are defined as: respectively or equivalently respectively, where The Bonferroni and Gini indices are thus defined as: respectively.Using PDF of AGSD (2.1), we get 3 2 1 2 3 1 6 ;, 2 q q q q q q q q e x f x dx Now using equation (7.7) in (7.1) and ( 7.2), we get Now using equations (7.8) and (7.9) in (7.5) and (7.6), the Bonferroni and Gini indices of AGSD (2.1) are thus obtained as:

STRESS-STRENGTH RELIABILITY
The stress-strength reliability of a component illustrates the life of the component which has random strength X that is subjected to a random stress Y .When the stress   Then, the stress-strength reliability R of AGSD (2.1) can be obtained as: It can be easily verified that the above expression reduces to the corresponding expression of Sujatha distribution introduced by Shanker (2016 b) at 12 1   .

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS
Let   1 2 3 , , , ... , n x x x x be random sample from AGSD (2.1).The likelihood function, L is given by


The natural log likelihood function is thus obtained as where x is the sample mean.
The maximum likelihood estimates (MLEs)  and  of  and  are then the solutions of the following non-linear equations These two natural log likelihood equations do not seem to be solved directly.However, the Fisher's scoring method can be applied to solve these equations.We have The following equations can be solved for MLEs  and  of  and  of AGSD (2.1) where 0  and 0  are the initial values of  and  , respectively.These equations are solved iteratively till sufficiently close values of  and  are obtained.

APPLICATIONS AND GOODNESS OF FIT
The goodness of fit of a generalization Sujatha distribution (AGSD) using maximum likelihood estimation has been discussed with four real lifetime data sets and the fit has been compared with one parameter Sujatha, Aradhana, Lindley and exponential distributions.Fuller et al. (1994): 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 4: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm (Bader and Priest, 1982) The density (1.1) is a two-component mixture of an exponential distribution having scale parameter  and a gamma distribution having shape parameter 2 and scale parameter  with their c) has shown that the Sujatha distribution is a three-component mixture of an exponential distribution having scale parameter  , a gamma distribution having shape parameter 2 and scale parameter  , and a gamma distribution with shape parameter 3 and scale parameter  Fig. 1 (a) Graphs of the PDF of AGSD for varying values of parameters  and  .

Fig
Fig. 2 (a) Graphs of   hx of AGSD for selected values of parameters  and  .

Table 1 . C.V of AGSD for varying values of parameters θ and α.
For a given value of  , C.V increases as the value of  increases.Again for a given value of  

Table 2 .
1 of AGSD for varying values of parameters θ and α.

Table 3 .
  , AGSD is always leptokurtic, which means that AGSD is more peaked than the normal curve.
2 of AGSD for varying values of parameters θ and α.

Table 4 . of AGSD for varying values of parameters θ and α
HAZARD RATE FUNCTION AND MEAN RESIDUAL LIFE FUNCTIONLet X be a continuous random variable with PDF   fxand CDF  Fx.The hazard rate function (also known as the failure rate function),   hx and the mean residual life function,   mxof X are respectively defined as Smith and Naylor (1987) lifetime data sets have been considered for the goodness of fit of considered distributions 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 fromSmith and Naylor (1987)This data set represents the lifetime's data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported byGross and Clark (1975, P. 105).This data set is the strength data of glass of the aircraft window reported by : 1.312 1.314 1.479 1.552 1.700 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 2.140 2.179 2.224 2.240 2.253 2.270 2.272 2.274 2.301 2.301 2.359 2.382 2.382 2.426 2.434 2.435 2.478 2.490 2.511 2.514 2.535 2.554 2.566 2.570 2.586 2.629 2.633 2.642 2.648 2.684 2.697 2.726 2.770 2.773 2.800 2.809 2.818 2.821 2.848 2.880 2.954 3.012 3.067 3.084 3.090 3.096 3.128 3.233 3.433 3.585 3.585 In order to compare the goodness of fit of AGSD, Sujatha, Aradhana, Lindley and exponential distributions, 2ln L  , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected) and BIC (Bayesian Information Criterion) of distributions for four real lifetime data sets have been computed and presented in table 5.The formulae for computing AIC, AICC and BIC are as follows: , where k = the number of parameters, n  the sample size.Table 5 .MLE's 2ln L  , AIC, AICC and BIC of the fitted distributions of data sets 1, 2, 3 and 4.