Integrated Pearson family and orthogonality of the Rodrigues polynomials: A review including new results and an alternative classification of the Pearson system

Abstract

An alternative classification of the Pearson family of probability densities is related to the orthogonality of the corresponding Rodrigues polynomials. This leads to a subset of the ordinary Pearson system, the Integrated Pearson Family. Basic properties of this family are discussed and reviewed, and some new results are presented. A detailed comparison between the integrated Pearson family and the ordinary Pearson system is presented, including an algorithm that enables to decide whether a given Pearson density belongs to the integrated system, or not. Recurrences between the derivatives of the corresponding orthonormal polynomial systems are also given.MSC: Primary 62E15, 60E05; Secondary 62-00. /∼npapadat/ This fact will be denoted by X ∼ IP(µ; q) or f ∼ IP(µ; q) or, more explicitly, X or f ∼ IP(µ; δ , β , γ).Despite the fact that the integrated Pearson family is quite restricted, compared to the usual Pearson system -see Proposition 2.1(iii), below -we believe that the reader will find here some interesting observations that are worth to be highlighted. The integrated Pearson system satisfies many interesting properties, like recurrences on moments and on Rodrigues polynomials, covariance identities, closeness of each type under particularly useful transformations etc.; such properties are by far more complicated (if they are, at all, true) for distributions outside the Integrated Pearson system. These features should be combined with the fact that the Rodrigues polynomials form an orthogonal system for the corresponding Pearson density if and only if the density belongs to the Integrated Pearson family. In other words, the Rodrigues polynomials and, consequently, the ordinary Pearson densities, are useful only if they are considered in the framework of the Integrated Pearson system. To our knowledge, these facts have not been written explicitly elsewhere.The paper is organized as follows: In Section 2 we provide a detailed classification of the integrated Pearson family. It turns out that, up to an affine transformation, there are six different types of densities, included in Table 2.1. We also provide conditions guaranteeing the existence of moments, and we give recurrences as long as these moments exist. In Section 3, a detailed comparison between the integrated Pearson family and the ordinary Pearson system is presented. Interestingly enough, there exist a simple algorithm that enables one to decide whether a given ordinary Pearson density belongs to the integrated

Strengthened Chernoff-type variance bounds

“…In the present section, we assume that X ∼ IP(µ; δ, β, γ) with δ ≤ 0. The well-known Normal, Gamma and Beta random variables and their affine transformations are of this form -see [2], Table 2.1. In this case the orthonormal polynomial system {φ k } ∞ k=0 is complete in L 2 (R, X) and, therefore, the following result holds.…”

“…Theorem A.1 ([16], page 401; [6], pages 99-100; [15], page 295; [2], Theorem 4.1). Assume that f is the density of a random variable X ∼ IP(µ; q) ≡ IP(µ; δ, β, γ) with support (α, ω).…”

“…Theorem A.2 ([3], pages 515-516; [2], Theorem 5.1). Let X ∼ IP(µ; δ, β, γ) ≡ IP(µ; q) with density f and support (α, ω).…”

“…In the particular case where X ∼ IP(µ; δ, β, γ) and δ ≤ 0 (i.e., if X is of Normal, Gamma or Beta-type), it follows that E|X| n < ∞ for all n. Moreover, there exists an ε > 0 such that Ee tX < ∞ for |t| < ε (see types 1-3 of Table 2.1 in [2]). Hence, the polynomials {φ k } ∞ k=0 , given by (A.6) (with n = ∞), form a complete orthonormal system in L 2 (R, X); see, for example, [3,7].…”

“…Turn now to the case where X ∼ IP(µ; δ, β, γ) with δ ≤ 0. It follows that all moments exist and, moreover, the moment generating function of X is finite in a neighborhood of zero (see [2], Table 2.1, types 1-3). Then, it is well known that the orthonormalized polynomial system {φ k } ∞ k=0 , given by (A.6) (with n = ∞), is complete in L 2 (R, X); see, for example, [3,7]; see also Remark A.3, below.…”