Exact formula for the cumulative distribution function of the quasi-range from the logistic distribution

In this paper we obtain an exact formula for the cumulative distribution function of the rth quasi-range from the logistic distribution. For the special case r = 0, the result agrees for the rangegiven by Gupta and Shah (1965).     

On the existence and uniqueness of the MLEs of the parameters of a general class of exponentiated distributions

In this paper, we present a formal simple proof for the existence and uniqueness of the maximum likelihood estimates (MLEs) of the parameters of a general class of exponentiated distributions. This class includes the exponentiated (general) exponential, exponentiated Rayleigh (scaled Burr X) and exponentiated Pareto distributions, as special cases, and thus the proof given here establishes the existence and uniqueness of the MLEs for these important special cases as well.     

A new approximation for the distribution of the difference of two t-variables

A mixture representation for the distribution of the difference of two independent t-varlables is provided to approximate the probabilities and percentiles The mixture of normal and standardized t is found to be quite suitable in terms of the accuracy and simplicity as it compares favorably to the best known approximation namelyt that due to Ghosh (1975). The idea of the mixture distribution is also extended to provide an approximation to the distribution of a linear combination of independent t-variables which provides an approximation to the Behrens-Fisher distribution in particular.     

Approximation of the Distribution of the Location Parameter in the Growth Curve Model

Abstract.  In this paper an Edgeworth-type approximation of order O(n ⁻² ) to the density of the estimator of the location parameter in the growth curve model has been derived. The approximation is a mixture of a normal and a Kotz-type distribution, thus being an elliptical distribution. A condition for unimodality of the mixture was found and marginal distribution of a subvector of the mixture distribution was derived. Finally, a small example was given to demonstrate an application of the approximation.     

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.     

An extension of the exponential distribution

A generalization of the exponential distribution is presented. The generalization always has its mode at zero and yet allows for increasing, decreasing and constant hazard rates. It can be used as an alternative to the gamma, Weibull and exponentiated exponential distributions. A comprehensive account of the mathematical properties of the generalization is presented. A real data example is discussed to illustrate its applicability.     

On the asymptotic efficiency of certain correlation tests of fit

Tests of fit based on correlation-type statistics are investigated for the exponential, extreme-value, and logistic distributions. The statistics are shown to be asymptotically normal at the rate log^½n. The result is used to show that such tests have 0 asymptotic relative efficiency.     

The shape of the hazard function: Does the generalized gamma have the last word?

This paper compares the five-parameter beta generalized gamma (BGG) distribution to the three-parameter generalized gamma (GG). Both distributions include the four standard hazard shapes that we believe is an important property for any parametric family. For several BGG distributions, we select matching GGs and compute the Kullback-Liebler distance, observing remarkable agreement. We explore the beta parameters' influence on the matched GG parameters, detecting a strong connection between the distributions. Lastly, we compare the distributions using two real-data examples. We conclude from these comparisons that the BGG is not likely to be more useful for analytical purposes than the simpler GG.     

On the moments of record values

In this paper some general relations for expectations of functions of record values are established. It is seen that these relations may be used to obtain recurrence relations for moments of record values. Bounds on expectations of record values with numerical computations are presented. Applications to the characterizations of the generalizeed exponential distribution are also given.     

A simple formula based on quantiles for the moments of beta generalized distributions

In this article, we derive explicit expansions for the moments of beta generalized distributions from power series expansions for the quantile functions of the baseline distributions. We apply our formula to the beta normal, beta Student t, beta gamma and beta beta generalized distributions. We propose a simple way to express the quantile function of any beta generalized distribution as a power series expansion with known coefficients.     

On a geometrical derivation of the upper tail of the null distribution of w-exponential

Shapiro and Wilk (1972) proposed a goodness of fit test for the exponential distribution. Carrie (1980) obtained an explicit expression of the null distribution of their test statistic W (n) E in a neighbourhood of its upper tail. His derivation uses a certain transformation involving the order statistics from the standard exponential distribution. In this paper we present an alternative derivation of this distribution using an elementary geometrical argument.     

