Parameter-based Fisher's information of orthogonal polynomials

Abstract

The Fisher information of the classical orthogonal polynomials with respect to a parameter is introduced, its interest justified and its explicit expression for the Jacobi, Laguerre, Gegenbauer and Grosjean polynomials found.

Relative Fisher information of discrete classical orthogonal polynomials

“…Moreover, like in the Kravchuk case, the maximum is obtained for n = N − 1, whose value depends on α, β and N . This behaviour can be observed in Figure 7, where the relative Fisher information is represented as a function of the degree n for the polynomials h 0,0 n (20), h 0,0 n (30) and h 3,−1/2 n (20). The dependence of I ω [h α,β n (N )] on the parameter N is considered in Figure 8, where the relative Fisher information is represented as a function of N for the polynomials h 0,0 2 (N ), h 0,0 10 (N ) and h 3,−1/2 2 (N ).…”

“…Moreover, these behaviours are much more emphasized for β than for α. This can be observed in Figure 10, where the relative Fisher information is represented as a function of β for the polynomials h 0,β 2 (20), h 3,β 2 (20), h 0,β 10 (20) and h 0,β 2 (30).…”

“…The dependence of I ω [K p n (N )] on the parameter p when n and N are fixed is further studied for the polynomials K p 2 (15), K p 4 (15) and K p 2 (20) in Figure 5. We remark that it has a concave shape, with a minimum around p = 1/2 but with a different asymptotic behaviour at its two extremes; namely, the asymptotic behaviour when p tends to 0 is…”

“…Although the notion of Fisher information was initially introduced by Ronald A. Fisher as a measure to estimate a parameter of a probability density, we shall follow B. Roy Frieden who realized that the locality Fisher information (also called intrinsic accuracy) for continuous distributions is much more useful for applications in science and technology [26]. For completeness we note that the parameter-based Fisher information was calculated for various continuous classical real orthogonal polynomials [20] as well as for the discrete Hahn polynomials [25], while the locality or shift-invariant Fisher information has been exactly calculated for the Hermite, Laguerre and Jacobi polynomials. The discretization of the locality Fisher information is discussed in detail in [46], where the definition of the Fisher information for discrete orthogonal polynomials is introduced.…”

Fisher information of special functions and second-order differential equations

“…(3), and then the corresponding Cramer-Rao information plane [16] was analyzed. See also [15], where the parameter-based Fisher information of classical orthogonal polynomials was calculated by use of their algebraic properties other than the differential equation. Now, our purpose is the analytic evaluation of the Fisher information of special functions other than the classical orthogonal polynomials, in terms of the expectation values of the coefficients which characterize the differential equations (1), (2) or (3) that they satisfy: This is the main goal of this work.…”