Moments of the function of non-normal random vector with applications to econometric estimators and test statistics 1

In this paper we have provided a general result on the moments of a function of nonnormal random vector. The results for the normal case follow as a special case of this result. It is also indicated that the moments of a large class of econometric estimators and test statistics can be obtained by using our general result. This includes least squares estimator in the dynamic model, unit root tests, and the two step semiparametric estimators, among others.     

Asymptotic Normality of Extreme Quantile Estimators Based on the Peaks-Over-Threshold Approach

The POT (Peaks-Over-Threshold) approach consists of using the generalized Pareto distribution (GPD) to approximate the distribution of excesses over thresholds. In this article, we establish the asymptotic normality of the well-known extreme quantile estimators based on this POT method, under very general assumptions. As an illustration, from this result, we deduce the asymptotic normality of the POT extreme quantile estimators in the case where the maximum likelihood (ML) or the generalized probability-weighted moments (GPWM) methods are used. Simulations are provided in order to compare the efficiency of these estimators based on ML or GPWM methods with classical ones proposed in the literature.     

Estimating expected sojourn times for a markov renewal process

We consider the estimation of the expected sojourn time in a Markov renewal process under the data condition that only the counts of the exits from the states are available for fixed intervals of time. For analytical and illustrative purposes we concentrate on the two-state process case. We present least squares and method of moments estimators and compare their statistical properties both analytically and empirically. We also present modified estimators with improved properties based upon an overlapping interval sampling strategy. The major results indicate that the least squares estimator is biased in general with the bias depending on the size of the sampling interval and the first two moments of the sojourn time distribution function. The bias becomes negligible as the size of the sampling interval increases. Analytical and empirical results indicate that the method of moments estimator is less sensitive to the size of the sampling interval and has slightly better mean squared error properties than the least squares estimator.     

Computational comparison of the general weighted moments estimators of the scale parameter of a Pareto distribution with a known shape parameter with other estimators based on a multiply type II-censored sample

Wu et al. [Computational comparison for weighted moments estimators and BLUE of the scale parameter of a Pareto distribution with known shape parameter under type II multiply censored sample, Appl. Math. Comput. 181 (2006), pp. 1462–1470] proposed the weighted moments estimators (WMEs) of the scale parameter of a Pareto distribution with known shape parameter on a multiply type II-censored sample. They claimed that some WMEs are better than the best linear unbiased estimator (BLUE) based on the exact mean-squared error (MSE). In this paper, the general WME (GWME) is proposed and the computational comparison of the proposed estimator with the WMEs and BLUE is done on the basis of the exact MSE for given sample sizes and different censoring schemes. As a result, the GWME is performing better than the best estimator among 12 WMEs and BLUE for all cases. Therefore, GWME is recommended for use. At last, one example is given to demonstrate the proposed GWME.     

Properties of estimators for the gamma distribution

Accurate moments of maximum likelihood and moment estimators for the scale and shape parameters of a two parameter gamma density are given, the former being tabulated over a segment of the parameter space. In addition, joint acceptance regions are given for a particular case. The three parameter model is also considered and comments made on second order asymptotics for the maximum likelihood estimators     

On multiple change-point estimation for Poisson process

This work is devoted to the problem of change-point parameter estimation in the case of the presence of multiple changes in the intensity function of the Poisson process. It is supposed that the observations are independent inhomogeneous Poisson processes with the same intensity function and this intensity function has two jumps separated by a known quantity. The asymptotic behavior of the maximum-likelihood and Bayesian estimators are described. It is shown that these estimators are consistent, have different limit distributions, the moments converge and that the Bayesian estimators are asymptotically efficient. The numerical simulations illustrate the obtained results.     

Moments of truncated normal/independent distributions

In this work we have considered the problem of finding the moments of a doubly truncated member of the class of normal/independent distributions. We obtained a general result and then use it to derive the moments in the case of doubly truncated versions of Pearson type VII distribution, slash distribution, contaminated normal distribution, double exponential distribution and variance gamma distribution. We also give an application of some actuarial data.     

Efficient Estimation of Parameters of the Negative Binomial Distribution

In this article we investigate a class of moment-based estimators, called power method estimators, which can be almost as efficient as maximum likelihood estimators and achieve a lower asymptotic variance than the standard zero term method and method of moments estimators. We investigate different methods of implementing the power method in practice and examine the robustness and efficiency of the power method estimators.     

Robust fitting of claim severity distributions and the method of trimmed moments

Many quantities arising in non-life insurance depend on claim severity distributions, which are usually modeled assuming a parametric form. Obtaining good estimates of the quantities, therefore, reduces to having good estimates of the model parameters. However, the notion of ‘good estimate’ depends on the problem at hand. For example, the maximum likelihood estimators (MLEs) are efficient, but they generally lack robustness. Since outliers are common in insurance loss data, it is therefore important to have a method that allows one to balance between efficiency and robustness. Guided by this philosophy, in the present paper we suggest a general estimation method that we call the method of trimmed moments (MTM). This method is appropriate for various model-fitting situations including those for which a close fit in one or both tails of the distribution is not required. The MTM estimators can achieve various degrees of robustness, and they also allow the decision maker to easily see the actions of the estimators on the data, which makes them particularly appealing. We illustrate these features with detailed theoretical analyses and simulation studies of the MTM estimators in the case of location–scale families and several loss distributions such as lognormal and Pareto. As a further illustration, we analyze a real data set concerning hurricane damages in the United States from 1925 to 1995.     

