A Double Recursion for Calculating Moments of the Truncated Normal Distribution and its Connection to Change Detection

Methodology and Computing in Applied Probability - The integral ${int }_{0}^{infty }x^{m} e^{-frac {1}{2}(x-a)^{2}}dx$ appears in likelihood ratios used to detect a change in the parameters of a...     

Integral Moments of L-Functions

We give a new heuristic for all of the main terms in the integralmoments of various families of primitive L-functions. The resultsagree with previous conjectures for the leading order terms.Our conjectures also have an almost identical form to exactexpressions for the corresponding moments of the characteristicpolynomials of either unitary, orthogonal, or symplectic matrices,where the moments are defined by the appropriate group averages.This lends support to the idea that arithmetical L-functionshave a spectral interpretation, and that their value distributionscan be modelled using Random Matrix Theory. Numerical examplesshow good agreement with our conjectures. 2000 Mathematics SubjectClassification 11M26, 15A52.     

Moments of wishart distribution

Moments of central and noncentral Wishart distributions are obtained by differentiating their characteristic functions applying matrix derivative techniques, using a special operator which takes into account the symmetry of the matrices. As a special case, higher moments of the multivariate normal distribution are obtained, arranged automatically in a square matrix form     

Moments of square-integrable functions

This paper is motivated by [2], where we have given necessary and sufficient conditions for a given basis P in the space of polynomials to be orthogonal with respect to the measure ϱdφ for a certain function ϱ ϵ L₂(dφ). Let P = {p_i: i = 0, 1, …}, p₀ = 1. Then the conditions are (1) a multivariate analog of the three-term recurrence relation holds, see Section 4 for details; and (2) {q_i = ∑_{j = 0}^∞ c_ij P_j, i = 0, 1, …} is a φ-orthonormal basis in the space of polynomials for some coefficients c_ij such that ∑_{i = 0}^∞ c_i0² <-∞. This paper provides an algebraic condition (a condition on the coefficients c_i0) such that ϱ satisfies ∥p∥ <B, Bϵ (0, ∞], and has a cone-positivity property. In particular, our results imply that ϱ is nonnegative a.e. if ∑_{i = 0}^∞ c_i0² < ∞ and ∑_{jϵ Sk} c_j0 q_j defines nonnegative polynomials for certain finite sets S₁, S₂, … of integers.     

Stochastic Generalized Method of Moments

The generalized method of moments (GMM) is a very popular estimation and inference procedure based on moment conditions. When likelihood-based methods are difficult to implement, one can often derive various moment conditions and construct the GMM objective function. However, minimization of the objective function in the GMM may be challenging, especially over a large parameter space. Due to the special structure of the GMM, we propose a new sampling-based algorithm, the stochastic GMM sampler, which replaces the multivariate minimization problem by a series of conditional sampling procedures. We develop the theoretical properties of the proposed iterative Monte Carlo method, and demonstrate its superior performance over other GMM estimation procedures in simulation studies. As an illustration, we apply the stochastic GMM sampler to a Medfly life longevity study. Supplemental materials for the article are available online.     

Moments of Markov Random Evolutions

We find moments of a process of Markov random evolutions in a finite-dimensional space.     

Moments of Generalized Wishart Distributions

Moments of generalized Wishart distributions are obtained through tensor differential forms and reindexing.     

Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S _T |^γ<∞, γ>2 whereT is a stopping time and {S _n=∑ ₁ ⁿ ,X _j ℱ _n ,n⩾1} is a martingale and these ensure when (**)X _n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET ^γ/2<∞, sup _n≥1 E|X _n |^γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S _k,T |^γ<∞ and of the second moment equationES _k,T ² =σ ² EΣ _j=k ^T S _k−1,j−1 ² where and {X _n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X _n , n≥1} withEX=0,EX ²=1 that the a.s. limit set of {(n log logn)^−k/2 S _k,n ,n≥k} is [0,2 ^k/2/k!] or [−2 ^k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .     

