scholarly journals An Asymmetric Bimodal Distribution with Application to Quantile Regression

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 899 ◽  
Author(s):  
Yolanda M. Gómez ◽  
Emilio Gómez-Déniz ◽  
Osvaldo Venegas ◽  
Diego I. Gallardo ◽  
Héctor W. Gómez
Keyword(s):  
Distribution Function ◽  
Maximum Likelihood ◽  
Quantile Regression ◽  
Simulation Study ◽  
Cumulative Distribution Function ◽  
Maximum Likelihood Estimators ◽  
Real Data ◽  
Bimodal Distribution ◽  
Cumulative Distribution ◽  
Cauchy Model

In this article, we study an extension of the sinh Cauchy model in order to obtain asymmetric bimodality. The behavior of the distribution may be either unimodal or bimodal. We calculate its cumulative distribution function and use it to carry out quantile regression. We calculate the maximum likelihood estimators and carry out a simulation study. Two applications are analyzed based on real data to illustrate the flexibility of the distribution for modeling unimodal and bimodal data.

NONPARAMETRIC KERNEL DISTRIBUTION FUNCTION ESTIMATION NEAR ENDPOINTS

2021 ◽  
Vol 10 (12) ◽  
pp. 3679-3697
Author(s):  
N. Almi ◽  
A. Sayah
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Kernel Estimator ◽  
Real Data ◽  
Cumulative Distribution ◽  
Function Estimation ◽  
Distribution Function Estimation ◽  
The Right ◽  
Nonparametric Kernel ◽  
Better Than

In this paper, two kernel cumulative distribution function estimators are introduced and investigated in order to improve the boundary effects, we will restrict our attention to the right boundary. The first estimator uses a self-elimination between modify theoretical Bias term and the classical kernel estimator itself. The basic technique of construction the second estimator is kind of a generalized reflection method involving reflection a transformation of the observed data. The theoretical properties of our estimators turned out that the Bias has been reduced to the second power of the bandwidth, simulation studies and two real data applications were carried out to check these phenomena and are conducted that the proposed estimators are better than the existing boundary correction methods.

THE EXACT CUMULATIVE DISTRIBUTION FUNCTION OF A RATIO OF QUADRATIC FORMS IN NORMAL VARIABLES, WITH APPLICATION TO THE AR(1) MODEL

2002 ◽  
Vol 18 (4) ◽  
pp. 823-852 ◽  
Author(s):  
G. Forchini
Keyword(s):  
Distribution Function ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Quadratic Forms ◽  
Cumulative Distribution Function ◽  
Closed Form Solution ◽  
Form Solution ◽  
Cumulative Distribution ◽  
Likelihood Estimator ◽  
Start Up

Often neither the exact density nor the exact cumulative distribution function (c.d.f.) of a statistic of interest is available in the statistics and econometrics literature (e.g., the maximum likelihood estimator of the autocorrelation coefficient in a simple Gaussian AR(1) model with zero start-up value). In other cases the exact c.d.f. of a statistic of interest is very complicated despite the statistic being “simple” (e.g., the circular serial correlation coefficient, or a quadratic form of a vector uniformly distributed over the unit n-sphere). The first part of the paper tries to explain why this is the case by studying the analytic properties of the c.d.f. of a statistic under very general assumptions. Differential geometric considerations show that there can be points where the c.d.f. of a given statistic is not analytic, and such points do not depend on the parameters of the model but only on the properties of the statistic itself. The second part of the paper derives the exact c.d.f. of a ratio of quadratic forms in normal variables, and for the first time a closed form solution is found. These results are then specialized to the maximum likelihood estimator of the autoregressive parameter in a Gaussian AR(1) model with zero start-up value, which is shown to have precisely those properties highlighted in the first part of the paper.

scholarly journals A New Approach to Assess Wind Potential

2021 ◽  
Keyword(s):  
Distribution Function ◽  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Wind Power ◽  
Cumulative Distribution Function ◽  
Maximum Likelihood Method ◽  
New Method ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Wind Distribution

<p>Weibull Cumulative Distribution Function (C.D.F.) has been employed to assess and compare wind potentials of two wind stations Europlatform and Stavenisse of The Netherland. Weibull distribution has been used for accurate estimation of wind energy potential for a long time. The Weibull distribution with two parameters is suitable for modeling wind data if wind distribution is unimodal. Whereas wind distribution is generally unimodal, random weather changes can make the distribution bimodal. It is always desirable to find a method that accurately represents actual statistical data. Some well-known statistical methods are Method of Moment (MoM), Linear Least Square Method (LLSM), Maximum Likelihood Method (M.L.M.), Modified Maximum Likelihood Method (MMLM), Energy Pattern Factor Method (EPFM), and Empirical Method (E.M.), etc. All these methods employ Probability Density Function (PDF) of Weibull distribution, except LLSM, which uses Cumulative Distribution Function (C.D.F.). In this communication, we are presenting a newly proposed method of evaluating Weibull parameters. Unlike most methods, this new method employs a cumulative distribution function. A MATLAB® GUI-based simulation is developed to estimate Weibull parameters using the C.D.F. approach. It is found that the Mean Square Error (M.S.E.) is the lowest when using the new method. The new method, therefore, estimates wind power density with reasonable accuracy. Wind Power (W.P.) is estimated by considering four different Wind Turbine (W.T.) models for two sites, and maximum W.P. is found using Evance R9000.</p>

