Higher-order moments,cumulants and spectral densities of the NGINAR(1) process

The motivation for time series with geometric marginal distributions arises from noting that the Poisson distribution is not always suitable for the modeling and analysis of integer-valued time series. The NGINAR(1) process that has been introduced by Risti? et al. (2009) represents a class of such time series. Joint higher-order (factorial) moments and cumulants with some other related statistical measures of the NGINAR(1) process are constructed. Also, the spectral and bispectral density functions of this process are investigated, including their nonparametric estimators, using the multitapering method. A real data example of the nonparametric multitaper spectral estimates is investigated, with a discussion of the results obtained.     

Progressive type II censored order statistics from exponential distributions

In the model of progressive type II censoring, point and interval estimation as well as relations for single and product moments are considered. Based on two-parameter exponential distributions, maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators (UMVUEs) and best linear unbiased estimators (BLUEs) are derived for both location and scale parameters. Some properties of these estimators are shown. Moreover, results for single and product moments of progressive type II censored order statistics are presented to obtain recurrence relations from exponential and truncated exponential distributions. These relations may then be used to compute all the means, variances and covariances of progressive type II censored order statistics based on exponential distributions for arbitrary censoring schemes. The presented recurrence relations simplify those given by Aggarwala and Balakrishnan (1996)     

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.     

Minimum contrast estimation of random processes based on information of second and third orders

This paper presents the results on consistency and asymptotic normality of a class of minimum contrast estimators for random processes with short- or long-range dependence based on the second- and third-order cumulant spectra. Asymptotic properties of sample spectral functionals of second and third orders, which are of independent interest in view of their possible use for nonparametric estimation of processes with short- or long-range dependence, are also provided.     

Asymptotic cumulants of the estimator of the canonical parameter in the exponential family

Asymptotic cumulants of the maximum likelihood estimator of the canonical parameter in the exponential family are obtained up to the fourth order with the added higher-order asymptotic variance. In the case of a scalar parameter, the corresponding results with and without studentization are given. These results are also obtained for the estimators by the weighted score, especially for those using the Jeffreys prior. The asymptotic cumulants are used for reducing bias and mean square error to improve a point estimator and for interval estimation to have higher-order accuracy. It is shown that the kurtosis to squared skewness ratio of the sufficient statistic plays a fundamental role.     

Binomial AR(1) processes: moments,cumulants, and estimation

The modelling and analysis of count-data time series are areas of emerging interest with various applications in practice. We consider the particular case of the binomial AR(1) model, which is well suited for describing binomial counts with a first-order autoregressive serial dependence structure. We derive explicit expressions for the joint (central) moments and cumulants up to order 4. Then, we apply these results for expressing moments and asymptotic distribution of the squared difference estimator as an alternative to the sample autocovariance. We also analyse the asymptotic distribution of the conditional least-squares estimators of the parameters of the binomial AR(1) model. The finite-sample performance of these estimators is investigated in a simulation study, and we apply them to real data about computerized workstations.     

CUMULANTS AND PARTITION LATTICES1

The (joint) cumulant of a set of (possibly coincident) random variables is defined as an alternating sum of moments with appropriate integral coefficients. By exploiting properties of the Mobius function of a partition lattice some basic results concerning cumulants are derived and illustrations of their use given.     

Asymptotic expansions of moments and cumulants

This paper shows how procedures for computing moments and cumulants may themselves be computed from a few elementary identities.Many parameters, such as variance, may be expressed or approximated as linear combinations of products of expectations. The estimates of such parameters may be expressed as the same linear combinations of products of averages. The moments and cumulants of such estimates may be computed in a straightforward way if the terms of the estimates, moments and cumulants are represented as lists and the expectation operation defined as a transformation of lists. Vector space considerations lead to a unique representation of terms and hence to a simplification of results. Basic identities relating variables and their expectations induce transformations of lists, which transformations may be computed from the identities. In this way procedures for complex calculations are computed from basic identities.The procedures permit the calculation of results which would otherwise involve complementary set partitions, k-statistics, and pattern functions. The examples include the calculation of unbiased estimates of cumulants, of cumulants of these, and of moments of bootstrap estimates.     

Inferences for generalized exponential distribution based on record statistics

In this paper we consider three parameter generalized exponential distribution. Exact expressions for single and product moments of record statistics are derived. These expressions are written in terms of Riemann zeta and polygamma functions. Recurrence relations for single and product moments of record statistics are also obtained. These relations can be used to obtain the higher order moments from those of the lower order. The means, variances and covariances of the record statistics are computed for various values of the shape parameter and for some record statistics. These values are used to compute the coefficients of the best linear unbiased estimators of the location and scale parameters. The variances of these estimators are also presented. The predictors of the future record statistics are also discussed.     

