Computation Of High-order Moments Research Articles

A Stein characterisation of the distribution of the product of correlated normal random variables

We obtain a Stein characterisation of the distribution of the product of two correlated normal random variables with non-zero means, and more generally the distribution of the sum of independent copies of such random variables. Our Stein characterisation is shown to naturally generalise a number of other Stein characterisations in the literature. From our Stein characterisation we derive recursive formulas for the moments of the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, which allows for efficient computation of higher order moments.

Markov-switching threshold stochastic volatility models with regime changes

<abstract><p>This paper introduces a comprehensive class of models known as Markov-Switching Threshold Stochastic Volatility (MS-TSV) models, specifically designed to address asymmetry and the leverage effect observed in the volatility of financial time series. Extending the classical threshold stochastic volatility model, our approach expresses the parameters governing log-volatility as a function of a homogeneous Markov chain with a finite state space. The primary goal of our proposed model is to capture the dynamic behavior of volatility driven by a Markov chain, enabling the accommodation of both gradual shifts due to economic forces and sudden changes caused by abnormal events. Following the model's definition, we derive several probabilistic properties of the MS-TSV models, including strict (or second-order) stationarity, causality, ergodicity, and the computation of higher-order moments. Additionally, we provide the expression for the covariance function of the squared (or powered) process. Furthermore, we establish the limit theory for the Quasi-Maximum Likelihood Estimator (QMLE) and demonstrate the strong consistency of this estimator. Finally, a simulation study is presented to assess the performance of the proposed estimation method.</p></abstract>

Some computational aspects of Tchebichef moments for higher orders

In this work, we propose a new algorithm for the computation of Tchebichef moments by means of a recurrence relation with respect to order and the Gram–Schmidt process, which reduces the numerical instability and the carry error caused by the computation of high-order moments. Results and comparison with other methods are presented.

Recurrence Relations for Moments of Generalized Order Statistics for Inverse Weibull Distribution and Some Characterizations

We present recurrence relations for single, inverse, product and ratio moments of Generalized Order Statistics when sample is available from an Inverse Weibull Distribution. These relations enable computation of higher order moments using corresponding lower order moments. Some Characterizations of the distribution are also given in terms of conditional moments.

Convergence of rational Bernstein operators

In this paper we discuss convergence properties and error estimates of rational Bernstein operators introduced by Piţul and Sablonnière. It is shown that the rational Bernstein operators converge to the identity operator if and only if the maximal difference between two consecutive nodes is converging to zero. Further a Voronovskaja theorem is given based on the explicit computation of higher order moments for the rational Bernstein operator.

A Recursive Formula for Moments of a Binomial Distribution

While teaching a course in probability and statistics, one of the authors came across an apparently simple question about the computation of higher order moments of a random variable. The topic of moments of higher order is rarely emphasized when teaching a statistics course. The textbooks we came across in our classes, for example, treat this subject rather scarcely; see [3, pp. 265–267], [4, pp. 184–187], also [2, p. 206]. Most of the examples given in these books stop at the second moment, which of course suffices if one is only interested in finding, say, the dispersion (or variance) of a random variable X , D2(X) = M2(X)− M(X)2. Nevertheless, moments of order higher than 2 are relevant in many classical statistical tests when one assumes conditions of normality. These assumptions may be checked by examining the skewness or kurtosis of a probability distribution function. The skewness, or the first shape parameter, corresponds to the the third moment about the mean. It describes the symmetry of the tails of a probability distribution. The kurtosis, also known as the second shape parameter, corresponds to the fourth moment about the mean and measures the relative peakedness or flatness of a distribution. Significant skewness or kurtosis indicates that the data is not normal. However, we arrived at higher order moments unintentionally. The discussion that follows is the outcome of a question raised by our students without knowing much about the motivation for studying moments of higher order. It has to do with discrete random variables

A concurrent system for the computation of higher-order moments

The cumulants defined in terms of moments are basic to the study of higher-order statistics (HOS) of a stationary stochastic process. This paper presents a concurrent systolic array system for the computation of higher-order moments. The system allows for the simultaneous computation of the second-, third-, and fourth-order moments. The architecture achieves good speedup through its excellent exploitation of parallelism, pipelining, and reusability of some intermediate results. The computational complexity and system performance issues related to the architecture are discussed. The concurrent system is designed with the CMOS VLSI technology and is capable of operating at 3.9 MHz.

The single server queue in discrete time‐numerical analysis II

AbstractThis paper deals with the numerical problems arising in the computation of higher order moments of the busy period for certain classical queues of the M|G|I type, both in discrete and in continuous timeThe classical functional equation for the moment generating function of the busy period is used. The higher order derivatives at zero of the moment generating function are computed by repeated use of the classical differentiation formula of Fá di Bruno. Moments of order up to fifty may be computed in this mannerA variety of computational aspects of Fá di Bruno's formula, which may be of use in other areas of application, are also discussed in detail.