For an absolutely continuous (integer-valued) r.v. X of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order k holds. This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. X, expressions that seem to be known only in particular cases.
Afendras, G., Papadatos, N., Papathanasiou, V., An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds, Bernoulli, 17 (2011), 507-529.
Goodarzi,F. (2023). An Extended Covariance Identity for the Pearson Family
with Applications to Lower Variance Bounds. Mathematical Culture and Thought, 41(2), 199-222. doi: 10.30504/mct.2022.1125.1788
MLA
Goodarzi,F. . "An Extended Covariance Identity for the Pearson Family
with Applications to Lower Variance Bounds", Mathematical Culture and Thought, 41, 2, 2023, 199-222. doi: 10.30504/mct.2022.1125.1788
HARVARD
Goodarzi F. (2023). 'An Extended Covariance Identity for the Pearson Family
with Applications to Lower Variance Bounds', Mathematical Culture and Thought, 41(2), pp. 199-222. doi: 10.30504/mct.2022.1125.1788
CHICAGO
F. Goodarzi, "An Extended Covariance Identity for the Pearson Family
with Applications to Lower Variance Bounds," Mathematical Culture and Thought, 41 2 (2023): 199-222, doi: 10.30504/mct.2022.1125.1788
VANCOUVER
Goodarzi F. An Extended Covariance Identity for the Pearson Family
with Applications to Lower Variance Bounds. An Expository Journal of the Iranian Mathematical Society, 2023; 41(2): 199-222. doi: 10.30504/mct.2022.1125.1788