Moments of Some Stopping Rules

Let X_n for n1 be independent random variables with . Set . Define T_k,c,m=inf{nm:|k!S_k,n|>cn^k/2}.We study critical values c_k,p for k2 and p>0, such that for c<c_k,p and all m, and for c>c_k,p and all sufficientlylarge m. In particular, c_1,1=c_2,1=1, c_3,1=2 and c_4,1=3 undercertain moment conditions on X₁, when X_n are identically distributed.We also investigate perturbed stopping rules of the form T_h,m=inf{nm:h(S_1,n/n^1/2)<_nor >_n} for continuous functions h and random variables _naand _nb with a<b. Related stopping rules of the Wiener processare also considered via the Uhlenbeck process.     

Moments on Catalan numbers

By combining inverse series relations with binomial convolutions and telescoping method, moments of Catalan numbers are evaluated, which resolves a problem recently proposed by Gutiérrez et al. [J.M. Gutiérrez, M.A. Hernández, P.J. Miana, N. Romero, New identities in the Catalan triangle, J. Math. Anal. Appl. 341 (1) (2008) 52-61].     

Moments over short intervals

An asymptotic result for the kth moment (k 9) of the error term in the Dirichlet divisor problem over short intervals is obtained, which improves on an earlier result of Nowak.Received: 22 April 2004     

Moments of products of elliptic integrals

We consider the moments of products of complete elliptic integrals of the first and second kinds. In particular, we derive new results using elementary means, aided by computer experimentation and a theorem of W. Zudilin. Diverse related evaluations, and two conjectures, are also given.     

Moments of the generalized hyperbolic distribution

In this paper we demonstrate a recursive method for obtaining the moments of the generalized hyperbolic distribution. The method is readily programmable for numerical evaluation of moments. For low order moments we also give an alternative derivation of the moments of the generalized hyperbolic distribution. The expressions given for these moments may be used to obtain moments for special cases such as the hyperbolic and normal inverse Gaussian distributions. Moments for limiting cases such as the skew hyperbolic t and variance gamma distributions can be found using the same approach.     

Moments of graphs in monotone families

The kth moment of the degree sequence d₁ ≥ d₂ ≥ …d_n of a graph G is . We give asymptotically sharp bounds for μ_k(G) when G is in a monotone family. We use these results for the case k = 2 to improve a result of Pach, Spencer, and Tóth [15]. We answer a question of Erd?s [9] by determining the maximum variance of the degree sequence when G is a triangle‐free n‐vertex graph. © 2005 Wiley Periodicals, Inc.     

On Moments of Bivariate Order Statistics

In the present paper, we give the exact explicit expression for the product moments (of any order) of bivariate order statistics (o.s.) from any arbitrary continuous bivariate distribution function (d.f.). Furthermore, for any arbitrary bivariate uniform d.f., universal distribution-free bounds for the differences of any two different product moments (of order (1,1) or (-1,1)) are given.     

Average Approximations and Moments of Measures

We investigate average approximations of infinite dimensional mappings and related problems connected with moments of measures on linear spaces. A conjecture stated by J. F. Traub and A. G. Werschulz (1994, Math. Intelligencer16, 42–48) is settled. Several positive results concerning average approximations of Banach space valued mappings are obtained. Some related open problems are discussed.     

Problème des Moments

Abstract. On se propose dans ce papier de donner une caractérisation des fonctions moments définies sur un * -semigroupe qui n'est pas forcément de type fini. Il s'agit en fait d'une généralisation du résultat de Putinar et de Vasilescu donnant une solution du problème des moments dans R ⁿ . On donnera alors en application de ce résultat une caractérisation des suites indéfinies ayant une singularité au point ξ=(1,. . ., 1)∈ C ^p d'ordre au moins 2κ .     

Moments of elliptically distributed random variates

This paper presents an inductive method for computing the moments of an elliptically distributed random variate. A sequence of new parameters relating higher-order to second moments is introduced. The known kurtosis parameter is shown to be a member of this sequence.