Asymptotic cumulants of the minimum phi-divergence estimator for categorical data under possible model misspecification

Abstract

The asymptotic cumulants of the minimum phi-divergence estimators of the parameters in a model for categorical data are obtained up to the fourth order with the higher-order asymptotic variance under possible model misspecification. The corresponding asymptotic cumulants up to the third order for the studentized minimum phi-divergence estimator are also derived. These asymptotic cumulants, when a model is misspecified, depend on the form of the phi-divergence. Numerical illustrations with simulations are given for typical cases of the phi-divergence, where the maximum likelihood estimator does not necessarily give best results. Real data examples are shown using log-linear models for contingency tables.     

Estimation of the Length Distribution of Marine Populations in the Gaussian-multinomial Setting using the Method of Moments

In this paper, the problem of estimation of the length distribution of marine populations in the Gaussian-multinomial model is considered. For the purpose of the mean and covariance parameter estimation, the method of moments estimators are developed. That is, minimum variance linear unbiased estimator for the mean frequency vector is derived and a consistent estimator for the covariance matrix of the length observations is presented. The usefulness of the proposed estimators is illustrated with an analysis of real cod length measurement data.     

Recurrence relations for single and product moments of order statistics from a generalized half logistic distribution with applications to inference

In this paper, we derive several recurrence relations satisfied by the single and product moments of order statistics from a generalized half logistic distribution. These generalize the corresponding results for the half logistic distribution established by Balakrishnan (1985). The relations established in this paper will enable one to compute the single and product moments of all order statistics for all sample sizes in a simple recursive manner; this may be done for any choice of the shape parameter k. These moments can then be used to determine the best linear unbiased estimators of location and scale parameters from complete as well as Type-I1 censored samples.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from generalized logistic distribution with applications to inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a generalized logistic distribution. The use of these relations in a systematic manner allow us to compute all the means, variances, and covariances of progressively Type-II right censored order statistics from the generalized logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁, …, R_m). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the generalized logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made through Monte Carlo simulations. Finally, the best linear unbiased predictors of censored failure times is discussed briefly.     

Almost unbiased ratio and product estimators based on interpenetrating subsamples

This paper proposes two classes of almost unbiased ratio and product estimators, in the case of interpenetrating subsample designs, which include the estimators considered by earlier authors. Several other almost unbiased ratio and product estimatbrs are given whlch are particular members of the proposed classes of estimators. Optimum estimators are also identified in these classes.     

Inference for Random Coefficient INAR(1) Process Based on Frequency Domain Analysis

In this article, we present the explicit expressions for the higher-order moments and cumulants of the first-order random coefficient integer-valued autoregressive (RCINAR(1)) process. The spectral and bispectral density functions are also obtained, which can characterize the RCINAR(1) process in the frequency domain. We use a frequency domain approach which is named Whittle criterion to estimate the parameters of the process. We propose a test statistic which is based on the frequency domain approach for the hypothesis test, H₀: α = 0?H₁: 0 < α < 1, where α is the mean of the random coefficient in the process. The asymptotic distribution of the test statistic is obtained. We compare the proposed test statistic with other statistics that can test serial dependence in time series of count via a typically numerical simulation, which indicates that our proposed test statistic has a good power.     

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.     

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.     

Concomitants of k-record values arising from Morgenstern family of distributions and their applications in parameter estimation

In this paper, we have obtained the marginal and joint distributions of concomitants of k-record values for the Morgenstern family of distributions (MFD) and hence obtained the moments and product moments of concomitants of k-record values. Applying this results we have derived the best linear unbiased estimators of some parameters involved in Morgenstern type bivariate logistic distribution which belongs to MFD based on concomitants of k-record values.     

Asymptotically Unbiased Estimation of the Coefficient of Tail Dependence

Abstract. We introduce and study a class of weighted functional estimators for the coefficient of tail dependence in bivariate extreme value statistics. Asymptotic normality of these estimators is established under a second‐order condition on the joint tail behaviour, some conditions on the weight function and for appropriately chosen sequences of intermediate order statistics. Asymptotically unbiased estimators are constructed by judiciously chosen linear combinations of weighted functional estimators, and variance optimality within this class of asymptotically unbiased estimators is discussed. The finite sample performance of some specific examples from our class of estimators and some alternatives from the recent literature are evaluated with a small simulation experiment.     

Recurrence relations for moments of progressively censored order statistics from logistic distribution with applications to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a logistic distribution. The use of these relations in a systematic manner allows us to compute all the means, variances and covariances of progressively Type-II right censored order statistics from the logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁,…,R_m). The results established here generalize the corresponding results for the usual order statistics due to [Shah, 1966] and [Shah, 1970]. These moments are then utilized to derive best linear unbiased estimators of the location and scale parameters of the logistic distribution. A comparison of these estimators with the maximum likelihood estimations is then made. The best linear unbiased predictors of censored failure times are briefly discussed. Finally, an illustrative example is presented.