Some Properties of the Size-Biased Janardan Distribution

Janardan Distribution is one of the important distributions from lifetime models and it has many applications in real life data. A size-biased form of the two parameter Janardan distribution has been introduced in this paper, of which the size-biased Lindley distribution is a special case. Its moments, median, skewness, kurtosis and Fisher index of dispersion are derived and compared with the size-biased Lindley distribution. The shape of the size-biased Janardan distribution is also discussed with graphs. The survival function and hazard rate of the size-biased Janardan distribution have been derived and it is concluded that the hazard rate of the distribution is monotonically increasing. The flexibility in the reliability measures of the size-biased Janardan distribution have been discussed by stochastic ordering. To estimate the parameters of the size-biased Janardan distribution maximum likelihood equations are developed.


Introduction
Size-biased distributions are the special cases of the weighted distributions. [6] introduced the weighted distributions to model ascertainment bias and later was discussed by [13]. [11] & [12] discussed the applications of weighted distributions and size biased sampling in real life. These distributions arise in practice when observations from a sample are recorded with unequal probability and provide a unifying approach for the problems where the observations fall in the non-experimental, non-replicate, and non-random categories. If the random variable X has the probability distributions function (pdf), ( ) α α = = we get the size-biased and area-biased distributions respectively. [3] proposed a weighted Lindley distribution by using a new weight function. Various properties of the model have been derived and the shape of the hazard rate is also discussed. [1] derived size-biased gamma distribution (SBGMD). They derived the characterizing properties of the SBGMD including Shannon entropy and Fisher's information matrix. They also derived Baye's estimator of the SBGMD using different priors. [5] examined the size-biased versions of the generalized beta of the first kind, generalized beta of the second kind and generalized gamma distributions. They discussed broader applications of the size-biased distributions in forestry sampling, modeling and analysis. [2] derived sizebiased Pareto distribution and discussed upper record values of the size-biased Pareto distribution. They proposed some recurrence relations satisfied by the single and product moments of upper record values form size-biased Pareto distribution.
[17] derived size-biased Poisson Lindley distribution (SBPLD) and its moments. They estimated parameter of the SBPLD and apply the model on thunderstorms. They concluded that the size-biased Poisson Lindley distribution (SBPLD) gives much closer fit than the sizebiased Poisson distribution (SBPD). [10] derived some size-biased probability distributions and their generalizations. These distributions provide a unifying approach for the problems where the observations fall in the non-experimental, non-replicated, and nonrandom categories.
[9] introduced one parameter Lindley distribution (LD) as ; , It is observed that for 1 α = , the SBJD (5) approaches to size-biased Lindley distribution (SBLD) with probability density function Some basic measures (moments, skewness and kurtosis) of the SBJD (5) and SBLD (6), are given in the following table   Table 1. Some measures of the SBJD and SBLD.

Reliability Measures of the SBJD
The survival function of the SBJD is

Fig. 4. Graph of the survival function for SBJD for different values of parameters.
The hazard rate function of the SBJD is Then suppose the derivative of ( ) It shows that hazard function of the SBJD is monotonically increasing (IFR)

Stochastic Ordering
A random variable X is said to be smaller than a random variable Y in the [14] considered the following results for establishing stochastic ordering of distributions  1 2 ; , log 0 ; , It means that This theorem shows the flexibility of the SBJD in the context of reliability measures (stochastic ordering, hazard rate ordering, mean residual ordering and likelihood ratio ordering).

Estimation of Parameters
Maximum Likelihood Estimates (MLE): Let 1 2 , , , n x x x ⋯ be random samples from the size-biased Janardan distribution in (2.1) then the likelihood estimates function of the SBJD is The equations (23) and (24) cannot be solved directly. However in order to solve these equations we derive the derivatives ( )