A Simplification and Implementation of Random-effects Meta-analyses Based on the Exact Distribution of Cochran’s Q

2014 ◽  
Vol 53 (01) ◽  
pp. 54-61 ◽  
Author(s):  
M. Preuß ◽  
A. Ziegler
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Type I Error ◽  
Meta Analysis ◽  
Real Data ◽  
R Package ◽  
Cumulative Distribution ◽  
Data Sets ◽  
Type I ◽  
Simulation Studies

SummaryBackground: The random-effects (RE) model is the standard choice for meta-analysis in the presence of heterogeneity, and the stand ard RE method is the DerSimonian and Laird (DSL) approach, where the degree of heterogeneity is estimated using a moment-estimator. The DSL approach does not take into account the variability of the estimated heterogeneity variance in the estimation of Cochran’s Q. Biggerstaff and Jackson derived the exact cumulative distribution function (CDF) of Q to account for the variability of Ť 2.Objectives: The first objective is to show that the explicit numerical computation of the density function of Cochran’s Q is not required. The second objective is to develop an R package with the possibility to easily calculate the classical RE method and the new exact RE method.Methods: The novel approach was validated in extensive simulation studies. The different approaches used in the simulation studies, including the exact weights RE meta-analysis, the I 2 and T 2 estimates together with their confidence intervals were implemented in the R package metaxa.Results: The comparison with the classical DSL method showed that the exact weights RE meta-analysis kept the nominal type I error level better and that it had greater power in case of many small studies and a single large study. The Hedges RE approach had inflated type I error levels. Another advantage of the exact weights RE meta-analysis is that an exact confidence interval for T 2is readily available. The exact weights RE approach had greater power in case of few studies, while the restricted maximum likelihood (REML) approach was superior in case of a large number of studies. Differences between the exact weights RE meta-analysis and the DSL approach were observed in the re-analysis of real data sets. Application of the exact weights RE meta-analysis, REML, and the DSL approach to real data sets showed that conclusions between these methods differed.Conclusions: The simplification does not require the calculation of the density of Cochran’s Q, but only the calculation of the cumulative distribution function, while the previous approach required the computation of both the density and the cumulative distribution function. It thus reduces computation time, improves numerical stability, and reduces the approximation error in meta-analysis. The different approaches, including the exact weights RE meta-analysis, the I 2 and T 2estimates together with their confidence intervals are available in the R package metaxa, which can be used in applications.

scholarly journals A New Kernel Estimator of Copulas Based on Beta Quantile Transformations

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1078
Author(s):  
Catalina Bolancé ◽  
Carlos Alberto Acuña
Keyword(s):  
Distribution Function ◽  
Data Analysis ◽  
Simulation Study ◽  
Financial Risk ◽  
Cumulative Distribution Function ◽  
Kernel Estimator ◽  
Cumulative Distribution ◽  
Nonparametric Estimator ◽  
Marginal Distributions ◽  
Kernel Approach

A copula is a multivariate cumulative distribution function with marginal distributions Uniform(0,1). For this reason, a classical kernel estimator does not work and this estimator needs to be corrected at boundaries, which increases the difficulty of the estimation and, in practice, the bias boundary correction might not provide the desired improvement. A quantile transformation of marginals is a way to improve the classical kernel approach. This paper shows a Beta quantile transformation to be optimal and analyses a kernel estimator based on this transformation. Furthermore, the basic properties that allow the new estimator to be used for inference on extreme value copulas are tested. The results of a simulation study show how the new nonparametric estimator improves alternative kernel estimators of copulas. We illustrate our proposal with a financial risk data analysis.

scholarly journals ANALISIS KUMULATIF COVID-19 PROVINSI PAPUA TAHUN 2020 MENGGUNAKAN MODEL DISTRIBUSI JOHNSON SB

2021 ◽  
Vol 21 (1) ◽  
pp. 21-28
Author(s):  
Felix Reba ◽  
Alvian Sroyer
Keyword(s):  
Distribution Function ◽  
Maximum Likelihood ◽  
Cumulative Distribution Function ◽  
Data Distribution ◽  
Secondary Data ◽  
Cumulative Distribution ◽  
Distribution Model ◽  
Cumulative Probability ◽  
Number Of Patients ◽  
Kolmogorov Smirnov

