Unified extension of variance bounds for integrated Pearson family

We use some properties of orthogonal polynomials to provide a class of upper/lower variance bounds for a function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$g(X)$$</EquationSource> </InlineEquation> of an absolutely continuous random variable <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$X$$</EquationSource> </InlineEquation>, in terms of the derivatives of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$g$$</EquationSource> </InlineEquation> up to some order. The new bounds are better than the existing ones. Copyright The Institute of Statistical Mathematics, Tokyo 2013