Heine-euler extensions of the poisson distribution

The paper shows that the Heine and Euler distributions (Benkherouf and Bather, 1988) are members of a family of q-series anologues of the Poisson distribution, with similar probability mass functions, but different restrictions on their parameters, and different modes of genesis and properties. The relationships between the Heine, Euler, pseudo-Euler, Poisson and geometric distributions are explored. Illustrative data sets are discussed.     

Time series models based on the unrestricted skew-normal process

The standard location and scale unrestricted (or unified) skew-normal (SUN) family studied by Arellano-Valle and Genton [On fundamental skew distributions. J Multivar Anal. 2005;96:93–116] and Arellano-Valle and Azzalini [On the unification of families of skew-normal distributions. Scand J Stat. 2006;33:561–574], allows the modelling of data which is symmetrically or asymmetrically distributed. The family has a number of advantages suitable for the analysis of stochastic processes such as Auto-Regressive Moving-Average (ARMA) models, including being closed under linear combinations, being able to satisfy the consistency condition of Kolmogorov’s theorem and providing the guarantee of the existence of such a SUN stochastic process. The family is able to be represented in a hierarchical form which can be used for the ease of simulation. In addition, it facilitates an EM-type algorithm to estimate the model parameters. The performances and suitability of the proposed model are demonstrated on simulations and using two real data sets in applications.     

On the comparison of the Fisher information of the log-normal and generalized Rayleigh distributions

Surles and Padgett recently considered two-parameter Burr Type X distribution by introducing a scale parameter and called it the generalized Rayleigh distribution. It is observed that the generalized Rayleigh and log-normal distributions have many common properties and both distributions can be used quite effectively to analyze skewed data set. In this paper, we mainly compare the Fisher information matrices of the two distributions for complete and censored observations. Although, both distributions may provide similar data fit and are quite similar in nature in many aspects, the corresponding Fisher information matrices can be quite different. We compute the total information measures of the two distributions for different parameter ranges and also compare the loss of information due to censoring. Real data analysis has been performed for illustrative purposes.     

Asymptotic distributions of the sphericity test in a complex multivariate normal distribution

In this paper, asymptotic expansions of the null and non-null distributions of the sphericity test criterion in the case of a complex multivariate normal distribution are obtained for the first time in terms of beta distributions. In the null case, it is found that the accuracy of the approximation by taking the first term alone in the asymptotic series is sufficient for practical purposes. In fact for p - 2. the asymptotic expansion reduces to the first term which is also the exact distribution in this case. Applications of the results to the area of inferences on multivariate time series are also given.     

On the distribution of quotient of random variables conditioned to the positive quadrant

Abstract

Motivated by Caginalp and Caginalp [Physica A—Statistical Mechanics and Its Applications, 499, 2018, 457–471], we derive the exact distribution of X/Y conditioned on X?>?0, Y?>?0 for more than ten classes of distributions, including the bivariate t, bivariate Cauchy, bivariate Lomax, Arnold and Strauss’ bivariate exponential, Balakrishna and Shiji’s bivariate exponential, Mohsin et al.’s bivariate exponential, Morgenstern type bivariate exponential, bivariate gamma exponential and bivariate alpha skew normal distributions. The results can be useful in finance and other areas.     

A Simpler Approximation for Areas under the Standard Normal Curve

A simple approximation for areas under the standard normal curve is presented that is suitable for use when tables and/or calculators are not available or not permitted.     

A new goodness of fit test for the equality of multinomial cell probabilities versus trend alternatives

In this paper, a new test statistic is presented for testing the null hypothesis of equal multinomial cell probabilities versus various trend alternatives. Exact asymptotic critical values are obtained, The power of the test is compared with several other statistics considered by Choulakian et al (1995), The test is shown to have better power for certain trend alternatives.     

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.