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|Title:||Limit theorems for functions of marginal quantiles|
|Keywords:||Central limit theorem|
Strong law of large numbers
Weak convergence of a process
|Source:||Babu, G.J., Bai, Z., Choi, K.P., Mangalam, V. (2011-05). Limit theorems for functions of marginal quantiles. Bernoulli 17 (2) : 671-686. ScholarBank@NUS Repository. https://doi.org/10.3150/10-BEJ287|
|Abstract:||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 Bahadur's representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that √ n(1/nσ n i=1φ(X(1) n : i, ⋯ , X(d) n : i) - ȳ)=1/√nσn i=1 Zn,i + oP (1) as n→ ∞, where ȳ is a constant and Zn,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. © 2011 ISI/BS.|
|Appears in Collections:||Staff Publications|
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