The generalized two-sided power distribution

The generalized standard two-sided power (GTSP) distribution was mentioned only in passing by Kotz and van Dorp Beyond Beta, Other Continuous Families of Distributions with Bounded Support and Applications, World Scientific Press, Singapore, 2004. In this paper, we shall further investigate this three-parameter distribution by presenting some novel properties and use its more general form to contrast the chronology of developments of various authors on the two-parameter TSP distribution since its initial introduction. GTSP distributions allow for J-shaped forms of its pdf, whereas TSP distributions are limited to U-shaped and unimodal forms. Hence, GTSP distributions possess the same three distributional shapes as the classical beta distributions. A novel method and algorithm for the indirect elicitation of the two-power parameters of the GTSP distribution is developed. We present a Project Evaluation Review Technique example that utilizes this algorithm and demonstrates the benefit of separate powers for the two branches of activity GTSP distributions for project completion time uncertainty estimation.     

The two-sided power-binomial distribution for modeling binary count data

The two-sided power (TSP) distribution is a flexible two-parameter distribution having uniform, power function and triangular as sub-distributions, and it is a reasonable alternative to beta distribution in some cases. In this work, we introduce the TSP-binomial model which is defined as a mixture of binomial distributions, with the binomial parameter p having a TSP distribution. We study its distributional properties and demonstrate its use on some data. It is shown that the newly defined model is a useful candidate for overdispersed binomial data.     

The distribution of the product of independent beta variables

We investigate the classic distribution and approximate distribution of the product of Beta variables which are independent. We show that the product of independent Beta variables is a Beta variable under the some assumptions. We also obtain the approximate distribution of the product of independent Beta variables.     

Moments of order statistics of the standard two-sided power distribution

The standard two-sided power distribution is a flexible distribution having uniform, power function and triangular as subdistributions, and it is a reasonable alternative to the Laplace distribution in some cases. In this work, computationally efficient expressions for moments of order statistics, expressions for L-moments, and asymptotic results for sample extrema are derived. Then a simulation study is performed for the location-scale estimation problem of a small data set by considering the maximum likelihood estimation method and the best linear unbiased estimation method based on the moments of order statistics.     

Minimum variance unbiased estimation for modified power series distribution

In this paper we study the problem of finding the minimum variance unbiased (MVU) estimators of the functions of the para-meters of the modified power series distributions (MPSD). A theorem giving the necessary and sufficient conditions for the existence of the MVU estimators has been proved. Also, the estimators for a number of estimable functions of a parameter are obtained. Two other theorems dealing with the MVU estimation of the left truncated MPSD with unknown truncation point are also given. The particular case of the Lagrangian Poisson, the Lagrangian binomial and the Borel-Tanner distributions are considered and tables are also provided for the MVU estimators for some functions of the parameters. The variances of the estimators are also given for some cases.     

The beta skew-generalized normal distribution

In this article, we develop the skew-generalized normal distribution introduced by Arellano-Valle et al. (2004 Arellano-Valle, R.B., Gomez, H.W., Quintana, F.A. (2004). A new class of skew-normal distribution. Commun. Stat. - Theory Methods. 33(7):1465–1480.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) to a new family of the Beta skew-generalized normal (BSGN) distribution . Here, we present some theorems and properties of BSGN distribution and obtain its moment-generating function.     

The Kumaraswamy GEV distribution

The popular generalized extreme value (GEV) distribution has not been a flexible model for extreme values in many areas. We propose a generalization – referred to as the Kumaraswamy GEV distribution – and provide a comprehensive treatment of its mathematical properties. We estimate its parameters by the method of maximum likelihood and provide the observed information matrix. An application to some real data illustrates flexibility of the new model. Finally, some bivariate generalizations of the model are proposed.     

Distribution of the product of a pair of independent two-sided power variates

ABSTRACT

The distributions of algebraic functions of random variables are important in theory of probability and statistics and other areas such as engineering, reliability, and actuarial applications, and many results based on various distributions are available in the literature. The two-sided power distribution is defined on a bounded range, and it is a generalization of the uniform, triangular, and power-function probability distributions. This paper gives the exact distribution of the product of two independent two-sided power-distributed random variables in a computable representation. The percentiles of the product are then computed, and a real data application is given.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

On the distribution of lrt for testing the equality of exponential distributions

In this paper, an asymptotic expansion of the distribution' of the likelihood ratio criterion for testing the equality of p one-parameter exponential distributions is obtained for unequal sample sizes. The expansion is obtained up to the order of n^-3 with the second term of the order of n^-2 so that the first term of this expansion alone should provide an excellent approximation to the distribution for moderately large values of n, where n is the combined sample size.     

The exact distribution and density functions of the stein-type estimator for normal variance

In this paper, we derive the exact distribution and density functions of the Stein-type estimator for the normal variance. It is shown by numerical evaluation that the density function of the Stein-type estimator is unimodal and concentrates around the mode more than that of the usual estimator.     

An approximation of the inverse moments of the positive hypergeometric distribution

The negative moments of the positive hyper geometric distribution are often approximated by the inverse of the positive moments of this distribution. In this paper, a suitable approximation to the positive hypergeometric distribution is used to obtain the negative moments.     

