Hermite polynomials and truncated trivariate normal distributions

The moments of a trivariate and in general of a multivariate normal distribution, which is truncated with respect to a single variable, are obtained by using properties of Hermite polynomials. An expression for the truncated correlation coefficient is derived in terms of the true population correlation coefficient and the truncation point. The values of this truncated correlation coefficient are tabulated for given values of the true correlation coefficient and a few selected values of the truncation point. A listing of the computer program for this purpose is also given.     

MOMENTS OF A LINEARLY TRUNCATED BIVARIATE NORMAL DISTRIBUTION1

The moments are obtained for a bivariate normal distribution which is linearly truncated with respect to both variables; the variables may be correlated. From these moments the parameters of the distribution can be estimated.     

On multivariate truncated generalized Cauchy distribution

In this paper, a multivariate form of truncated generalized Cauchy distribution (TGCD), which is denoted by (MVTGCD), is introduced. The joint density function, conditional density function, moment generating function and mixed moments of order ${b=\sum_{i=1}^{k}b_{i}}$ are obtained. Making use of the mixed moments formula, skewness and kurtosis in case of the bivariate case are obtained. Also, all parameters of the distribution are estimated using the maximum likelihood and Bayes methods. A real data set is introduced and analyzed using three models. The first model is the bivariate Cauchy distribution, the second is the truncated bivariate Cauchy distribution and the third is the bivariate truncated generalized Cauchy distribution. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the bivariate truncated generalized Cauchy model fits the data better than the other models.     

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.     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.     

Recurrence relations for the moments of truncated multinormal distribution

In this paper some recurrence relations of moments of doubly truncated multivariate normal distribution are obtained. The bivariate case is given as an example and some applications are indicated.     

Some results on the truncated multivariate t distribution

The use of truncated distributions arises often in a wide variety of scientific problems. In the literature, there are a lot of sampling schemes and proposals developed for various specific truncated distributions. So far, however, the study of the truncated multivariate t (TMVT) distribution is rarely discussed. In this paper, we first present general formulae for computing the first two moments of the TMVT distribution under the double truncation. We formulate the results as analytic matrix expressions, which can be directly computed in existing software. Results for the left and right truncation can be viewed as special cases. We then apply the slice sampling algorithm to generate random variates from the TMVT distribution by introducing auxiliary variables. This strategic approach can result in a series of full conditional densities that are of uniform distributions. Finally, several examples and practical applications are given to illustrate the effectiveness and importance of the proposed results.     

On the sample correlation coefficient in the truncated bivariate normal population

The truncated bivariate normal distribution (TBVND) with truncation in both variables on the left is studied here. The behaviour of the sample correlation coefficient is assessed through its moments when the sample is from such a population. Some inequalities established by Rao et al. (1968) are extended     

Inverse moments of discrete distributions

In this paper an expression for the inverse moment of order r is given for the truncated binomial and Poisson distributions. This enables one to obtain inverse moments in a finite series. Some applications and multivariate generalizations are also given. The method also enables one to obtain relations between inverse moments and factorial moments and distributions of sums of variables.     

The Poisson Maximum Entropy Model for Homogeneous Poisson Processes

Our main interest is parameter estimation using maximum entropy methods in the prediction of future events for Homogeneous Poisson Processes when the distribution governing the distribution of the parameters is unknown. We intend to use empirical Bayes techniques and the maximum entropy principle to model the prior information. This approach has also been motivated by the success of the gamma prior for this problem, since it is well known that the gamma maximizes Shannon entropy under appropriately chosen constraints. However, as an alternative, we propose here to apply one of the often used methods to estimate the parameters of the maximum entropy prior. It consists of moment matching, that is, maximizing the entropy subject to the constraint that the first two moments equal the empirical ones and we obtain the truncated normal distribution (truncated below at the origin) as a solution. We also use maximum likelihood estimation (MLE) methods to estimate the parameters of the truncated normal distribution for this case. These two solutions, the gamma and the truncated normal, which maximize the entropy under different constraints are tested as to their effectiveness for prediction of future events for homogeneous Poisson processes by measuring their coverage probabilities, the suitably normalized lengths of their prediction intervals and their goodness-of-fit measured by the Kullback–Leibler criterion and a discrepancy measure. The estimators obtained by these methods are compared in an extensive simulation study to each other as well as to the estimators obtained using the completely noninformative Jeffreys’ prior and the usual frequency methods. We also consider the problem of choosing between the two maximum entropy methods proposed here, that is, the gamma prior and the truncated normal prior, estimated both by matching of the first two moments and, by maximum likelihood, when faced with data and we advocate the use of the sample skewness and kurtosis. The methods are also illustrated on two examples: one concerning the occurrence of mammary tumors in laboratory animals taking part in a carcinogenicity experiment and the other, a warranty dataset from the automobile industry.     

