Janardan Distribution and its Application to Waiting Times Data

A two parameter continuous distribution named “Janardan distribution (JD)”, of which the Lindley distribution (LD) is a particular case, has been introduced. Its moments, failure rate function, mean residual life function and stochastic orderings have been discussed. The maximum likelihood method and the method of moments have been discussed for estimating its parameters. The superiority of the proposed distribution has been illustrated with an application to a real data set.

In this paper, a two parameter continuous distribution named "Janardan distribution (JD)", of which the Lindley distribution (LD) (1.1) is a particular case, has been suggested. Its first four moments and some of the related measures have been obtained. Its failure rate, mean residual rate and stochastic ordering have also been studied. Estimation of its parameters has been discussed and the distribution has been fitted to some of those data sets where the Lindley distribution has earlier been fitted by others, to test its goodness of fit.

JANARDAN DISTRIBUTION
A two parameter continuous distribution with parameters α and θ is defined by its probability density function (p.d.f.) ; , 1 We would call this two parameter continuous distribution as "Janardan distribution (JD)". It can easily be seen that Lindley distribution (LD) is a particular case of (2.

MOMENTS AND RELATED MEASURES
The rth moment about origin of the Janardan distribution has been obtained as

ReseaRch PaPeR
Taking 1, 2, 3 r = and 4 in (3.1), the first four moments about origin of Janardan distribution are thus obtained as It can be easily verified that for 1 α = , the moments about origin of the Janardan distribution reduce to the respective moments of the Lindley distribution. Further, the mean of the distribution is always greater than the mode, the distribution is positively skewed.

FAILURE RATE AND MEAN RESIDUAL LIFE
The corresponding failure rate function, ( ) h x and the mean residual life function, ( ) m x of the Janardan distribution are thus given by It can be easily verified that .It is also obvious that ( ) h x is an increasing function of x , α and θ , whereas ( ) m x is a decreasing function of x , αandθ . For A random variable X is said to be smaller than a random variable Y in the The following results due to Shaked and Shanthikumar (1994) are well known for establishing stochastic ordering of distributions The Janardan distribution is ordered with respect to the strongest 'likelihood ratio' ordering as shown in the following theorem: Theorem: Let X ∼ Janardan distribution( )  This theorem shows the flexibility of the Janardan distribution over Lindley and exponential distributions. , which is the mean of the Janardan distribution. The two equations (6.1.3) and (6.1.4) do not seem to be solved directly. However, the Fisher's scoring method can be applied to solve these equations. For, we have ( ) The following equations for θ and α can be solved  , we finally get

APPLICATION
The Janardan distribution has been fitted to a number of data-sets to which earlier the Lindley distribution has been fitted by others and to almost all these data-sets the Ja-nardan distribution provides closer fits than the Lindley distribution.
Here the fittings of the Janardan distribution to data-set relating to waiting times (in minutes) of 100 bank customers reported by Ghitany et al (2008) have been presented in the following table. The expected frequencies according to the Lindley distribution have also been given for ready comparison with those obtained by the Janardan distribution. The estimates of the parameters have been obtained by the method of moments.
It can be seen from table 1 that the Janardan distribution gives much closer fits than the Lindley distribution and thus provides a better alternative to the Lindley distribution for modeling waiting times data. In this paper, a two-parameter continuous distribution, named "Janardan distribution (JD)", of which the Lindley distribution (LD) is a particular case, has been proposed. Several properties of the Janardana distribution such as moments, failure rate function, mean residual life function, stochastic orderings, estimation of parameters by the method of maximum likelihood and the method of moments have been discussed. Finally, an application of the proposed distribution has been given by fitting to data sets relating to waiting times to test its goodness of fit to which earlier the LD has been fitted and it is found that the Janardan distribution provides better fits than those by the LD. An application to a real data set indicates that the fit of the proposed distribution is superior to the fit of the Lindley distribution and we hope that the proposed distribution may be interesting for a wider range of statistics research.