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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Official URL: http://linkinghub.elsevier.com/retrieve/pii/016771...

Related URL: http://dx.doi.org/10.1016/0167-7152(90)90047-B

## Abstract

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 f_{h}(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 = (μ_{1},…,μ_{p}) of a random vector X = (X_{1},…,X_{p}) 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), σ^{2} = 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, σ^{2}). Finally Chernoff-type inequality and an identity are derived for a multivariate random vector X = (X_{1},…,X_{p}) when the components X_{i}, 1 ≤ i ≤ p, are independent univariate elliptical random variables.

Item Type: | Article |
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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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