Coronavirus belongs to the coronaviridae family. The coronavirus family groups are alpha (α), beta (β), gamma (γ) and delta (δ) coronavirus. Although research related to covid-19 in several provinces in Indonesia has been conducted by several researchers so far there has been no research related to the Covid-19 model in Papua province. One of the obstacles faced by some researchers is related to the Covid-19 data parameters which are difficult to estimate, so that the model formulated could not describe the outbreak well. Therefore the aim of this study is to conduct a cumulative analysis of the 2020 Papua province Covid-19 using the Johnson SB distribution model. The methods used to perform the analysis are Kolmogorov Smirnov for testing the suitability of the Covid-19 data to the model, Johnson SB to show the data distribution model, Maximum Likelihood to estimate the parameters and the Johnson SB cumulative distribution function to describe the probability of Covid-19 data. 19 Papua Province in 2020. The secondary data on the number of Covid-19 cases in Papua, obtained from the Papua Provincial Health Office is used in this research. The results showed that, the highest increase in the number of patients every day, starting from September 1 2020 to October 31, 2020 for infected cases was on 16-17 September, by 274 patients. Meanwhile, most recovery (308 patients) happened to be on 30-31 October and the highest death (5 people) was on 27-28 September. The highest cumulative probability for cases of infection, recovery and death were (Confirmed <4965) = 0.3, Prob(Cured <6408) = 0.9 and Prob(died <91) = 0.4 respectively.

scholarly journals An Asymmetric Bimodal Double Regression Model

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2279
Author(s):  
Yolanda M. Gómez ◽  
Diego I. Gallardo ◽  
Osvaldo Venegas ◽  
Tiago M. Magalhães
Keyword(s):  
Maximum Likelihood ◽  
Regression Model ◽  
Simulation Study ◽  
Maximum Likelihood Estimators ◽  
Real Data ◽  
Cauchy Distribution ◽  
Scale Parameters ◽  
Finite Samples ◽  
Data Application ◽  
Different Shapes

In this paper, we introduce an extension of the sinh Cauchy distribution including a double regression model for both the quantile and scale parameters. This model can assume different shapes: unimodal or bimodal, symmetric or asymmetric. We discuss some properties of the model and perform a simulation study in order to assess the performance of the maximum likelihood estimators in finite samples. A real data application is also presented.

scholarly journals The Odd Log-Logistic Poisson Inverse Rayleigh Distribution: Statistical Properties and Applications

2020 ◽  
pp. 775-787
Author(s):  
Ibrahim Elbatal
Keyword(s):  
Maximum Likelihood ◽  
Simulation Study ◽  
Maximum Likelihood Estimators ◽  
Real Data ◽  
Rayleigh Distribution ◽  
Statistical Properties ◽  
Data Sets ◽  
New Model ◽  
Fundamental Properties

In this work, a new extension of the Inverse Rayleigh model is proposed and studied. We derive some of its fundamental properties. We assess the performance of the maximum likelihood estimators via a simulation study. The importance of the new model is shown via two applications to real data sets. The new model is better fit than other important competitive models based on two real data sets.

A new generalized Lindley-Weibull class of distributions: Theory, properties and applications

2021 ◽  
Vol 71 (1) ◽  
pp. 211-234
Author(s):  
Boikanyo Makubate ◽  
Thatayaone Moakofi ◽  
Broderick Oluyede
Keyword(s):  
Maximum Likelihood ◽  
Power Series ◽  
Order Statistics ◽  
Simulation Study ◽  
Maximum Likelihood Estimators ◽  
Real Data ◽  
Maximum Likelihood Estimates ◽  
Mean Square ◽  
Lorenz Curves ◽  
Special Case

Abstract We propose a new generalized class of distributions called Lindley-Weibull Power Series (LWPS) distributions and their special case called Lindley-Weibull logarithmic (LWL) distributions. Structural properties of the LWPS class of distributions and its sub-model LWL distribution including moments, order statistics, Rényi entropy, mean and median deviations, Bonferroni and Lorenz curves, and maximum likelihood estimates are derived. A simulation study to examine the bias and mean square error of the maximum likelihood estimators for each parameter is presented. Finally, real data examples are presented to illustrate the applicability and usefulness of the proposed class of distributions.

On Log-q-Gaussian Distribution

2018 ◽  
Vol 70 (2) ◽  
pp. 105-121
Author(s):  
Ken-ichi Koike ◽  
Yuya Shimegi
Keyword(s):  
Distribution Function ◽  
Gaussian Distribution ◽  
Cumulative Distribution Function ◽  
Extreme Value Distribution ◽  
Likelihood Estimation ◽  
Real Data ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Ams Classification ◽  
New Distribution

We propose the log- q-Gaussian distribution which is obtained as the distribution of a random variable whose logarithm is q-Gaussian. Various types of properties of the new distribution are given such as the moments, the cumulative distribution function, the extreme value distribution, the likelihood estimation, and so on. Some examples for real data are also given. AMS Classification: 60Exx 62Exx.

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