Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high-dimensional setting optimal estimation of the...

We consider minimax shrinkage estimation of location for spherically symmetric distributions under a concave function of the usual squared error loss. Scale mixtures of normal distributions and losses with completely monotone derivatives are featured.

<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>

In this short note the closed form of the soft wavelet shrinkage estimator is derived, extending the work of Huang (2002) for the scale mixture of normal distributions.