Posterior probability intervals for wavelet thresholding

Summary. We use cumulants to derive Bayesian credible intervals for wavelet regression estimates. The first four cumulants of the posterior distribution of the estimates are expressed in terms of the observed data and integer powers of the mother wavelet functions. These powers are closely approximated by linear combinations of wavelet scaling functions at an appropriate finer scale. Hence, a suitable modification of the discrete wavelet transform allows the posterior cumulants to be found efficiently for any given data set. Johnson transformations then yield the credible intervals themselves. Simulations show that these intervals have good coverage rates, even when the underlying function is inhomogeneous, where standard methods fail. In the case where the curve is smooth, the performance of our intervals remains competitive with established nonparametric regression methods.     

Moments of L-Statistics: A Divided Differences Approach

The derivation of the distributions of linear combinations of order statistics or L-statistics and the computation of their moments has been approached in the literature several ways. In this paper we use the properties of divided differences to obtain expressions for moments of some order statistics that arise as special cases of L-statistics. Expectations of some well-known L-statistics such as the trimmed mean and the winsorised mean for the pareto distribution are computed. The study also undertakes the computation of L-moments that are expectations of certain linear combinations of order statistics. The algorithms have been implemented using some well-known continuous distributions as examples.     

Symbolic cumulant calculations for frequency domain time series

With time series data, there is often the issue of finding accurate approximations for the variance of such quantities as the sample autocovariance function or spectral estimate. Smith and Field (J. Time. Ser. Anal 14: 381–395, 1993) proposed a variance estimate motivated by resampling in the frequency domain. In this paper we present some results on the cumulants of this and other frequency domain estimates obtained via symbolic computation. The statistics of interest are linear combinations of products of discrete Fourier transforms. We describe an operator which calculates the joint cumulants of such statistics, and use the operator to deepen our understanding of the behaviour of the resampling based variance estimate. The operator acts as a filter for a general purpose operator described in Andrews and Stafford (J.R. Statist. Soc. B55, 613–627).     

Optimal bounds on l-statistics based on samples drawn with replacement from finite populations

We describe a method of establishing optimal bounds on the expectations of arbitrary linear combinations of order statistics based on iid samples drawn with replacement from finite populations of a fixed size. The bounds are expressed in terms of the population size, mean, central absolute moments, and coefficients of the combination. The bounds are precisely determined for the trimmed means and their differences, and single order statistics and their differences in particular. We also show that with increase in population size, our bounds approach the respective universal ones for arbitrary iid samples.     

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.     

A new nonparametric procedure designed for simulation studies

This paper demonstrates how certain statistics, computed from a sample of size n (from almost any distribution) may be simulated by using a sequence of substantially less than n random normal variates. Many statistics, θ, including almost all maximum likelihood estimates, can be expressed in terms of the sample trigonometric moments, STM. The STM are asymptotically multivariate normal with a mean vector and variance-covariance matrix easily expressible in terms of equally spaced characteristic function evaluations. Thus one only needs to know the Fourier transform or equivalently the characteristic function associated with elements of any moderate to large i. i. d. sample and have access to a normal random number generator to generate a sequence of STM with distributional properties almost identical to those of STM computed from that sample. These STM can in turn be used to compute the desired statistic θ.     

Comments on “Unbiased estimates for moments and cumulants in linear regression”

In a recent number of the Journal of Statistical Planning and Inference, a short communication by Withers and Nadarajah (2011) on the unbiased estimation of the moments and cumulants for linear regression models was published. However, some of the results presented in this paper, as well as those closely related, were partially obtained in other works, which were not referenced or were referenced in other contexts. In addition, the consistency of the derived estimates was not studied. The aim of these short comments is to briefly present these results, as well as some omitted works, in order to give larger representation of the state of art in this field.     

Functional marginality and response-surface fitting

Well-formed polynomials contain the marginal terms of all terms; for example, they contain both x 1 and x 2 if x 1 x 2 is present. Such models have a goodness of fit that is invariant to linear transformations of the x variables. Recently, selection procedures have been proposed which may not give well-formed polynomials. Analysis of two data sets for which non-well-formed polynomials have been selected shows that conversion to well-formed polynomials is beneficial in terms of goodness of fit, as well as giving fits invariant to linear transformation of the x variables. It is concluded that selection procedures should search among well-formed polynomials only.     

On characterizing the normal and poisson distributions

Multivariate normal, correlated multivariate Poisson and multiple Poisson distributions are characterized, in the class of exponential-type distributions, by the properties of the linear combinations of the variables, the properties of their cumulants and the recurance relation between the cumulants.     

The credibility premiums based on estimated moment-generating function

Traditionally, experience ratemaking is in principle based on the idea of Bühlmann’s credibility theory that, except for net premiums, was rarely applied to other premium calculation principles. This article uses Bühlmann’s credibility procedure to estimate moment-generating functions (MGFs) of risks and then deduces estimates of moments of those risks. For the premium calculation principles that can be expressed as functions of certain moments or more directly of the MGFs, this article develops a new type of experience ratemaking methods by means of the estimated MGFs and discusses their consistency and asymptotic normality. Numerical simulation shows that, under the Esscher and exponential premium principles, the new credibility estimates are better than existing credibilityestimates in the literature.     

