Approximating the difference of two t-variables for all degrees of freedom using truncated variables

A scaled t‐distribution is used to approximate the distribution of a linear combination of two independent t‐variables for any number of degrees of freedom, and in particular for low degrees of freedom where moments do not exist. The approximation is the method‐of‐moments solution to the analogous problem with truncated t‐variables. The approximation exists for all degrees of freedom, is very accurate for more than two degrees of freedom, and performs as well as other approximations of this form when they exist.     

Fractional Moments and Maximum Entropy: Geometric Meaning

We consider the problem of recovering a probability density on a bounded or unbounded subset D of [0, ∞), from the knowledge of its sequence of fractional moments within a maximum entropy (MaxEnt) setup. Based upon entropy convergence results previously formulated, the fractional moments are selected so that the entropy of the MaxEnt approximation be minimum. A geometric interpretation of the reconstruction procedure is formulated as follows: the two moment curves generated by the unknown density and its MaxEnt approximation are interpolating in Hermite-Birkoff sense; that is, they are both interpolating and tangent at the selected nodes.     

Moments of order statistics of the standard two-sided power distribution

The standard two-sided power distribution is a flexible distribution having uniform, power function and triangular as subdistributions, and it is a reasonable alternative to the Laplace distribution in some cases. In this work, computationally efficient expressions for moments of order statistics, expressions for L-moments, and asymptotic results for sample extrema are derived. Then a simulation study is performed for the location-scale estimation problem of a small data set by considering the maximum likelihood estimation method and the best linear unbiased estimation method based on the moments of order statistics.     

Population moments and moments of the sample mean

This paper presents new formulae which simultaneously express and estimate moments of the sample mean and estimate population moments, from a simple random sample drawn without replcement from a finite population. By avoiding the generality of the multivariate case, these two problems are not only unified but are made significantly more tractable. Explicit solution are given up to eighth moments. Asymptotic results for infinite populations are also given.     

Characterizing parameters of multivariate elliptical distributions*

This paper defines new parameters characterizing multivariate elliptical distributions. Mardia's coefficient of multivariate kurtosis is shown to be essentially one of these parameters. A simple relation is established between centered multivariate product moments and second moments of the variables. The general results are verified on the contaminated normal distribution as an example.     

Moments of a ratio of two quadratic forms

In this paper, we give the exact moments of a ratio of qua- dratic forms in normal variables, where the quadratic forms are not assumed to be independent. This problem was tackled by other authors who gave approximations and partial results. Here we will give the exact moments for the general case.     

Discrimination between the binomial and hypergeometric models

This paper examines the problem of discerning whether sampling is being done with or without replacement. It is shown that with just one sample in the absence of additional information only a rather trivial solution is possible. For replicated samples, method of moments and graphically oriented procedures are considered. A simulation study reveals that there appears to be little advantage to employing any more sophisticated approach than the method of moments.     

Generalized franklin's model for randomized response sampling

In this paper we introduce two estimators of a population proportion when randomized response sampling with a normal randomizing distribution is used* The estimators have been obtained by using the method of moments. Both of the proposed estimators are shown to be more efficient than the corresponding estimators of Eranklin (1989 b).     

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.     

An alternative estimator for randomized response sampling with continuous distributions from a dichotomous population

In this paper, we introduce an alternative estimator of a population proportion from a dichotomous population when using randomized response sampling with continuous randomizing distributions. We also propose the alternative use of exponential randomizing densities. The estimator is obtained by method of moments and is compared with Franklin's (1989) estimator using normal and exponential distributions. The proposed estimator is more efficient than Franklin's (1989) estimator under suitable conditions for the two distributions.     

Skewness properties of asymmetric foems of tu1cey lambda distributions

Some measures of skewness are the third central moments the mean minus the median, arid measures in terms of percentiles. The signs of these measures are analysed for two asymmetric generalisations of Tukey's lambda distribution. The results expand and explain some results of Ramberg et al.1979, and illustrate the problem of choosing an appropriate measure of skewness.     

Parameter estimation in multivariate logit models with many binary choices

Multivariate Logit models are convenient to describe multivariate correlated binary choices as they provide closed-form likelihood functions. However, the computation time required for calculating choice probabilities increases exponentially with the number of choices, which makes maximum likelihood-based estimation infeasible when many choices are considered. To solve this, we propose three novel estimation methods: (i) stratified importance sampling, (ii) composite conditional likelihood (CCL), and (iii) generalized method of moments, which yield consistent estimates and still have similar small-sample bias to maximum likelihood. Our simulation study shows that computation times for CCL are much smaller and that its efficiency loss is small.     

