Remarks on univariate elliptical distributions

Prakasa Rao, B. L. S. (1990) Remarks on univariate elliptical distributions Statistics & Probability Letters, 10 (4). pp. 307-315. ISSN 0167-7152

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A random variable X is said to have an univariate elliptical distribution (or an elliptical density) with parameters μ and γ > 0 if it has a density of the form fh(x|μ, γ) = γ-1/2 h((x − μ)2/γ) for some function h(·). If μ = 0 and γ = 1, then X is said to have a spherical density corresponding to radial function h(·). Here we derive Chernoff-type inequality and a natural identity for univariate elliptical distribution. The problems of estimation of mean vector μ = (μ1,…,μp) of a random vector X = (X1,…,Xp) with independent components is discussed as an application of the identity. The risk of the improved estimator for p ≥ 3 dominating the unbiased estimator X of μ under squared error loss is derived. Locally asymptotic minimax estimation for a function g(θ) of θ = (μ, σ), σ2 = γ is discussed. Special case of g(θ) = μ + cσ is discussed in detail and as a further special case, a locally asymptotic minimax estimator of μ + cσ is derived for normal distribution with parameters (μ, σ2). Finally Chernoff-type inequality and an identity are derived for a multivariate random vector X = (X1,…,Xp) when the components Xi, 1 ≤ i ≤ p, are independent univariate elliptical random variables.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Univariate Elliptical Distribution; Chernoff-type Inequality; Local Asymptotic Minimax Estimator; Estimation Of Linear Function Of Mean And Standard Deviation
ID Code:37196
Deposited On:26 Apr 2011 10:30
Last Modified:26 Apr 2011 10:30

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