On the Ratio of two Independent Exponentiated Pareto Variables

where β and c are, respectively, the threshold and shape parameters. The distribution is found to coincide with many social, scientific, geophysical, actuarial, and various other types of observable phenomena. Some examples where the Pareto distribution gives good fit are the sizes of human settlements, the values of oil reserves in oil fields, the standardized price returns on individual stocks, sizes of meteorites, etc. An extension/generalization of the Pareto distribution is the exponentiated Pareto distribution with cdf G(x) = F(x) , α > 0 . (2)


Introduction
The Pareto distribution is a power law probability distribution having cumulative distribution function (cdf) where β and c are, respectively, the threshold and shape parameters.The distribution is found to coincide with many social, scientific, geophysical, actuarial, and various other types of observable phenomena.Some examples where the Pareto distribution gives good fit are the sizes of human settlements, the values of oil reserves in oil fields, the standardized price returns on individual stocks, sizes of meteorites, etc.An extension/generalization of the Pareto distribution is the exponentiated Pareto distribution with cdf G(x) = F α (x) , α > 0 . (2) Here α denotes the exponentiating parameter, and for α = 1 the distribution reduces to the standard Pareto distribution (1).R. C. Gupta, Gupta, and Gupta (1998) introduced the exponentiated exponential distribution, and its properties were studied by R. D. Gupta and Kundu (2001).Pal, Ali, and Woo (2006) studied certain aspects of the exponentiated Weibull distribution.M. M. Ali, Pal, and Woo (2007) studied several exponentiated distributions, including the exponentiated Pareto distribution, and discussed their properties.They showed that the exponentiated Pareto distribution gives a good fit to the tail-distribution of Nasdaq data.
The problem of estimating the probability that a random variable X is less than another random variable Y arises in many practical situations, like biometry, reliability study, etc.This problem has been studied by many authors for different distributions of X and Y , see, for example Pal, Ali, and Woo (2005), M. Ali, Pal, and Woo (2005), and M. M. Ali, Pal, and Woo (2009).
In this paper, we find the distribution of the ratio of two independent exponentiated Pareto random variables X and Y and study its properties.Some special functions have been used to evaluate terms that cannot be expressed in closed form.We also find the UMVUE of Pr(X < Y ) and the UMVUE of its variance.Finally, we obtain the UMVUEs of Pr(X < Y ) and its variance by analyzing a simulated data set and a reallife data set.

Distribution of the Ratio Y/X
Let X and Y be independent exponentiated Pareto random variables having cdf (2) with parameters α 1 , β 1 , c and α 2 , β 2 , c, respectively.Using formula 3.197(3) in Gradshteyn and Ryzhik (1965), the cdf of the ratio Q = Y /X is obtained as where From (3) and formula 15.2.1 in Abramowitz and Stegun (1972), the pdf of Q is therefore given by The kth moment of Q about the origin is obtained as where is the generalized hypergeometric function in Gradshteyn and Ryzhik (1965).
3 Distribution of the Ratio X/(X+Y) Consider the ratio R = X/(X +Y ), where X and Y are independent exponentiated Pareto random variables having the cdf (2) with parameters α 1 , β 1 , c and α 2 , β 2 , c, respectively.
From (3) we obtain the cdf of the ratio R as and its density function is Using formula 7.512(5) in Gradshteyn and Ryzhik (1965), the kth moment of R about the origin is obtained as ) and (a) P 0 = 0.
Figure 2 shows the density curves and Table 2 provides the asymptotic means, variances, and coefficients of skewness for the distribution (7) for different combinations of α 1 , α 2 , when c = 5, β 1 = 1, and β 2 = 2.The figure also shows that the distribution is skewed to the left.
Table 2 indicates that • the distribution is skewed to the left; • its mean and variance increase slightly as α 1 increases when α 1 > α 2 but the mean slightly decreases and the variance slightly increases as α 1 increases when α 1 < α 2 .Now we attempt to find the uniformly minimum-variance unbiased estimator (UMVUE) of ξ = Pr(X < Y ), where X and Y are independently distributed as exponentiated Pareto, having cdf's From (3) we have ( where F (a, b; c; z) is the hypergeometric function.For which depends only on the exponentiating parameters.We shall assume that β 1 , β 2 and c are known.
An unbiased estimator of ξ is given by Since the complete sufficient statistics for α 1 and α 2 are where a = V * /U * .

Data Analysis
We now analyze two sets of data, a simulated one and a real life one, to give UMVUEs of ξ = Pr(X < Y ) and its variance.
We first fitted exponentiated Pareto distributions with parameters α 1 , β 1 , c and α 2 , β 2 , c, respectively, to the data on X and Y and then checked the goodness of fit using exponentiated Pareto probability plots.

Conclusion
The paper studies the distributions of the ratios Y /X and X/(X + Y ), when X and Y are distributed independently as exponentiated Pareto.These distributions find importance in many situations, like studying the proportion of human settlement in a place, proportion of oil reserves in an oil field, etc.They are also useful in computing the reliability function Pr(X < Y ), when X and Y denote the stress and strength variables, respectively.The paper, further, finds the UMVUE of Pr(X < Y ), and also UMVUE of its variance.A simulated data set and a real data set are analyzed to estimate Pr(X < Y ) and its variance.

Figure 3 :
Figure 3: Probability plots for the distributions fitted to data set 1 (left) and 2 (right).

Table 3 :
The Major Rice Crop in Kilograms from the two Tambols for the CropYear  2001-2002 (April 1, 2001 to March 31, 2002)., . . ., X m ) and (Y 1 , . . ., Y n ) denote the random samples on X and Y , and let the corresponding ordered statistics be X