On a Logarithmic Weighted Power Distribution: Theory, Modelling and Applications

Engineers, economists, hydrologists, social scientists, and behavioural scientists often deal with data belonging to the unit interval. One of the most common approaches for modeling purposes is the use of unit distributions, beginning with the classical power distribution. A simple way to improve its applicability is proposed by the transmuted scheme. We propose an alternative in this article by slightly modifying this scheme with a logarithmic weighted function, thus creating the log-weighted power distribution. It can also be thought of as a variant of the log-Lindley distribution, and some other derived unit distributions. We investigate its statistical and functional capabilities, and discuss how it distinguishes between power and transmuted power distributions. Among the functions derived from the log-weighted distribution are the cumulative distribution, probability density, hazard rate, and quantile functions. When appropriate, a shape analysis of them is performed to increase the exibility of the proposed modelling. Various properties are investigated, including stochastic ordering (first order), generalized logarithmic moments, incomplete moments, Rényi entropy, order statistics, reliability measures, and a list of new distributions derived from the main one are offered. Subsequently, the estimation of the model parameters is discussed through the maximum likelihood procedure. Then, the proposed distribution is tested on a few data sets to show in what concrete statistical scenarios it may outperform the transmuted power distribution.

Asymptotically Normal Estimators for the Parameters of the Gamma-Exponential Distribution

Currently, much research attention has focused on generalizations of known mathematical objects in order to obtain adequate models describing real phenomena. An important role in the applied theory of probability and mathematical statistics is the gamma class of distributions, which has proven to be a convenient and effective tool for modeling many real processes. The gamma class is quite wide and includes distributions that have useful properties such as, for example, infinite divisibility and stability, which makes it possible to use distributions from this class as asymptotic approximations in various limit theorems. One of the most important tasks of applied statistics is to obtain estimates of the parameters of the model distribution from the available real data. In this paper, we consider the gamma-exponential distribution, which is a generalization of the distributions from the gamma class. Estimators for some parameters of this distribution are given, and the asymptotic normality of these estimators is proven. When obtaining the estimates, a modified method of moments was used, based on logarithmic moments calculated on the basis of the Mellin transform for the generalized gamma distribution. On the basis of the results obtained, asymptotic confidence intervals for the estimated parameters are constructed. The results of this work can be used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support.

Nonparametric Blind Signal Detection Based on Logarithmic Moments in Very Impulsive Noise

The detection problem in impulsive noise modeled by the symmetric alpha stable SαS distribution is studied. The traditional detectors based on the second or higher order moments fail in SαS noise, and the method based on the fractional lower order moments (FLOMs) performs poorly when the noise distribution has small values of characteristic exponent. In this paper, a detector based on the logarithmic moments is investigated. The analytical expressions of the false alarm and detection probabilities are derived in nonfading channels as well as Rayleigh fading channels. The effect of noise uncertainty on the performance is discussed. Simulation results show that the logarithmic detector performs better than the FLOM and Cauchy detectors in very impulsive noise. In addition, the logarithmic detector is a nonparametric method and avoids estimating the parameter of the noise distribution, which makes the logarithmic detector easier to implement than the FLOM detector.

Application of Generalized Student’s T-Distribution In Modeling The Distribution of Empirical Return Rates on Selected Stock Exchange Indexes

This paper examines the application of the so called generalized Student’s t-distribution in modeling the distribution of empirical return rates on selected Warsaw stock exchange indexes. It deals with distribution parameters by means of the method of logarithmic moments, the maximum likelihood method and the method of moments. Generalized Student’s t-distribution ensures better fitting to empirical data than the classical Student’s t-distribution.