Let Fn(x) be the empirical distribution function based on n independent random variables X1,...,Xn from a common distribution function F(x), and let be the sample mean. We derive the rate of convergence of to normality (for the regular as well as nonregular cases), a law of iterated logarithm,...

We consider estimation of a location vector in the presence of known or unknown scale parameter in three dimensions. The technique of proof is Stein's integration by parts and it is used to cover several cases (e.g., non-unimodal distributions) for which previous results were known only in the...

<Para ID="Par1">From an observable <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$(X,U)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathbb R^p \times \mathbb R^k$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mi mathvariant="double-struck">R</mi> <mi>p</mi> </msup> <mo>×</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>k</mi> </msup> </mrow> </math> </EquationSource> </InlineEquation>, we consider estimation of an unknown location parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\theta \in \mathbb R^p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">θ</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>p</mi> </msup> </mrow> </math> </EquationSource> </InlineEquation> under two distributional settings: the density of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$(X,U)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> is...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></para>

We study existence of unbiased estimators of risk for estimators of the location parameter of a spherically symmetric distribution, when a residual vector is available to estimate scale, under invariant quadratic loss. We show such existence often characterizes normality.

For a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under...

We consider estimation of the mean vector, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\theta $$</EquationSource> </InlineEquation>, of a spherically symmetric distribution with known scale parameter under quadratic loss and when a residual vector is available. We show minimaxity of generalized Bayes estimators corresponding to superharmonic priors with a non...</equationsource></inlineequation>