The exponentiated discrete Weibull Distribution

In this paper, the exponentiated discrete Weibull distribution is introduced. This new generalization of the discrete Weibull distribution can also be considered as a discrete analogue of the exponentiated Weibull distribution. A special case of this exponentiated discrete Weibull distribution defines a new generalization of the discrete Rayleigh distribution for the first time in the literature. In addition, discrete generalized exponential and geometric distributions are some special sub-models of the new distribution. Here, some basic distributional properties, moments, and order statistics of this new discrete distribution are studied. We will see that the hazard rate function can be in- creasing, decreasing, bathtub, and upside-down bathtub shaped. Estimation of the parameters is illustrated using the maximum likelihood method. The model with a real data set is also examine

On Methods of Estimation for the Type II Discrete Weibull Distribution

In this paper, we describe and analyze several methods of estimation for the type II discrete Weibull distribution, outlining their applicability and properties, assessing and comparing their performance via intensive Monte Carlo simulation experiments. We consider the standard maximum likelihood method, a method of proportion, and two variants of the least-squares method. The type II discrete Weibull distribution can be used in reliability engineering for modeling count data or discrete lifetimes and its use is theoretically motivated by its capability of modeling either bounded or unbounded support, and either increasing or decreasing failure rate. Statistical analyses of real datasets are presented to show the capability of the distribution in fitting reliability data and illustrate the application of the proposed inferential techniques

Parameter estimation for type 3 discrete weibull distribution : a comparative study

The type III discrete Weibull distribution can be used in reliability analysis for modeling failure data such as the number of shocks, cycles, or runs a component or a structure can overcome before failing. This paper describes three methods for estimating its parameters: two customary techniques and a technique particularly suitable for discrete distributions, which, in contrast to the two other techniques, provides analytical estimates, whose derivation is detailed here. The techniques\u2019 peculiarities and practical limits are outlined. A Monte Carlo simulation study has been performed to assess the statistical performance of these methods for different parameter combinations and sample sizes and then give some indication for their mindful use. Two applications of real data are provided with the aim of showing how the type III discrete Weibull distribution can fit real data, even better than other popular discrete models, and how the inferential procedures work. A software implementation of the model is also provided

Seismic hazard assessment in the Northern Aegean Sea (Greece) through discrete Semi-Markov modeling.

Οι ημι-Μαρκοβιανές αλυσίδες χρησιμοποιούνται για τη μελέτη της σεισμικότητας στο Βόρειο  Αιγαίο.  Η  βασική  τους  διαφορά  από  τις  Μαρκοβιανές  αλυσίδες  είναι  ότι επιτρέπουν  μια  οποιαδήποτε  αυθαίρετη  κατανομή  για  τους  χρόνους  παραμονής (χρόνοι μεταξύ διαδοχικών σεισμών). Υποθέτουμε ότι η χρονοσειρά των σεισμών που έχουν  γίνει  στο  Βόρειο  Αιγαίο  αποτελεί  μια  διακριτή  ημι-Μαρκοβιανή  αλυσίδα. Θεωρείται ότι οι χρόνοι παραμονής ακολουθούν γεωμετρικές ή διακριτές κατανομές Weibull. Πρώτα ταξινομήθηκαν τα δεδομένα σε δυο κατηγορίες, όπου κατάσταση 1: Μέγεθος 6.5 -7 και κατάσταση 2 Μέγεθος>7, και στη συνέχεια σε τρεις κατηγορίες, όπου   κατάσταση 1: μέγεθος 6.5 -6.7,   κατάσταση 2 :   Μέγεθος 6.8 -7.1   και κατάσταση 3 : Μέγεθος 7.2 -7.4 . Εκτιμήθηκαν οι παράμετροι των συναρτήσεων πιθανότητας των χρόνων  παραμονής  και  υπολογίστηκαν  οι  πίνακες  πυρήνες  της  ημι-Μαρκοβιανής αλυσίδας για όλες τις παραπάνω περιπτώσεις. Έγινε σύγκριση των πινάκων πυρήνων και   προέκυψαν συμπεράσματα για  τη μελλοντική σεισμική επικινδυνότητα στην υπό μελέτη περιοχή.Semi-Markov  chains  are  used  for  studying  the  evolution  of  seismicity  in  the Northern Aegean Sea (Greece). Their main difference from the Markov chains is that   they  allow the sojourn times (i.e. the time between successive earthquakes), to follow any arbitrary distribution. It is assumed that the time series of earthquakes that  occurred  in  Northern  Aegean  Sea  form  a  discrete  semi-Markov  chain. The probability law of the sojourn times, is considered to be the geometric distribution or   the   discrete  Weibull  distribution. Firstly,  the  data  are  classified  into  two categories that is, state 1: Magnitude 6.5  -7 and state 2 Magnitude>7, and secondly into three categories , that is    state 1: Magnitude 6.5-6.7, state 2: Magnitude 6.8-7.1 and state 3: Magnitude 7.2-7.4 . This methodology is followed in order to obtain more accurate results and find out whether there exists an impact of the different classification on the results. The parameters of the probability laws of the sojourn times are estimated and the semi-Markov kernels are  evaluated for all the above cases  .  The  semi-Markov  kernels  are  compared and  the   conclusions  are  drawn relatively to future seismic hazard in the area under study

A bivariate count model with discrete Weibull margins

Multivariate discrete data arise in many fields (statistical quality control, epidemiology, failure and reliability analysis, etc.) and modelling such data is a relevant task. Here we consider the construction of a bivariate model with discrete Weibull margins, based on Farlie-Gumbel-Morgenstern copula, analyse its properties especially in terms of attainable correlation, and propose several methods for the point estimation of its parameters. Two of them are the standard one-step and two-step maximum likelihood procedures; the other two are based on an approximate method of moments and on the method of proportion, which represent intuitive alternatives for estimating the dependence parameter. A Monte Carlo simulation study is presented, comprising more than one hundred artificial settings, which empirically assesses the performance of the different estimation techniques in terms of statistical properties and computational cost. For illustrative purposes, the model and related inferential procedures are fitted and applied to two datasets taken from the literature, concerning failure data, presenting either positive or negative correlation between the two observed variables. The applications show that the proposed bivariate discrete Weibull distribution can model correlated counts even better than existing and well-established joint distributions