Explicit formulas for the cumulants and the vector-valued odd moments of the multivariate linearly skewed elliptical distributions

In this letter explicit expressions are derived for the cumulants and the vector-valued odd moments of the multivariate linearly skewed elliptical family of distributions. The general calculations of such moments are described by multivariate integrals which complicate the calculations. We show how such multivariate computations can be projected into a univariate framework, which extremely simplifies the computations.     

Lower bounds on expectations of positive L-statistics from without-replacement models

We consider samples drawn without replacement from finite populations. We establish optimal lower non-negative and upper non-positive bounds on the expectations of linear combinations of order statistics centered about the population mean in units generated by the population central absolute moments of various orders. We also specify the general results for important examples of sample extremes, Gini mean differences and sample range. The paper completes the results of Papadatos and Rychlik [2004. Bounds on expectations of L-statistics from without replacement samples. J. Statist. Plann. Inference 124, 317–336], where sharp negative lower and positive upper bounds on the expectations of the combinations were presented for the without-replacement samples.     

On fractional moments of quadratic expressions in normal variables1

Fractional moments, product cumulants and product moments of general quadratic expressions in singular and nonsingular normal variables are explicitly evaluated. A general method of deriving such moments is also indicated. Particular cases art; shown to agree with known results.     

Correlation is first order independent of transformation

We show that the correlation between the estimates of two parameters is almost unchanged if they are each transformed in an arbitrary way. To be more specific, the correlation of two estimates is invariant (except for a possible sign change) up to a first order approximation, to smooth transformations of the estimates. There is a sign change if exactly one of the transformations is decreasing in a neighborhood of its parameter. In addition, we approximate the variance, covariance and correlation between functions of sample means and moments.     

Bias-reduced estimates for skewness, kurtosis, L-skewness and L-kurtosis

Estimates based on L-moments are less non-robust than estimates based on ordinary moments because the former are linear combinations of order statistics for all orders, whereas the later take increasing powers of deviations from the mean as the order increases. Estimates based on L-moments can also be more efficient than maximum likelihood estimates. Similarly, L-skewness and L-kurtosis are less non-robust and more informative than the traditional measures of skewness and kurtosis. Here, we give nonparametric bias-reduced estimates of both types of skewness and kurtosis. Their asymptotic computational efficiency is infinitely better than that of corresponding bootstrapped estimates.     

Econometrics exams and round numbers: Use or misuse of indirect estimation methods?

The use of Monte Carlo methods to generate exam datasets is nowadays a well-established practice among econometrics and statistics examiners all over the world. Its advantages are well known: providing each student a different data set ensures that estimates are actually computed individually, rather than copied from someone sitting nearby. The method however has a major fault: initial “random errors,” such as mistakes in downloading the assigned dataset, might generate downward bias in student evaluation. We propose a set of calibration algorithms, typical of indirect estimation methods, that solve the issue of initial “random errors” and reduce evaluation bias. Ensuring round initial estimates of the parameters for each individual dataset, our calibration procedures allow the students to determine if they have started the exam correctly. When initial estimates are not round numbers, this random error in the initial stage of the exam can be corrected for immediately, thus reducing evaluation bias. The procedure offers the further advantage of rounding markers’ life by allowing them to check round numbers answers only, rather than lists of numbers with many decimal digits¹.     

The effect of dependence on distribution of the functions of random variables

In this article, we study the effect of dependence on the distributional properties of functions of two random variables. Expressions for the cumulative distribution functions of the linear combinations, products, and ratios of two dependent random variables in terms of their associated copula are derived. We discuss the effect of dependence on quantities such as the variances of linear combinations of functions, the value-at-risk measure, and the stress–strength parameter. Several examples, a simulation study, and a real data analysis are provided to illustrate the result.     

Generalized elliptical distributions

A family of distributions generated by an operator acting on generalized normal density is introduced. This family contains as particular cases many known distributions, including the generalized normal, generalized t, and generalized gamma distributions. Several mathematical properties of the family (including expansions, characteristic function, moments, cumulants, and order statistics properties) are derived. Estimation procedures are derived too by the method of moments, method of maximum likelihood, and the method of empirical characteristic function. A real data application is presented. Finally, extensions to the multivariate case are outlined.     

Some results for multidimensional stationary independent increment processes

The characteristic function, cumulants and moments of vector-valued multidimensional processes, satisfying properties similar to stationary independent increments, are derived. By considering a set of additional postulates for such processes, it is shown that the marginal distribution of such processes is multivariate Poisson. Some of the results in this paper are extensions of the properties of the first two moments of a univariate one-dimensional process with stationary independent increments.