The Pearson and Johnson Systems

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Likelihood Estimation ◽  
Real Data ◽  
Numerical Implementation ◽  
Pearson System ◽  
Type Iv ◽  
Score Statistics ◽  
Parameter Values ◽  
Johnson System

This chapter re-examines two of the best-known systems of parametric distributions: the Pearson and the Johnson. It is shown that, in the Pearson system, Pearson Types III and V are boundary embedded models of the main Types I, IV, and VI. A comprehensive way of finding the best type to fit is given using appropriate score statistics to guide a systematic search of all model types, including symmetric boundary models. Maximum likelihood estimation is used and details of its numerical implementation are given. Type IV can be a difficult model to fit. A method is discussed for this model that is reasonably robust, subject to certain restrictions on the parameter values. The same examination is made of the Johnson system where the lognormal, SL family is shown to be an embedded subsystem of both the main subsystems SB and SU. Two real data examples are given.

scholarly journals Bayesian Estimation of The Ex-Gaussian Distribution

2021 ◽  
Vol 9 (4) ◽  
pp. 809-819
Author(s):  
Abir El Haj ◽  
Yousri Slaoui ◽  
Clara Solier ◽  
Cyril Perret
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Gaussian Distribution ◽  
Estimation Method ◽  
Reaction Times ◽  
Simulated Data ◽  
Likelihood Estimation ◽  
Real Data ◽  
Estimated Parameters ◽  
Parameter Values

Fitting of the exponential modified Gaussian distribution to model reaction times and drawing conclusions from its estimated parameter values is one of the most popular method used in psychology. This paper aims to develop a Bayesian approach to estimate the parameters of the ex-Gaussian distribution. Since the chosen priors yield to posterior densities that are not of known form and that they are not always log-concave, we suggest to use the adaptive rejection Metropolis sampling method. Applications on simulated data and on real data are provided to compare this method to the standard maximum likelihood estimation method as well as the quantile maximum likelihood estimation. Results shows the effectiveness of the proposed Bayesian method by computing the root mean square error of the estimated parameters using the three methods.

scholarly journals The Marshall-Olkin Extended Power Lomax Distribution with Applications

2020 ◽  
pp. 331-341
Author(s):  
JIJU GILLARIOSE ◽  
Lishamol Tomy
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Simulated Data ◽  
Likelihood Estimation ◽  
Real Data ◽  
Limiting Distributions ◽  
Lomax Distribution ◽  
New Distribution ◽  
Extended Power

In this article, we dened a new four-parameter model called Marshall-Olkin extended power Lomax distribution and studied its properties. Limiting distributions of sample maxima and sample minima are derived. The reliability of a system when both stress and strength follows the new distribution is discussed and associated characteristics are computed for simulated data. Finally, utilizing maximum likelihood estimation, the goodness of the distribution is tested for real data.

Topp–Leone Linear Exponential Distribution

2018 ◽  
Vol 33 (1) ◽  
pp. 31-43
Author(s):  
Bol A. M. Atem ◽  
Suleman Nasiru ◽  
Kwara Nantomah
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Exponential Distribution ◽  
Likelihood Estimation ◽  
Real Data ◽  
Simulation Studies ◽  
Finite Sample ◽  
Data Set ◽  
Finite Sample Properties ◽  
Linear Exponential Distribution

Abstract This article studies the properties of the Topp–Leone linear exponential distribution. The parameters of the new model are estimated using maximum likelihood estimation, and simulation studies are performed to examine the finite sample properties of the parameters. An application of the model is demonstrated using a real data set. Finally, a bivariate extension of the model is proposed.

scholarly journals Cubic Transformation of Exponential Weibull and its Statistical Properties

2020 ◽  
pp. 39-50
Author(s):  
Haiyue Wang ◽  
Zhenhua Bao
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Weibull Distribution ◽  
Likelihood Estimation ◽  
Real Data ◽  
Statistical Properties ◽  
Reasoning Process ◽  
The Family

In this paper, a cubic transformation exponential Weibull distribution is proposed by using the family of cubic transformation distributions introduced by Rahman et al.The reasoning process of the proposed cubic transformation exponential Weibull distribution is discussed in detail, and its statistical properties and parameter estimation are also discussed.Using real data, the maximum likelihood estimation is used to simulate. Through the comparison of fitting results, it is concluded that the cubic transformation exponential Weibull distribution proposed in this paper has stronger applicability.

scholarly journals The Burr X Fréchet Model for Extreme Values: Mathematical Properties, Classical Inference and Bayesian Analysis

2019 ◽  
pp. 797-818 ◽  
Author(s):  
Haitham Yousof ◽  
S. Jahanshahi ◽  
Vikas Kumar Sharma
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Extreme Values ◽  
Likelihood Estimation ◽  
Real Data ◽  
Mcmc Methods ◽  
Bayes Estimators ◽  
Mathematical Properties ◽  
Asymptotic Confidence Intervals ◽  
Classical Inference

In this paper, we investigate a new model based on Burr X and Fréchet distribution forextreme values and derive some of its properties. Maximum likelihood estimation alongwith asymptotic confidence intervals is considered for estimating the parameters of thedistribution. We demonstrate empirically the flexibility of the distribution in modelingvarious types of real data. Furthermore, we also provide Bayes estimators and highestposterior density intervals of the parameters of the distribution using Markov ChainMonte Carlo (MCMC) methods.