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.     

Combining least-squares and quantile regressions

Least-squares and quantile regressions are method of moments techniques that are typically used in isolation. A leading example where efficiency may be gained by combining least-squares and quantile regressions is one where some information on the error quantiles is available but the error distribution cannot be fully specified. This estimation problem may be cast in terms of solving an over-determined estimating equation (EE) system for which the generalized method of moments (GMM) and empirical likelihood (EL) are approaches of recognized importance. The major difficulty with implementing these techniques here is that the EEs associated with the quantiles are non-differentiable. In this paper, we develop a kernel-based smoothing technique for non-smooth EEs, and derive the asymptotic properties of the GMM and maximum smoothed EL (MSEL) estimators based on the smoothed EEs. Via a simulation study, we investigate the finite sample properties of the GMM and MSEL estimators that combine least-squares and quantile moment relationships. Applications to real datasets are also considered.     

A Proposal for a New Bound for Discrete Distributions

In this work we re-examine some classical bounds for non negative integer-valued random variables by means of information theoretic or maxentropic techniques using fractional moments as constraints. The proposed new bound, no more analytically expressible in terms of moments or moment generating function (mgf), is built by mixing classical bounds and the Maximum Entropy (ME) approximant of the underlying distribution; such a new bound is able to exploit optimally all the information content provided by the sequence of given moments or by the mgf. Particular care will be devoted to obtain fractional moments from the available information given in terms of integer moments and/or moment generating function. Numerical examples show clearly that the bound improvement involving the ME approximant based on fractional moments is not trivial.     

Moments for matrix normal variables

In order to obtain moments for matrix normally distributed variables the moment generating function is differentiated by aid of matrix derivatives. Moments of arbitrary order as well as a recursive relation are obtained. Further, some more details are given for the first four moments     

A new bivariate beta distribution

A new bivariate beta distribution capable of providing better fits than all its competitors is introduced. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities and conditional moments. The method of maximum likelihood is used to derive the associated estimation procedure. Applications to six bivariate data sets are illustrated.     

Relations and identities for the moments of order statistics from a sample containing a single outlier

In this paper, we derive several new recurrence relations and indentities satisfied by the single and the product moments of order statistics from a sample of size n in the presence of an outlier. These recurrence relations involve the first two single moments and the product moments in samples of sized n?1 and less. By making use of these recurrence relations we show that it is sufficient to evaluate at most two single rerents and (n?2)/2 product moment when n is even and two single moments and (n?2)/2 product moments when n is odd, in order to evaluate the first and second single moments and product moments of all order statistics in a sample of size n comprising an outlier, given these moments for the all sample of size less than n. These generalize the results of Govindarajulu (1963), Joshi (1971), and Joshi and Balakrishnan (1982) to the case when the sample includes a single outlier. We also establish some simple identitites involving linear combination of convariances of order statistics. These results can be effectively used to reduce the amount of numerical computation considerably and also to check the accuracy of the computations while evaluating means, variances and covariances of order statistics from an outlier model.     

A comparison of the parameter estimation methods for bimodal mixture Weibull distribution with complete data

Bimodal mixture Weibull distribution being a special case of mixture Weibull distribution has been used recently as a suitable model for heterogeneous data sets in many practical applications. The bimodal mixture Weibull term represents a mixture of two Weibull distributions. Although many estimation methods have been proposed for the bimodal mixture Weibull distribution, there is not a comprehensive comparison. This paper presents a detailed comparison of five kinds of numerical methods, such as maximum likelihood estimation, least-squares method, method of moments, method of logarithmic moments and percentile method (PM) in terms of several criteria by simulation study. Also parameter estimation methods are applied to real data.     

Propriétés dans L2 et estimation des processus purement bilinéaires et strictement superdiagonaux à coefficients périodiques

L₂‐properties and estimation of purely bilinear and strictly superdiagonal time series models with periodic coefficients The authors consider the subclass of purely bilinear and strictly superdiagonal time series models with periodic coefficients. Indeed, thanks to their possible application to a wide variety of fields including economics and finance, bilinear time series models with time‐dependent coefficients have recently been the object of attention in the statistical literature. The authors give conditions ensuring the existence of a causal solution in L², the invertibility and the existence of higher‐order moments. The problem of estimating the parameters is also investigated through an approach based on second and third empirical moments. The authors numerically illustrate their theoretical results via Monte Carlo simulations.