A unified approach to minimum variance unbiased estimation of a probability function belonging to an exponential family

The problem of minimum variance unbiased estimation of the probability density function of a random variable belonging to an exponential family is considered. The method of estimation proposed in this paper requires the solution of a certain integral equation. For many probability distributions the solution of this equation is given by a known result in integral transform theory.     

Change PP plot and continous sample quantile function

Functional inference recommends data analysis of a sample of n observations by functional and graphical representations of its probability models using various functions on 0 < u < 1, including the quantile function. This paper discusses: charge PP plots and a continuous version of the sample quantile function which use the mid-distinct value probability integral transform; comparison density functions; comparison interpretation of probability integral transform; maximum spacings method of one sample parameter estimation.     

On integrals of dirichlet distributions and their applications

The purpose of this paper is to extend results obtained by Sahai and Anderson (1973) and Hurlburt and Spiegel (1976) computationally as well as theoretically. We apply results given in Yassaee (1974), (1976) and apply our computer program to show that one can evaluate the probability of type one error for conditional tests in linear model, in general. We show that one can compute the probability integral of Dirichlet distribution by the use of our program which computes the probability integral of inverted Dirichlet distribution.     

SERIES REPRESENTATION OF DOUBLY NONCENTRAL t DISTRIBUTION BY USE OF THE MELLIN INTEGRAL TRANSFORM

ABSTRACT

The Mellin integral transform is widely used to find the distributions of products and quotients of independent random variables defined over the positive domain. But it is hardly used to derive the distributions defined over both positive and negative values of the random variables. In this paper, the Mellin integral transform is applied to obtain the doubly noncentral t density and its distribution function in convergent series forms.     

The probability integral for a bivariate generalization of the non-central t

An expression for the probability integral of a bivariate generalization of the non-central t distribution is derived. This expression is then used to construct tables for various combinations of parameter values     

Transforming the exponential by minimizing the sum of the absolute differences

This work presents an optimal value to be used in the power transformation to transform the exponential to normality for statistical process control (SPC) applications. The optimal value is found by minimizing the sum of absolute differences between two distinct cumulative probability functions. Based on this criterion, a numerical search yields a proposed value of 3.5142, so the transformed distribution is well approximated by the normal distribution. Two examples are presented to demonstrate the effectiveness of using the transformation method and its applications in SPC. The transformed data are almost normally distributed and the performance of the individual charts is satisfactory. Compared to charts that use the original exponential data and probability control limits, the individual charts constructed using the transformed distribution are superior in appearance, ease of interpretation and implementation by practitioners.     

Bootstrapping a Universal Pivot When Nuisance Parameters are Estimated

In complete samples from a continuous cumulative distribution with unknown parameters, it is known that various pivotal functions can be constructed by appealing to the probability integral transform. A pivotal function (or simply pivot) is a function of the data and parameters that has the property that its distribution is free of any unknown parameters. Pivotal functions play a key role in constructing confidence intervals and hypothesis tests. If there are nuisance parameters in addition to a parameter of interest, and consistent estimators of the nuisance parameters are available, then substituting them into the pivot can preserve the pivot property while altering the pivot distribution, or may instead create a function that is approximately a pivot in the sense that its asymptotic distribution is free of unknown parameters. In this latter case, bootstrapping has been shown to be an effective way of estimating its distribution accurately and constructing confidence intervals that have more accurate coverage probability in finite samples than those based on the asymptotic pivot distribution. In this article, one particular pivotal function based on the probability integral transform is considered when nuisance parameters are estimated, and the estimation of its distribution using parametric bootstrapping is examined. Applications to finding confidence intervals are emphasized. This material should be of interest to instructors of upper division and beginning graduate courses in mathematical statistics who wish to integrate bootstrapping into their lessons on interval estimation and the use of pivotal functions.

[Received November 2014. Revised August 2015.]     

On the hyper-dirichlet type 1 and hyper-liouville distributions

This paper concerns the characterization of a new family of multivariate beta distribution functions - the hyper-Dirichlet type 1 distribution. This family describes the joint density function of the terminal variates of an arbitrary tree constructed from finite sequences of probability vectors having independent Dirichlet type 1 distributions. Expressions for the general properties of the hyper-Dirichlet type 1 distribution are presented. In addition, the hyper-Liouville distribution is described and its properties are discussed as well as a generalization of the Liouville integral identity.     

The computation of standard normal distribution integral in any required precision based on reliability method

Based on reliability theory, the value of the standard normal distribution integral can be obtained by calculating the probability of the failure domain of the linear performance function. After the sample space is divided into some sub-sample spaces, a number of sub-failure domains are obtained. In the paper, the methods of computing the probabilities of sub-failure domains are discussed. All the formulae and the steps of computing the standard normal distribution integral which meet any required precision are given in the paper. Examples show that it is easy for the method to compute the standard normal distribution integral.     

Computation of multivariate normal probabilities with polar coordinate systems

We consider the problem of evaluation of the probability that all elements of a multivariate normally distributed vector have non-negative coordinates; this probability is called the non-centred orthant probability. The necessity for the evaluation of this probability arises frequently in statistics. The probability is defined by the integral of the probability density function. However, direct numerical integration is not practical. In this article, a method is proposed for the computation of the probability. The method involves the evaluation of a measure on a unit sphere surface in p-dimensional space that satisfies conditions derived from a covariance matrix. The required computational time for the p-dimensional problem is proportional to p²·2^p?1, and it increases at a rate that is lower than that in the case of the existing method.     