scholarly journals Scalable Bias-corrected Linkage Disequilibrium Estimation Under Genotype Uncertainty

2021 ◽  
Author(s):  
David Gerard
Keyword(s):  
Linkage Disequilibrium ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Maximum Likelihood Estimators ◽  
Likelihood Estimation ◽  
Real Data ◽  
Standard Errors ◽  
Posterior Distributions ◽  
Posterior Mean ◽  
Genome Wide

AbstractLinkage disequilibrium (LD) estimates are often calculated genome-wide for use in many tasks, such as SNP pruning and LD decay estimation. However, in the presence of genotype uncertainty, naive approaches to calculating LD have extreme attenuation biases, incorrectly suggesting that SNPs are less dependent than in reality. These biases are particularly strong in polyploid organisms, which often exhibit greater levels of genotype uncertainty than diploids. A principled approach using maximum likelihood estimation with genotype likelihoods can reduce this bias, but is prohibitively slow for genome-wide applications. Here, we present scalable moment-based adjustments to LD estimates based on the marginal posterior distributions of the genotypes. We demonstrate, on both simulated and real data, that these moment-based estimators are as accurate as maximum likelihood estimators, and are almost as fast as naive approaches based only on posterior mean genotypes. This opens up bias-corrected LD estimation to genome-wide applications. Additionally, we provide standard errors for these moment-based estimators. All methods are implemented in the ldsep package on the Comprehensive R Archive Network https://cran.r-project.org/package=ldsep.

scholarly journals The Exponentiated Burr XII Power Series Distribution: Properties and Applications

Stats ◽  
2018 ◽  
Vol 2 (1) ◽  
pp. 15-31
Author(s):  
Arslan Nasir ◽  
Haitham Yousof ◽  
Farrukh Jamal ◽  
Mustafa Korkmaz
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Power Series ◽  
Likelihood Estimation ◽  
Real Data ◽  
Estimation Procedure ◽  
Model Parameters ◽  
Data Sets ◽  
Power Series Distributions ◽  
New Distribution

In this work, we introduce a new Burr XII power series class of distributions, which is obtained by compounding exponentiated Burr XII and power series distributions and has a strong physical motivation. The new distribution contains several important lifetime models. We derive explicit expressions for the ordinary and incomplete moments and generating functions. We discuss the maximum likelihood estimation of the model parameters. The maximum likelihood estimation procedure is presented. We assess the performance of the maximum likelihood estimators in terms of biases, standard deviations, and mean square of errors by means of two simulation studies. The usefulness of the new model is illustrated by means of three real data sets. The new proposed models provide consistently better fits than other competitive models for these data sets.

scholarly journals Generalizations of Burr Type X Distribution with Applications

2020 ◽  
pp. 1-8
Author(s):  
Noor Akma Ibrahim ◽  
Mundher Abdullah Khaleel
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Density Function ◽  
Likelihood Estimation ◽  
Real Data ◽  
Data Sets ◽  
Lifetime Data ◽  
Simulation Studies ◽  
Cumulative Density Function ◽  
Two Parameters

We propose the generalizations of Burr Type X distribution with two parameters by using the methods of Beta-G, Gamma-G and Weibull-G families of distributions. We discuss maximum likelihood estimation of the model’s parameters. The performances of the parameter’s estimates are assessed via simulation studies under different sets of conditions. In the applications to real data sets, three sets of data are used whereby from the results we can deduce that these models can be used quite effectively in analysing lifetime data. Keywords: cumulative density function; lifetime data; maximum likelihood estimation

scholarly journals The Topp-Leone odd log-logistic Gumbel Distribution: Properties and Applications

2020 ◽  
Vol 9 (2) ◽  
pp. 288-310
Author(s):  
Fazlollah Lak ◽  
Morad Alizadeh ◽  
Hamid Karamikabir
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Hazard Rate ◽  
Gumbel Distribution ◽  
Likelihood Estimation ◽  
Real Data ◽  
Rate Function ◽  
Model Parameters ◽  
Data Sets ◽  
Hazard Rate Function

In this article, the Topp-Leone odd log-logistic Gumbel (TLOLL-Gumbel) family of distribution have beenstudied. This family, contains the very flexible skewed density function. We study many aspects of the new model like hazard rate function, asymptotics, useful expansions, moments, generating Function, R´enyi entropy and order statistics. We discuss maximum likelihood estimation of the model parameters. Further, we study flexibility of the proposed family are illustrated of two real data sets.

scholarly journals Half Logistic Exponential Extension Distribution with Properties and Applications

2020 ◽  
Vol 9 (3) ◽  
pp. 506-512
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Hazard Rate ◽  
Goodness Of Fit ◽  
Likelihood Estimation ◽  
Real Data ◽  
Model Parameters ◽  
Type I ◽  
Proposed Model ◽  
New Distribution

In this article, we have introduced a new distribution based on type I half logistic-G family and exponential extension as a base distribution known as Half Logistic Exponential Extension (HLEE) distribution. The statistical properties of this model are also explored, such as the behavior of probability density, hazard rate, and quantile functions are investigated. The Maximum likelihood estimation (MLE) method is used to estimate model parameters. For the potentiality of the proposed model we have compared the goodness of fit with some others models. We have proven the importance and flexibility of the new distribution in modeling with real data applications empirically.

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