Robust minimax Stein estimation under invariant data-based loss for spherically and elliptically symmetric distributions

<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 spherically symmetric with an unknown scale parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\sigma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">σ</mi> </math> </EquationSource> </InlineEquation> and is ellipically symmetric with an unknown covariance matrix <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\Sigma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="normal">Σ</mi> </math> </EquationSource> </InlineEquation>. Evaluation of estimators of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\theta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">θ</mi> </math> </EquationSource> </InlineEquation> is made under the classical invariant losses <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\Vert d - \theta \Vert ^2 / \sigma ^2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mrow> <mo stretchy="false">‖</mo> <mi>d</mi> <mo>-</mo> <mi mathvariant="italic">θ</mi> <mo stretchy="false">‖</mo> </mrow> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <msup> <mi mathvariant="italic">σ</mi> <mn>2</mn> </msup> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$(d - \theta )^t \Sigma ^{-1} (d - \theta )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mi mathvariant="italic">θ</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </msup> <msup> <mi mathvariant="normal">Σ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mi mathvariant="italic">θ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> as well as two respective data based losses <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$\Vert d - \theta \Vert ^2 / \Vert U\Vert ^2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mrow> <mo stretchy="false">‖</mo> <mi>d</mi> <mo>-</mo> <mi mathvariant="italic">θ</mi> <mo stretchy="false">‖</mo> </mrow> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">‖</mo> <mi>U</mi> <mo stretchy="false">‖</mo> </mrow> <mn>2</mn> </msup> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$(d - \theta )^t S^{-1} (d - \theta )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mi mathvariant="italic">θ</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </msup> <msup> <mi>S</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mi mathvariant="italic">θ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> where <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$\Vert U\Vert ^2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mo stretchy="false">‖</mo> <mi>U</mi> <mo stretchy="false">‖</mo> </mrow> <mn>2</mn> </msup> </math> </EquationSource> </InlineEquation> estimates <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$$\sigma ^2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mi mathvariant="italic">σ</mi> <mn>2</mn> </msup> </math> </EquationSource> </InlineEquation> while <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$$S$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>S</mi> </math> </EquationSource> </InlineEquation> estimates <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$$\Sigma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="normal">Σ</mi> </math> </EquationSource> </InlineEquation>. We provide new Stein and Stein–Haff identities that allow analysis of risk for these two new losses, including a new identity that gives rise to unbiased estimates of risk (up to a multiple of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$$1 / \sigma ^2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <msup> <mi mathvariant="italic">σ</mi> <mn>2</mn> </msup> </mrow> </math> </EquationSource> </InlineEquation>) in the spherical case for a larger class of estimators than in Fourdrinier et al. (J Multivar Anal 85:24–39, <CitationRef CitationID="CR8">2003</CitationRef>). Minimax estimators of Baranchik form illustrate the theory. It is found that the range of shrinkage of these estimators is slightly larger for the data based losses compared to the usual invariant losses. It is also found that <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$$X$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>X</mi> </math> </EquationSource> </InlineEquation> is minimax with finite risk with respect to the data-based losses for many distributions for which its risk is infinite when calculated under the classical invariant losses. In these cases, including the multivariate <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$$t$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>t</mi> </math> </EquationSource> </InlineEquation> and, in particular, the multivariate Cauchy, we find improved shrinkage estimators as well. Copyright Springer-Verlag Berlin Heidelberg 2015