Computation of the percentage points and the power for the two-sided Kolmogorov-Smirnov one sample test

Two recursive schemes are presented for the calculation of the probabilityP(g(x)≤S _n (x)≤h(x) for allx∈®), whereS _n is the empirical distribution function of a sample from a continuous distribution andh, g are continuous and isotone functions. The results are specialized for the calculation of the distribution and the corresponding percentage points of the test statistic of the two-sided Kolmogorov-Smirnov one sample test. The schemes allow the calculation of the power of the test too. Finally an extensive tabulation of percentage points for the Kolmogorov-Smirnov test is given.     

On optimal testing for the equality of equicorrelation: An example of loss in power

Sen Gupta (1988) considered a locally most powerful (LMP) test for testing nonzero values of the equicorrelation coefficient of a standard symmetric multivariate normal distribution. This paper constructs analogous tests for the symmetric multivariate normal distribution. It shows that the new test is uniformly most powerful invariant even in the presence of a nuisance parameter, σ². Further applications of LMP invariant tests to several equicorrelated populations have been considered and an extension to panel data modeling has been suggested.     

The inverse power Lindley distribution

Several probability distributions have been proposed in the literature, especially with the aim of obtaining models that are more flexible relative to the behaviors of the density and hazard rate functions. Recently, two generalizations of the Lindley distribution were proposed in the literature: the power Lindley distribution and the inverse Lindley distribution. In this article, a distribution is obtained from these two generalizations and named as inverse power Lindley distribution. Some properties of this distribution and study of the behavior of maximum likelihood estimators are presented and discussed. It is also applied considering two real datasets and compared with the fits obtained for already-known distributions. When applied, the inverse power Lindley distribution was found to be a good alternative for modeling survival data.     

On a bounded bimodal two-sided distribution fitted to the Old-Faithful geyser data

In this paper, we shall develop a novel family of bimodal univariate distributions (also allowing for unimodal shapes) and demonstrate its use utilizing the well-known and almost classical data set involving durations and waiting times of eruptions of the Old-Faithful geyser in Yellowstone park. Specifically, we shall analyze the Old-Faithful data set with 272 data points provided in Dekking et al. [3]. In the process, we develop a bivariate distribution using a copula technique and compare its fit to a mixture of bivariate normal distributions also fitted to the same bivariate data set. We believe the fit-analysis and comparison is primarily illustrative from an educational perspective for distribution theory modelers, since in the process a variety of statistical techniques are demonstrated. We do not claim one model as preferred over the other.     

The inverse power Lindley distribution in the presence of left-censored data

In this study, classical and Bayesian inference methods are introduced to analyze lifetime data sets in the presence of left censoring considering two generalizations of the Lindley distribution: a first generalization proposed by Ghitany et al. [Power Lindley distribution and associated inference, Comput. Statist. Data Anal. 64 (2013), pp. 20–33], denoted as a power Lindley distribution and a second generalization proposed by Sharma et al. [The inverse Lindley distribution: A stress–strength reliability model with application to head and neck cancer data, J. Ind. Prod. Eng. 32 (2015), pp. 162–173], denoted as an inverse Lindley distribution. In our approach, we have used a distribution obtained from these two generalizations denoted as an inverse power Lindley distribution. A numerical illustration is presented considering a dataset of thyroglobulin levels present in a group of individuals with differentiated cancer of thyroid.     

A semi-parametric method for estimating the beta coefficients of the hidden two-sided asset return jumps

ABSTRACT

We introduce a new methodology for estimating the parameters of a two-sided jump model, which aims at decomposing the daily stock return evolution into (unobservable) positive and negative jumps as well as Brownian noise. The parameters of interest are the jump beta coefficients which measure the influence of the market jumps on the stock returns, and are latent components. For this purpose, at first we use the Variance Gamma (VG) distribution which is frequently used in modeling financial time series and leads to the revelation of the hidden market jumps' distributions. Then, our method is based on the central moments of the stock returns for estimating the parameters of the model. It is proved that the proposed method provides always a solution in terms of the jump beta coefficients. We thus achieve a semi-parametric fit to the empirical data. The methodology itself serves as a criterion to test the fit of any sets of parameters to the empirical returns. The analysis is applied to NASDAQ and Google returns during the 2006–2008 period.     

The beta log-normal distribution

For the first time, we introduce the beta log-normal (LN) distribution for which the LN distribution is a special case. Various properties of the new distribution are discussed. Expansions for the cumulative distribution and density functions that do not involve complicated functions are derived. We obtain expressions for its moments and for the moments of order statistics. The estimation of parameters is approached by the method of maximum likelihood, and the expected information matrix is derived. The new model is quite flexible in analysing positive data as an important alternative to the gamma, Weibull, generalized exponential, beta exponential, and Birnbaum–Saunders distributions. The flexibility of the new distribution is illustrated in an application to a real data set.     

The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval

A new two-parameter distribution over the unit interval, called the Unit-Inverse Gaussian distribution, is introduced and studied in detail. The proposed distribution shares many properties with other known distributions on the unit interval, such as Beta, Johnson S_B, Unit-Gamma, and Kumaraswamy distributions. Estimation of the parameters of the proposed distribution are obtained by transforming the data to the inverse Gaussian distribution. Unlike most distributions on the unit interval, the maximum likelihood or method of moments estimators of the parameters of the proposed distribution are expressed in simple closed forms which do not need iterative methods to compute. Application of the proposed distribution to a real data set shows better fit than many known two-parameter distributions on the unit interval.