Moments of order statistics from doubly truncated continuous distributions and characterizations

In this paper, recurrence relations from a general class of doubly truncated continuous distributions which are satisfied by single as well as product moments of order statistics are obtained. Recurrence relations from doubly truncated generalized Weibull, exponential, Raleigh and logistic distributions have been derived as special cases of our result, Some previous results for doubly truncated Weibull, standard exponential, power function and Burr type XII distributions are obtained as special cases. The general recurrence relation of single moments has been used in the case of the left and right truncation to characterize the Weibull, Burr type XII and Pareto distributions.     

Elementary expressions for moments of truncated negative binomial random variables

Moments of truncated negative binomial random variables arise in many areas. But moments of general order do not appear to be available, even a correct expression for the variance of a truncated negative binomial random variable was derived only in 2016. Here, we derive the elementary expressions for the moments of general order for four different types of truncated negative binomial random variables. Computational issues are discussed for the expressions.     

Multivariate distributions involving ratios of normal variables

The papsr considers distributions of collections of ratios of normal variables, The derivation of the joint density is linked to SKI sting literature on absolute, incomplete or truncated moments of multinormals. The distribution function may be expressed as a sum of rectangular multi normal probabilities. When the coefficients of variation of the denominators are close to zero, then a simple transformation of the ratios is approximately inultinormal. An application to Bayesian analysis is included.     

The multivariate hazard gradient and moments of the truncated multinormal distribution

It is well known that the expectation and variance of a truncated normal distribution can be simply expressed in terms of the hazard rate function. This paper shows that it is possible to express the expectation and covariance matrices of a truncated multinormal distribution with similarly simple expressions in which the hazard rate function is generalized to thevector multivariate hazard rate(also: hazard gradient) of Johnson and Kotz. This provides a concise computational form for the mutivariate moments and lends support to the contention that the hazard gradient is the appropriate generalization of the univariate hazard rate.     

Truncated inverted generalized exponential distribution and its properties

The inverted generalized exponential distribution is defined as an alternative model for lifetime data. The existence of moments of this distribution is shown to hold under some restrictions. However, all the moments exist for the truncated inverted generalized exponential distribution and closed-form expressions for them are derived in this article. The distributional properties of this truncated distribution are studied. Maximum likelihood estimation method is discussed for the estimation of the parameters of the distribution both theoretically and empirically. In order to see the modeling performance of the distribution, two real datasets are analyzed.     

Order statistics from non-identical power function random variables

In this paper, we derive some recurrence relations satisfied by the single and the product moments of order statistics arising from n independent and non-identically distributed power function random variables. These recurrence relations will enable one to compute all the single and the product moments of all order statistics in a simple recursive manner. The results for the multiple-outlier model are deduced as special cases. The results are further generalized to the case of truncated power function random variables.     

Moment expressions and summary statistics for the complete and truncated weibull distribution

Traditionally, the moments of the Weibull distribution have been calculated using the standard Weibull (Johnson and Kotz, 1970) . This article will expand on that idea and cover the truncated cases for the standard Weibull distributions. Also, the same techniques used for the standard form will be used to derive the moment expressions for the three-parameter complete and truncated Weibull distributions. The summary statistics are then calculated from the moment expressions. Weibull moments involve the gamma and incomplete gamma functions.     

Higher order moments of the estimated tangency portfolio weights

In this paper, we consider the estimated weights of the tangency portfolio. We derive analytical expressions for the higher order non-central and central moments of these weights when the returns are assumed to be independently and multivariate normally distributed. Moreover, the expressions for mean, variance, skewness and kurtosis of the estimated weights are obtained in closed forms. Later, we complement our results with a simulation study where data from the multivariate normal and t-distributions are simulated, and the first four moments of estimated weights are computed by using the Monte Carlo experiment. It is noteworthy to mention that the distributional assumption of returns is found to be important, especially for the first two moments. Finally, through an empirical illustration utilizing returns of four financial indices listed in NASDAQ stock exchange, we observe the presence of time dynamics in higher moments.     

Recurrence relations between product moments of order statistics

A general result for obtaining recurrence relations between product moments of order statistics is established and this result is used to determine the recurrence relations between product moments of some doubly truncated distributions. The examples considered are Weibull, exponential, Pareto, power function and Cauchy distributions.     

Characterization of Lindley distribution by truncated moments

In the past few years, the Lindley distribution has gained popularity for modeling lifetime data as an alternative to the exponential distribution. This paper provides two new characterizations of the Lindley distribution. The first characterization is based on a relation between left truncated moments and failure rate function. The second characterization is based on a relation between right truncated moments and reversed failure rate function.