A new generalization of quasi Amarendra distribution with some characterizations and its applications

Weighted Quasi Amarendra (WQA) Distribution

The probability density function of quasi Amarendra distribution is given by

Consider X be the non-negative random variable with probability density function ) (x f .Let ) (x w be its non-negative weight function, then the probability density function of weighted random variable w X is given by )) ( ( For various choices of weight function w(x) especially if w(x) = x c , proposed distribution is termed as weighted distribution. In this paper, we have to introduce the weighted version of quasi Amarendra distribution called as weighted quasi Amarendra distribution. So, the weight function considered at w(x) = x c , resulting distribution is called weighted distribution and its probability density function is given by

Now substituting equations ( 1) and (4) in equation ( 3), we will get the required probability density function of weighted quasi Amarendra distribution.

 

And the cumulative distribution function of weighted quasi Amarendra distribution can be determined as

After the simplification of equation ( 6), we will obtain the cumulative distribution function of weighted quasi Amarendra distribution as

Reliability measures

In this section, we will discuss about the reliability function, hazard rate function, reverse hazard rate function and mills ratio of the proposed weighted quasi Amarendra distribution.

Reliability function

The reliability function or survival function of weighted quasi Amarendra distribution can be determined as

Hazard function

The hazard function is also known as hazard rate or failure rate and is given by

Reverse hazard function

The reverse hazard rate function is given by.

Now substituting equations ( 5) and (7) in equation ( 8), we will get the required probability density function of r th order statistics of weighted quasi Amarendra distribution.

Therefore for r = n, we will obtain the probability density function of higher order statistic X(n) of weighted quasi Amarendra distribution as

And for r = 1, we will obtain the probability density function of first order statistic X(1) of weighted quasi Amarendra distribution as

Statistical Properties

In this section, we will discuss about the various statistical properties of weighted quasi Amarendra distribution that include moments, harmonic mean, moment generating function and characteristic function.

Moments

Consider X be the random variable following weighted quasi Amarendra distribution with parameters α, θ and c, then the r th order moment E(X r ) of executed distribution can be determined as

After the simplification of equation ( 9), we obtain )

Now putting r = 1, 2, 3 and 4 in equation ( 10), we will obtain the first four moments of weighted quasi Amarendra distribution as

Harmonic mean

The harmonic mean for proposed weighted quasi Amarendra distribution can be determined as

After simplification, we obtain from equation ( 11)

Moment generating function and characteristic function

Consider X be the random variable following weighted quasi Amarendra distribution with parameters α, θ and c, then the moment generating function of introduced distribution can be determined as

Bonferroni and Lorenz Curves

The bonferroni and Lorenz curves are also known as classical curves are mostly being used to measure the inequality in income or how wealth is distributed. The bonferroni and Lorenz curves are given by.

Maximum Likelihood Estimation and Fisher's Information Matrix

In this section, we will discuss the technique of maximum likelihood estimation to estimate the parameters of weighted quasi Amarendra distribution. Consider X1, X2, …, Xn be the random sample of size n drawn from the weighted quasi Amarendra distribution, then the likelihood function can be defined as.

Now by differentiating the log likelihood equation (12) with respect to parameters α, θ and c and must satisfy the following normal equations 0 ) (

Where ψ (.) is the digamma function. Because of complicated form of above likelihood equations, algebraically it is very difficult to solve the system of nonlinear equations. Therefore, we use the technique like Newton-Raphson for estimating the parameters of introduced distribution. In order to use the asymptotic normality result for obtaining the obtain confidence interval.

Application

In this section, we have analyzed and inspected a real lifetime data set for fitting weighted quasi Amarendra distribution in order to show that weighted quasi Amarendra distribution provides a better fit over quasi Amarendra and Amarendra distributions. The following real data set is given below as The following real data set given below in table 1 represents the strength of 1.5cm glass fibres measured at the National physical laboratory England reported by Smith and Naylor (1987) [12] and the data set is given below as.

Table 1: Data represents the strength of 1.5cm glass fibres reported by Smith and Naylor (1987) [12] 0.55 To compute the model comparison criterions along with the estimation of unknown parameters, the technique of R software is applied. In order to compare the performance of weighted quasi Amarendra distribution over quasi Amarendra and Amarendra distributions, we use the criterion values AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), AICC (Akaike Information Criterion Corrected) and -2logL. The distribution is better which shows lesser the corresponding criterion values of AIC, BIC, AICC and -2logL. For determining the criterion values like AIC, BIC, AICC and -2logL following formulas are used. Where k is the number of parameters in the statistical model, n is the sample size and -2logL is the maximized value of the loglikelihood function under the considered model. From results given above in table 2, it has been clearly realized that the weighted quasi Amarendra distribution has the lesser AIC, BIC, AICC and -2logL values as compared to the quasi Amarendra and Amarendra distributions. Hence, it can be revealed that the weighted quasi Amarendra distribution provides a better fit over quasi Amarendra and Amarendra distributions.

Conclusion

In this paper, a new class of quasi Amarendra distribution called as weighted quasi Amarendra distribution has been thoroughly presented and studied. The proposed new distribution is introduced by using the weighted technique to its baseline distribution. Its different structural properties including shape of behaviour of pdf and cdf, raw moments, mean and variance, harmonic mean, order statistics, bonferroni and lorenz curves, survival function, hazard rate function, reverse hazard function and moment generating function have been discussed. For estimating the parameters of introduced distribution, the technique of maximum likelihood estimation has been used. Finally, a real lifetime data set has been examined and analyzed to demonstrate the

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