The abscissa of convergence of the laplace-stieltjes transform of a ph-distribution

If a probability distribution of phase type has an irreducible representation (α,T), the abscissa of convergence of its Laplace-Stieltjes transform is shown to be the eigenvalue of maximum real part of the matrix T.     

Log-gamma-generated families of distributions

We introduce two new general families of continuous distributions, generated by a distribution F and two positive real parameters α and β which control the skewness and tail weight of the distribution. The construction is motivated by the distribution of k-record statistics and can be derived by applying the inverse probability integral transformation to the log-gamma distribution. The introduced families are suitable for modelling the data with a significantly skewed and heavy-tailed distribution. Various properties of the introduced families are studied and a number of estimations and data fitness on real data are given to illustrate the results.     

Bayesian Analysis of the Censored Regression Model with an AEPD Error Term

We present the censored regression model with the error term following the asymmetric exponential power distribution. We propose three Markov chain Monte Carlo (MCMC) algorithms: the first one uses the probability integral transformation; the second one uses a combination of the probability integral transformation and random walk draws; while the third one uses random walk draws. Using simulated data we compare the performance of the three MCMC algorithms. Then we compare the posterior means, or Bayes estimates, with maximum likelihood estimates. We estimate the stock option portion of executive compensation as an example of the empirical application.     

On selecting a subset containing the most probable multinomial event

Panchapakesan's procedure is considered for the problem of selectinga subset containing the most probable multinomial event. We use the type-2 Dirichlet integral to express the probability of a correct selection and propose a much simpler proof for the worst configuration. We also show that the supremum of the expected subset size occurs at the equal configuration.     

Using mathematical programming to compute singlular multivariate normal probablities

This paper describes two new, mathematical programming-based approaches for evaluating general, one- and two-sidedp-variate normal probabilities where the variance-covariance matrix (of arbitrary structure) is singular with rankr(r<pand r and p can be of unlimited dimensions. In both cases, principal components are used to transform the original, ill-definedp-dimensional integral into a well-definedrdimensional integral over a convex polyhedron. The first algorithm that is presented uses linear programming coupled with a Gauss-Legendre quadrature scheme to compute this integral, while the second algorithm uses multi-parametric programming techniques in order to significantly reduce the number of optimization problems that need to be solved. The application of the algorithms is demonstrated and aspects of computational performance are discussed through a number of examples, ranging from a practical problem that arises in chemical engineering to larger, numerical examples.     

Pareto Poisson–Lindley distribution with applications

A new lifetime distribution is introduced based on compounding Pareto and Poisson–Lindley distributions. Several statistical properties of the distribution are established, including behavior of the probability density function and the failure rate function, heavy- and long-right tailedness, moments, the Laplace transform, quantiles, order statistics, moments of residual lifetime, conditional moments, conditional moment generating function, stress–strength parameter, Rényi entropy and Song's measure. We get maximum-likelihood estimators of the distribution parameters and investigate the asymptotic distribution of the estimators via Fisher's information matrix. Applications of the distribution using three real data sets are presented and it is shown that the distribution fits better than other related distributions in practical uses.     

Distribution of products of independently distributed pathway random variables

A lot of work has been done on products and ratios of random variables by Provost and his co-workers, see, for example, Provost [S.B. Provost, The exact distribution of the ratio of a linear combination of chi-square variables over the root of a product of chi-square variables, Canad. J. Statist. 14 (1986), pp. 61–67; S.B. Provost, The distribution function of a statistic for testing the equality of scale parameters in two gamma populations, Metrika 36 (1989), pp. 337–345]. Here, we extend this idea by introducing a pathway model. The exact density functions of the products of pathway random variables are obtained using the Mellin transform technique. Their computable series forms are derived. The particular cases of the derived results are shown to be associated with the thermonuclear functions and reaction rate probability integral in the theory of nuclear reaction rate, Krätzel integral in applied analyses and inverse Gaussian density in stochastic processes. Graphical representations of the density functions of the product of random variables for the different values of the pathway parameters are shown. The new probability model is fitted to revenue data.     

Modified skew-slash distribution

Abstract

In this work, we introduce a new skewed slash distribution. This modification of the skew-slash distribution is obtained by the quotient of two independent random variables. That quotient consists on a skew-normal distribution divided by a power of an exponential distribution with scale parameter equal to two. In this way, the new skew distribution has a heavier tail than that of the skew-slash distribution. We give the probability density function expressed by an integral, but we obtain some important properties useful for making inferences, such as moment estimators and maximum likelihood estimators. By way of illustration and by using real data, we provide maximum likelihood estimates for the parameters of the modified skew-slash and the skew-slash distributions. Finally, we introduce a multivariate version of this new distribution.     

On the exact distribution of the likelihood ratio criterion for testing

The exact null distribution of the likelihood ratio criter- 2 ion for testing the hypothesis H: y = y~; z = a I, a unknown and UQ a given known vector against the alternative A =f H in a p-vari- ate normal population N (y,z) has been derived in the form of Meijer's G-function using mellin integral transform and also in a chisquare series form. Asymptotic behavior of the distribution of -2 log L has also been discussed. Percentage points for p=2(l)10for various level of significance and various degrees of freedom have been computed, but only selected tables have been presented in this paper.     

Crossing probabilities for a square root boundary by a bessel process

Probabilities for Wiener and Bessel processes to cross a square root boundary are calculated and the Mellin transform of the distribution of the associated stopping time is given.The transform is inverted and a table of critical values for the crossing probability as a function of the observation time is included. Also an analytic expression for the crossing probability is computed for the case of constant drift.