Table 5 2: Expression for Gini index and income gap ratio
Related Figures (70)
A BIVARIATE PARETO DISTRIBUTION WITH FREUND'S DEPENDENCE STRUCTURE The probability density function f(x,,x,)in terms of A(x) is given by (Cox The unique expression for the joint density function in terms of mean residual life Another definition closely related to above bivariate mean residual life The bivariate mean residual life function is related to Cox’s failure rate by the These models have been extensively used. Jupp and Mardia (1982) Arnold (1983) further studied the properties of these models and Yeh (1994) obtained characterization results for the bivariate Pareto IV distributions. Muliere and Scarcini (1987) proposed a bivariate Pareto distribution with joint leads to Marshall-Olkin type dependent bivariate Pareto model, Figure 2.1 Plot of the bivariate survival function of a BP II distribution with shown in the following figure. l=2, o=3, 4 =2, a, =2.5, a =4 anda, =5. The graphical representation of the model BP II (1,0, 0,,Q,0,@) and the marginal survival function is obtained as bivariate Paretos exhibits marginals, which are mixture of Pareto distribution. then The correlation coefficient of the bivariate Pareto models discussed in Thus the joint density function for the bivariate random variable (X,, X,) is now known oO, the likelihood is now becomes Remark 2.1 For a known w and o, the density function given in (2.6) belongs to an exponential family. Hence 2.5.3 Estimation of Parameters of a BP J (o, ,A, a a | Distribution Table 2.1 MLE of a BP II (1,0,0,,0,,0,06,) when @,=2, a,=1.5, @ =1.75 Consider a random sample of size n from a population having bivariate Remark 2.2 The estimator of a and @, are independent ofa. The proof follows directly from Serfling (1980 p. 145). Once again differentiating (2.52) with respect to A, and simplifying, we get Table 2.3 Transformed data set The Kolmogorov-Smirnov test for the bivariate random variable is made Figure 2.3: Bivariate Empirical Survival Plot for the data given in Table 2.3. is fitted graphically and is given below. With this we conclude this chapter. Theorem 3.1 A random vector (X,,X,) with survival function F (x,,x,) in the support of (1,00) x(1,c) satisfies (3.3) if and only if F(2,,x,) belongs to the Corollary 3.1 Let (X,,X,) be a bivariate random variable with mixture Pareto Corollary 3.2 Let (X,,X.,) bea bivariate random variable with Pareto marginals then (X,,X,) has a Marshall-Olkin type bivariate Pareto I (Muliere and Scarcini Table 3.1 Bivariate Pareto model possessing dullness property As typical in the case of lack of memory property, the dullness property equivalent. Further from equation (3.16) and Corollary 3.2 we obtain the survival Also observing that (3.24) implies dullness property if and only if for all x, x,,x,,t>0, the equation (3.22) holds. In particular if we assume Pareto marginals for X,, we have the followin Now conversely assume that equation (4.39) holds and using (1.42) and (4.4), we bivariate residual entropy function and bivariate geometric vitality function we Differentiating (4.40), (4.41) and (4.42) with respect tot,t,,t,, we get Table 4.2 Class of distribution possessing BLMP denotes the proportion of population whose first resource has an income greater Inequality Measures for Bivariate Distributions with Load Sharing Dependence than x and up to ¢ while the second source generates an income x, <x. The are consistent with the univariate definition of the Lorenz curve and reflect Table 5.1 Values of the Lorenz functions Figure 5.1 Lorenz curve corresponding to the failure time data From the Figure 5.1, it can be inferred that there is less disparity among the samples when both the units have exceeded the truncation point (assumed as unity Theorem 5.2 Under the usual assumptions and conditions specified in Theorem deals with this. Inequality Measures for Bivariate Distributions with Load Sharing Dependence Substituting the above expression in (5.22), we get (5.19). Inequality Measures for Bivariate Distributions with Load Sharing Dependence Proceeding in similar arguments with (5.11), (5.7) and (5.8), the third equality car be claimed. Inequality Measures for Bivariate Distributions with Load Sharing Dependence Inequality Measures for Bivariate Distributions with Load Sharing Dependence which is the bivariate Pareto I (BPI(o,a,,@,,@,@,)) distribution (Asha and It should be observed that (6.5) is a finite mixture of two pseudo-survival uniform distribution. Some members belonging to this class are worked out Since A(u,v) is a pseudo-survival copula, evidently not a bivariate In this section we consider how total failure rate can be related to the operties of the general representation given in (6.5) in terms of genera Table 6.1 lists the failure rate for distributions defined by representation (6.5). Table 6.1 Total Failure Rate for the Uniform Representation researchers is the no-ageing property and its variants like the dullness property. The proof now follows from Theorem 6.1. Observe that the uniform representation (6.5) can be written as This is the class of exponential minima given in equation (3.5) characterized by This is class of Pareto minima given in (3.6) characterized by the bivariate characterization of a univariate Weibull distribution. In this chapter we have given a uniform representation of the Freund
