A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa

It seems difficult to find a formula in the literature that relates moments to cumulants (and vice versa) and is useful in computational work rather than in an algebraic approach. Hence I present four very simple recursive formulas that translate moments to cumulants and vice versa in the univariate and multivariate situations.     

Asymptotic Expansions for i.i.d. Sums Via Lower-order Convolutions

In this article, we introduce new asymptotic expansions for probability functions of sums of independent and identically distributed random variables. Results are obtained by efficiently employing information provided by lower-order convolutions. In comparison with Edgeworth-type theorems, advantages include improved asymptotic results in the case of symmetric random variables and ease of computation of main error terms and asymptotic crossing points. The first-order estimate can perform quite well against the corresponding renormalized saddlepoint approximation and, pointwise, requires evaluation of only a single convolution integral. While the new expansions are fairly straightforward, the implications are fortuitous and may spur further related work.     

Asymptotic Expansion and Conditional Robustness for the Sample Multiple Correlation Coefficient Under Nonnormality

ABSTRACT

Asymptotic distributions of the standardized estimators of the squared and non squared multiple correlation coefficients under nonnormality were obtained using Edgeworth expansion up to O(1/n). Conditions for the normal-theory asymptotic biases and variances to hold under nonnormality were derived with respect to the parameter values and the weighted sum of the cumulants of associated variables. The condition for the cumulants indicates a compensatory effect to yield the robust normal-theory lower-order cumulants. Simulations were performed to see the usefulness of the formulas of the asymptotic expansions using the model with the asymptotic robustness under nonnormality, which showed that the approximations by Edgeworth expansions were satisfactory.     

Ma system identification using higher order cumulants application to modelling solar radiation

The main goal of this investigation is to elaborate an accurate and efficient algorithm able to estimate the moving average (MA) parameters in noisy environment. So, we address the problem of estimating the parameters of a MA--non-minimum phase (NMP)-system from the output observation when the system is excited by an unobservable independent identically distributed (i.i.d.) sequence. A new procedure, based on third and fourth order cumulants, to estimate the parameters of MA process when the order is known, is presented. This procedure was tested for various examples of MA (NMP) system, and for different order and different signal-to-noise-ratio (SNR). A comparison with some existing methods was also performed and the result shows the efficiency of the proposed algorithm. For validation purpose this method is used to search for a model able to describe and to predict the data set representing the daily solar radiation.     

Estimation of product moments of a stationary stochastic process with application to estimation of cumulants and cumulant spectral densities

Inference concerning the structure of stationary stochastic processes can be investigated by looking at properties of various cumulant spectral densities of order two and higher. However, except for cases when cumulants and product moments are identical, estimation of higher-order cumulant spectral densities has been restricted by the dependence of higher-order cumulants on lower-order product moments. By first estimating product moments and then using an identity between product moments and cumulants, asymptotically unbiased and consistent estimates of cumulants are obtained. This in turn leads to asymptotically unbiased and consistent estimators of higher-order cumulant spectral densities. In addition, asymptotic normality of product-moment estimators is exhibited under weak dependence.     

Some alternatives to Edgeworth

The Edgeworth expansion is well known as a means for obtaining approximate tail probabilities from information concerning the moments of the distribution. Recent saddlepoint and asymptotic methods lead to several alternative approximations. These alternatives are developed and compared by means of average relative error.     

New Developments in Statistical Computing

In 1885, Sir Francis Galton first defined the term “regression” and completed the theory of bivariate correlation. A decade later, Karl Pearson developed the index that we still use to measure correlation, Pearson's r. Our article is written in recognition of the 100th anniversary of Galton's first discussion of regression and correlation. We begin with a brief history. Then we present 13 different formulas, each of which represents a different computational and conceptual definition of r. Each formula suggests a different way of thinking about this index, from algebraic, geometric, and trigonometric settings. We show that Pearson's r (or simple functions of r) may variously be thought of as a special type of mean, a special type of variance, the ratio of two means, the ratio of two variances, the slope of a line, the cosine of an angle, and the tangent to an ellipse, and may be looked at from several other interesting perspectives.     

Simulating Controlled Variate and Rank Correlations Based on the Power Method Transformation

The power method transformation is a popular algorithm used for simulating correlated non normal continuous variates because of its simplicity and ease of execution. Statistical models may consist of continuous and (or) ranked variates. In view of this, the methodology is derived for simulating controlled correlation structures between non normal (a) variates, (b) ranks, and (c) variates with ranks in the context of the power method. The correlation structure between variate-values and their associated rank-order is also derived for the power method. As such, a measure of the potential loss of information is provided when ranks are used in place of variate-values. The results of a Monte Carlo simulation are provided to confirm and demonstrate the methodology.     

Conditioning out rare events for exponential families has negligible effect on inference

ABSTRACT

Suppose that we observe X ～Binomial(n, p). Inference on p is difficult if X = 0 or n. One way around this is to condition on these events not happening. We show that this has only an exponentially small effect on its cumulants. This is also true if we condition away other rare events. Our results are presented for exponential families, with applications to the binomial, multinomial and negative multinomial distributions.     

Relationships Between Central Moments and Cumulants,with Formulae for the Central Moments of Gamma Distributions

Abstract

Simple expressions are presented that relate cumulants to central moments without involving moments about the origin. These expressions are used to obtain recursive formulae for the central moments of the gamma distribution, with exponential and chi-square distributions as special cases.