The johnson translation system in monte carlo studies

Multivariate lognormal and sinh^-1-normal random vectors, members of the Johnson family, can be generated by direct transformation of multinormal random vectors. Formulae for specifying the parameters of the multinomial distribution in terms of the first and second order moments of the desired multivariate lognormal or sinh^-1-normal distribution are given. These results facilitate the use of these distributions in Monte Carlo studies. In some previous studies employing these distributions, the effect of non-normality was confounded with that of unequal covariance structure.     

Robust Bayesian analysis of the binomial empirical Bayes problem

Empirical Bayes estimation is considered for an i.i.d. sequence of binomial parameters θ_i arising from an unknown prior distribution G(.). This problem typically arises in industrial sampling, where samples from lots are routinely used to estimate the lot fraction defective of each lot. Two related issues are explored. The first concerns the fact that only the first few moments of G are typically estimable from the data. This suggests consideration of the interval of estimates (e.g., posterior means) corresponding to the different possible G with the specified moments. Such intervals can be obtained by application of well-known moment theory. The second development concerns the need to acknowledge the uncertainty in the estimation of the first few moments of G. Our proposal is to determine a credible set for the moments, and then find the range of estimates (e.g., posterior means) corresponding to the different possible G with moments in the credible set.     

On the Parameter Estimation of the Asymmetric Multivariate Laplace Distribution

This article examines a family of three-parameter multivariate Laplace distributions ML _p (a, μ, Σ) which is closed under constant shifts. Parameter vectors a and μ are called shift and shape parameter, respectively, positive definite p × p-matrix Σ is a scale parameter. The first three moments are derived and used for estimating the parameters. The behavior of the obtained estimates is explored in a simulation experiment.     

The distribution of a moment estimator for a parameter of the generalized poisson distribution

The ratio of the sample variance to the sample mean estimates a simple function of the parameter which measures the departure of the Poisson-Poisson from the Poisson distribution. Moment series to order n?²⁴ are given for related estimators. In one case, exact integral formulations are given for the first two moments, enabling a comparison to be made between their asymptotic developments and a computer-oriented extended Taylor series (COETS) algorithm. The integral approach using generating functions is sketched out for the third and fourth moments. Levin's summation algorithm is used on the divergent series and comparative simulation assessments are given.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from a generalized half-logistic distribution with application 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 generalized half-logistic distribution. The use of these relations in a systematic recursive manner enables the computation of all the means, variances, and covariances of progressively Type-II right-censored order statistics from the generalized half-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁, …, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Sandhu [Recurrence relations for single and product moments of order statistics from a generalized half-logistic distribution with applications to inference, J. Stat. Comput. Simul. 52 (1995), pp. 385–398.]. The moments so determined are then utilized to derive the best linear unbiased estimators of the scale and location–scale parameters of the generalized half-logistic distribution. The best linear unbiased predictors of censored failure times are discussed briefly. Finally, a numerical example is presented to illustrate the inferential method developed here.     

Moments of the Product-Limit Estimator Under Left-Truncation and Right-Censoring

ABSTRACT

The product-limit estimator (PLE) is a well-known nonparametric estimator for the distribution function of the lifetime when data are left-truncated and right-censored. Much work has focused on developing its asymptotic properties. Finite sample results have been difficult to obtain. This article is concerned about finite moments of the PLE. The moments of the PLE can be represented as a power series in n ^?1. In addition, through the U-statistic mechanism, we obtain also computable formulas for the first, second, third, and fourth of the PLE up to o(n ^?2). Finally, a numerical example is presented.     

Linear estimation of location and scale parameters in the generalized gamma distribution

The expressions for moments of order statistics from the generalized gamma distribution are derived. Coefficients to get the BLUEs of location and scale parameters in the generalized gamma distribution are computed. Some simple alternative linear unbiased estimates of location and scale parameters are also proposed and their relative efficiencies compared to the BLUEs are studied.     

Higher order moments of order statistics from the Lindley distribution and associated inference

ABSTRACT

The Lindley distribution is an important distribution for analysing the stress–strength reliability models and lifetime data. In many ways, the Lindley distribution is a better model than that based on the exponential distribution. Order statistics arise naturally in many of such applications. In this paper, we derive the exact explicit expressions for the single, double (product), triple and quadruple moments of order statistics from the Lindley distribution. Then, we use these moments to obtain the best linear unbiased estimates (BLUEs) of the location and scale parameters based on Type-II right-censored samples. Next, we use these results to determine the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics to develop Edgeworth approximate confidence intervals of the location and scale Lindley parameters. In addition, we carry out some numerical illustrations through Monte Carlo simulations to show the usefulness of the findings. Finally, we apply the findings of the paper to some real data set.     

Adjusted Pearson residuals in beta regression models

In this paper, matrix formulae of order n^?1, where n is the sample size, for the first two moments of Pearson residuals are obtained in beta regression models. Adjusted Pearson residuals are also obtained, having, to this order, expected value zero and variance one. Monte Carlo simulation results are presented illustrating the behaviour of both adjusted and unadjusted residuals.     

