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. https://doi.org/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
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
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 (In Persian). doi: 10.30504/mct.2022.1125.1788