Estimation and Comparison of Signed Symmetric Covariation Coefficient and Generalized Association Parameter for Alpha-stable Dependence Modeling

In this paper, we study the estimators of two measures of dependence: the signed symmetric covariation coefficient (scov) proposed by Garel and Kodia and the generalized association parameter (g.a.p.) put forward by Paulauskas. In the sub-Gaussian case, the scov and the g.a.p. coincide. The estimator of the scov proposed here is based on fractional lower-order moments. The estimator of the g.a.p. is based on estimation of a stable spectral measure. We investigate the relative performance of these estimators by comparing results from simulations.     

A note on biases of jackknife estimators of third central moments

In this paper we study the biases of jackknife estimators of central third moments which play an important role in improving the accuracy of the normal approximation. It has been found in simulation studies that the jackknife estimator of the skewness coefficient, into which the jackknife variance and third moment estimators are substituted, have downward biases. For the jackknife variance estimators, their asymptotic properties are precisely studied and their biases are discussed theoretically, Here we study the biases of the jackknife estimators of the central third moments for U-statistics theoretically, The results show that the biases are not always downward.     

Almost Consistent Estimation of Panel Probit Models with “Small” Fixed Effects

We propose four different GMM estimators that allow almost consistent estimation of the structural parameters of panel probit models with fixed effects for the case of small Tand large N. The moments used are derived for each period from a first order approximation of the mean of the dependent variable conditional on explanatory variables and on the fixed effect. The estimators differ w.r.t. the choice of instruments and whether they use trimming to reduce the bias or not. In a Monte Carlo study, we compare these estimators with pooled probit and conditional logit estimators for different data generating processes. The results show that the proposed estimators outperform these competitors in several situations.     

Comparison of different estimation techniques for portfolio selection

The main problem in applying the mean-variance portfolio selection consists of the fact that the first two moments of the asset returns are unknown. In practice the optimal portfolio weights have to be estimated. This is usually done by replacing the moments by the classical unbiased sample estimators. We provide a comparison of the exact and the asymptotic distributions of the estimated portfolio weights as well as a sensitivity analysis to shifts in the moments of the asset returns. Furthermore we consider several types of shrinkage estimators for the moments. The corresponding estimators of the portfolio weights are compared with each other and with the portfolio weights based on the sample estimators of the moments. We show how the uncertainty about the portfolio weights can be introduced into the performance measurement of trading strategies. The methodology explains the bad out-of-sample performance of the classical Markowitz procedures.     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

Parameter estimation of state space models for univariate observations

This paper contributes to the problem of estimation of state space model parameters by proposing estimators for the mean, the autoregressive parameters and the noise variances which, contrarily to maximum likelihood, may be calculated without assuming any specific distribution for the errors. The estimators suggested widen the scope of the application of the generalized method of moments to some heteroscedastic models, as in the case of state-space models with varying coefficients, and give sufficient conditions for their consistency. The paper includes a simulation study comparing the proposed estimators with maximum likelihood estimators. Finally, these methods are applied to the calibration of the meteorological radar and estimation of area rainfall.     

Comparison of improved regression estimators with and without moments

For estimating the coefficients in a linear regression model, the double k–class estimators are considered and the small disturbance asymptotic approximation for their density function is obtained. Then employing the criterion of concentration probability around the true parameter values, a comparison is made between the estimators possessing finite moments and the estimators having no finite moments.     

The large-sample distribution of the most fundamental of statistical summaries

We consider one of the most fundamental of statistical problems, namely that of inference for the mean, standard deviation and coefficients of skewness and kurtosis of an unknown univariate distribution. Assuming the distributional form of the parent population to be unknown, we focus our attention on moment-based inference. As is well-known, the method of moments estimates of the population measures under consideration are the sample mean, standard deviation and coefficients of skewness and kurtosis. Despite being some of the most frequently used of all statistical summaries, it comes as a surprise to find that their full joint distribution has not previously been studied in the literature. We derive a very general theoretical result for the large-sample asymptotic joint distribution of the four estimators and use simulation to explore the validity of the result as a means of approximating the biases, variances and covariances of the estimators for finite sample sizes. The theoretical result is then used to obtain asymptotically distribution-free inferential procedures for the population measures of original interest. Specifically, we propose and investigate the efficacy of bias-corrected and non-bias-corrected methods for point estimation and confidence set construction. We also discuss the relevance of the developed methodology both as an end in itself and as an aid to model formulation.     

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).     

Panel data unit roots tests: The role of serial correlation and the time dimension

We investigate the influence of residual serial correlation and of the time dimension on statistical inference for a unit root in dynamic longitudinal data, known as panel data in econometrics. To this end, we introduce two test statistics based on method of moments estimators. The first is based on the generalized method of moments estimators, while the second is based on the instrumental variables estimator. Analytical results for the Instrumental Variables (IV) based test in a simplified setting show that (i) large time dimension panel unit root tests will suffer from serious size distortions in finite samples, even for samples that would normally be considered large in practice, and (ii) negative serial correlation in the error terms of the panel reduces the power of the unit root tests, possibly up to a point where the test becomes biased. However, near the unit root the test is shown to have power against a wide range of alternatives. These findings are confirmed in a more general set-up through a series of Monte Carlo experiments.     

Estimating Particle Shape and Orientation Using Volume Tensors

We develop statistical procedures for estimating shape and orientation of arbitrary three‐dimensional particles. We focus on the case where particles cannot be observed directly, but only via sections. Volume tensors are used for describing particle shape and orientation, and we derive stereological estimators of the tensors. These estimators are combined to provide consistent estimators of the moments of the so‐called particle cover density. The covariance structure associated with the particle cover density depends on the orientation and shape of the particles. For instance, if the distribution of the typical particle is invariant under rotations, then the covariance matrix is proportional to the identity matrix. We develop a non‐parametric test for such isotropy. A flexible Lévy‐based particle model is proposed, which may be analysed using a generalized method of moments in which the volume tensors enter. The developed methods are used to study the cell organization in the human brain cortex.