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.     

Riordan matrices and higher-dimensional lattice walks

An algebraic combinatorial method is used to count higher-dimensional lattice walks in Z^m${\mathbb{Z}}^m$ that are of length n ending at height k. As a consequence of using the method, Sands’ two-dimensional lattice walk counting problem is generalized to higher dimensions. In addition to Sands’ problem, another subclass of higher-dimensional lattice walks is also counted. Catalan type solutions are obtained and the first moments of the walks are computed. The first moments are then used to compute the average heights of the walks. Asymptotic estimates are also given.     

Sharp Bounds for Generalized Order Statistics via Logarithmic Moments

ABSTRACT

We present sharp bounds for expectations of generalized order statistics with random indices. The bounds are expressed in terms of logarithmic moments E X ^a (log max {1, X}) ^b of the underlying observation X. They are attainable and provide characterizations of some non trivial distributions. No restrictions are imposed on the parameters of the generalized order statistics model.     

THE GENERALIZED SECANT HYPERBOLIC DISTRIBUTION AND ITS PROPERTIES

ABSTRACT

The properties of a family of distributions generalizing the secant hyperbolic are developed. This family consists of symmetric distributions, with kurtosis ranging from 1.8 to infinity, and includes the logistic as a special case, the uniform as a limiting case, and closely approximates the normal and Student's t-distributions with corresponding kurtosis. A significant difference between this family and Student's t is that for any member of the generalized secant hyperbolic family, all moments are finite. Further, technical difficulties associated with evaluating moments of Student's t (especially for fractional degrees of freedom) are not present with this family. The properties of the maximum likelihood and modified maximum likelihood estimates of the location and scale parameters for complete samples are considered. Examples illustrate the methods developed in this work.     

Asymptotic expansions for the moments of serial correlation coefficients

This work is concerned with evaluating the moments of a number of serial correlation coefficients which arise in various ways and where the observations are from the first order autoregressive Gaussian process with known zero mean. The forms considered have biases whose main parts (of order 0(n^-1) , where n is the sample size) are substantially different. They are the intra-class correlation,the maximum likelihood estimators and an estimator whose main part of the bias is sere. The moments are obtained as asymptotic expansions in terms of the parameter of the process and to terms of order 0(n^-3). It is found that removing certain end terms in the denominator of a serial correlation has the effect of reducing the magnitude of the main part of its bias considerably and in one case completely eliminating it. This work extends the results of various authors,e.g.Kandall(1954), Marriott and pope(1954) and white (1961) in the special cases of the first order autogressive process.     

Negative moments of a modified power series distribution and bias of the maximum likelihood estimator

The class of Modified Power Series distributions (MPSD) containing Lagrangian Poisson (LPD) (Consul and Jain, 1973) and Lagrangian binomial distributions (LBD) (Jain and Consul, 1971) was studied by Gupta (1974). We investigate the problem of finding the negative momentsE[X^-r ], of displaced and decapitated Modified Power Series Distributions. We derive the relationship between rand (r-1) negative moments. The negative moments of the decapitated and displaced LPD are obtained. These results are, then, used to find the exact amount of bias in the ML estimators of the parameters in the LPD and the LBD. We have also given the variances of the ML estimator and the minimum variance unbiased estimator of the parameter in the LPD.     

Moments of the sampled autocovariances and autocorrelations for a Gaussian white-noise process

Cumulants, moments about zero, and central moments are obtained for the mean-corrected serial covariances and serial correlations for series realizations of length n from a white-noise Gaussian process. All first and second moments (and some third, fourth, and higher moments) are given explicitly for the serial covariances; and the corresponding moments for the serial correlations are derived either explicitly or implicitly.     

A new family of multivariate skew slash distribution

In this article, the new family of multivariate skew slash distribution is defined. According to the definition, a stochastic representation of the multivariate skew slash distribution is derived. The first four moments and measures of skewness and kurtosis of a random vector with the multivariate skew slash distribution are obtained. The distribution of quadratic forms for the multivariate skew slash distribution and the non central skew slash χ² distribution are studied. Maximum likelihood inference and real data illustration are discussed. In the end, the potential extension of multivariate skew slash distribution is discussed.     

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.     

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.     

A closed testing procedure for comparing successive exponential populations with respect to location parameter

Consider k independent random samples such that i^th sample is drawn from a two-parameter exponential population with location parameter μ_i and scale parameter θ_i,?i = 1, …, k. For simultaneously testing differences between location parameters of successive exponential populations, closed testing procedures are proposed separately for the following cases (i) when scale parameters are unknown and equal and (ii) when scale parameters are unknown and unequal. Critical constants required for the proposed procedures are obtained numerically and the selected values of the critical constants are tabulated. Simulation study revealed that the proposed procedures has better ability to detect the significant differences and has more power in comparison to exiting procedures. The illustration of the proposed procedures is given using real data.