Score functions, generalized relative Fisher information and applications

Abstract

Generalizations of the linear score function, a well-known concept in theoretical statistics, are introduced. As the Gaussian density and the classical Fisher information are closely related to the linear score, nonlinear (respectively fractional) score functions allow to identify generalized Gaussian densities (respectively Lévy stable laws) as the (unique) probability densities for which the score of a random variable X is proportional to $-X$. In all cases, it is shown that the variance of the relative to the generalized Gaussian (respectively Lévy) score provides an upper bound for $L^1$-distance from the generalized Gaussian density (respectively Lévy stable laws). Connections with nonlinear and fractional Fokker–Planck type equations are introduced and discussed.

Appendix

In this short appendix we summarize the mathematical notations and the meaning of the fractional derivative. Given a probability density f(x), $x \in {\mathbb {R}}^n$, we define its Fourier transform $\mathcal F(f)$ by

$$\begin{aligned} \mathcal F(f)(\xi ) = \widehat{f}(\xi ) := \int _{{\mathbb {R}}^n} e^{- i \, \xi \cdot \, x} f(x)\, dx, \qquad \forall \xi \in {\mathbb {R}}^n. \end{aligned}$$

Let us set $n=1$. Then, the one-dimensional derivative $\mathcal D_{\alpha }$ is defined as follows. For $0 <\alpha < 1$ we let $R_\alpha$ be the one-dimensional normalized Riesz potential operator defined for locally integrable functions by [22, 24]

$$\begin{aligned} R_\alpha (f)(x) = S(\alpha ) \int _{\mathbb {R}}\frac{f(y)\, dy}{|x-y|^{1-\alpha }}. \end{aligned}$$

The constant $S(\alpha )$ is chosen to have

$$\begin{aligned} \widehat{R_\alpha (f)}(\xi ) = |\xi |^\alpha \widehat{f}(\xi ). \end{aligned}$$

(39)

Since for $0 <\alpha < 1$ it holds [18]

$$\begin{aligned} \mathcal F |x|^{\alpha -1} = |\xi |^{-\alpha } \pi ^{1/2} \Gamma \left( \frac{1-\alpha }{2} \right) \Gamma \left( \frac{\alpha }{2} \right) , \end{aligned}$$

(40)

where, as usual $\Gamma (\cdot )$ denotes the Gamma function, the value of $S(\alpha )$ is given by

$$\begin{aligned} S(\alpha ) = \left[ \pi ^{1/2} \Gamma \left( \frac{1-\alpha }{2} \right) \Gamma \left( \frac{\alpha }{2} \right) \right] ^{-1}. \end{aligned}$$

Note that $S(\alpha ) = S(1-\alpha )$.

We then define the fractional derivative of order $\alpha$ of a real function f as ($0 {<}\alpha {<} 1$)

$$\begin{aligned} \frac{d^\alpha f(x)}{dx^\alpha } = {\mathcal {D}}_\alpha f(x) = \frac{d}{dx}R_{1-\alpha }(f)(x). \end{aligned}$$

(41)

Thanks to (39), in Fourier variables

$$\begin{aligned} \widehat{ {\mathcal {D}}}_\alpha f(\xi ) = i \frac{\xi }{|\xi |} |\xi |^\alpha \widehat{f}(\xi ). \end{aligned}$$

(42)