Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadurs representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that [sqrt{n}Biggl(frac{1}{n}sum_{i=1}^nphibigl(X_{n:i}^{(1)},...,X_{n:i}^{(d)}bigr)-bar{gamma}Biggr)=frac{1}{sqrt{n}}sum_{i=1}^nZ_{n,i}+mathrm{o}_P(1)] as $nrightarrowinfty$, where $bar{gamma}$ is a constant and $Z_{n,i}$ are i.i.d. random variables for each $n$. This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations.
Download