scholarly journals Inducing a desired value of correlation between two point-scale variables: a two-step procedure using copulas

2021 ◽  
Author(s):  
Alessandro Barbiero
Keyword(s):  
Computational Cost ◽  
Likelihood Estimation ◽  
Real Data ◽  
Discrete Distributions ◽  
Point Scale ◽  
Numerical Illustration ◽  
Monte Carlo Simulation Study ◽  
Marginal Distributions ◽  
Copula Parameter ◽  
Two Step Estimation

AbstractFocusing on point-scale random variables, i.e. variables whose support consists of the first m positive integers, we discuss how to build a joint distribution with pre-specified marginal distributions and Pearson’s correlation $$\rho $$ ρ . After recalling how the desired value $$\rho $$ ρ is not free to vary between $$-1$$ - 1 and $$+1$$ + 1 , but generally ranges a narrower interval, whose bounds depend on the two marginal distributions, we devise a procedure that first identifies a class of joint distributions, based on a parametric family of copulas, having the desired margins, and then adjusts the copula parameter in order to match the desired correlation. The proposed methodology addresses a need which often arises when assessing the performance and robustness of some new statistical technique, i.e. trying to build a huge number of replicates of a given dataset, which satisfy—on average—some of its features (for example, the empirical marginal distributions and the pairwise linear correlations). The proposal shows several advantages, such as—among others—allowing for dependence structures other than the Gaussian and being able to accommodate the copula parameter up to an assigned level of precision for $$\rho $$ ρ with a very small computational cost. Based on this procedure, we also suggest a two-step estimation technique for copula-based bivariate discrete distributions, which can be used as an alternative to full and two-step maximum likelihood estimation. Numerical illustration and empirical evidence are provided through some examples and a Monte Carlo simulation study, involving the CUB distribution and three different copulas; an application to real data is also discussed.

The Poisson Nadarajah–Haghighi Distribution: Properties and Applications to Lifetime Data

2019 ◽  
Vol 27 (01) ◽  
pp. 2050005
Author(s):  
Muhammad Mansoor ◽  
M. H. Tahir ◽  
Aymaan Alzaatreh ◽  
Gauss M. Cordeiro
Keyword(s):  
Maximum Likelihood ◽  
Censored Data ◽  
Survival Data ◽  
Maximum Likelihood Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Likelihood Method ◽  
Lifetime Data ◽  
Monte Carlo Simulation Study ◽  
Mathematical Properties

A new three-parameter compounded extended-exponential distribution “Poisson Nadarajah–Haghighi” is introduced and studied, which is quite flexible and can be used effectively in modeling survival data. It can have increasing, decreasing, upside-down bathtub and bathtub-shaped failure rate. A comprehensive account of the mathematical properties of the model is presented. We discuss maximum likelihood estimation for complete and censored data. The suitability of the maximum likelihood method to estimate its parameters is assessed by a Monte Carlo simulation study. Four empirical illustrations of the new model are presented to real data and the results are quite satisfactory.

scholarly journals Parameter Estimation for Type III Discrete Weibull Distribution: A Comparative Study

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Alessandro Barbiero
Keyword(s):  
Monte Carlo Simulation ◽  
Weibull Distribution ◽  
Real Data ◽  
Discrete Distributions ◽  
Software Implementation ◽  
Discrete Models ◽  
Monte Carlo Simulation Study ◽  
Type Iii ◽  
Failure Data ◽  
Better Than

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’ 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.

The copula-based method for statistical analysis of step-stress accelerated life test with dependent competing failure modes

2018 ◽  
Vol 233 (3) ◽  
pp. 401-418 ◽  
Author(s):  
Yicheng Zhou ◽  
Zhenzhou Lu ◽  
Yan Shi ◽  
Kai Cheng
Keyword(s):  
Statistical Analysis ◽  
Failure Modes ◽  
Likelihood Estimation ◽  
Real Data ◽  
Copula Function ◽  
Accelerated Life Test ◽  
Dependence Structure ◽  
Life Test ◽  
Copula Parameter ◽  
Accelerated Life

Competing risk usually exists in engineering applications; thus, the study of the statistical inference of accelerated life testing with competing failure modes is of great significance. In this article, we address the statistical analysis of a step-stress accelerated life test in the presence of dependent competing failure modes. The dependence structure among distributions of lifetimes is constructed by copula function with unknown copula parameter. The parametric maximum likelihood estimation is developed to obtain the estimates of underlying parameters. The asymptotic standard errors and asymptotic confidence interval of estimates are also obtained by missing information principle. An extensive simulation study and a real data analysis are carried out to observe the performance of the proposed method. The results of the case studies show that our proposed method is valid and effective for the statistical analysis of step-stress accelerated life test with dependent competing failure modes.

scholarly journals The Burr-Weibull Power Series Class of Distributions

2018 ◽  
Vol 48 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Broderick Oluyede ◽  
Precious Mdlongwa ◽  
Boikanyo Makubate ◽  
Shujiao Huang
Keyword(s):  
Power Series ◽  
Likelihood Estimation ◽  
Real Data ◽  
Model Parameters ◽  
Estimation Technique ◽  
Exponential Distributions ◽  
Data Set ◽  
Monte Carlo Simulation Study ◽  
Special Model ◽  
Mathematical Properties

A new generalized class of distributions called the Burr-Weibull Power Series (BWPS) class of distributions is developed and explored. This class of distributions generalizes the Burr power series and Weibull power series classes of distributions, respectively. A special model of the BWPS class of distributions, the new Burr-Weibull Poisson (BWP) distribution is considered and some of its mathematical properties are obtained. The BWP distribution contains several new and well known sub-models, including Burr-Weibull, Burr-exponential Poisson, Burr-exponential, Burr-Rayleigh Poisson, Burr-Rayleigh, Burr-Poisson, Burr, Lomax-exponential Poisson, Lomax-Weibull, Lomax-exponential, Lomax-Rayleigh, Lomax-Poisson, Lomax, Weibull, Rayleigh and exponential distributions. Maximum likelihood estimation technique is used to estimate the model parameters followed by a Monte Carlo simulation study. Finally an application of the BWP model to a real data set is presented to illustrate the usefulness of the proposed class of distributions.

scholarly journals A New Extension of Thinning-Based Integer-Valued Autoregressive Models for Count Data

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 62
Author(s):  
Zhengwei Liu ◽  
Fukang Zhu
Keyword(s):  
Likelihood Estimation ◽  
Real Data ◽  
Autoregressive Models ◽  
Superior Performance ◽  
Data Sets ◽  
Binomial Thinning ◽  
Free Case ◽  
Two Parameters ◽  
Conditional Maximum ◽  
Thinning Operator

The thinning operators play an important role in the analysis of integer-valued autoregressive models, and the most widely used is the binomial thinning. Inspired by the theory about extended Pascal triangles, a new thinning operator named extended binomial is introduced, which is a general case of the binomial thinning. Compared to the binomial thinning operator, the extended binomial thinning operator has two parameters and is more flexible in modeling. Based on the proposed operator, a new integer-valued autoregressive model is introduced, which can accurately and flexibly capture the dispersed features of counting time series. Two-step conditional least squares (CLS) estimation is investigated for the innovation-free case and the conditional maximum likelihood estimation is also discussed. We have also obtained the asymptotic property of the two-step CLS estimator. Finally, three overdispersed or underdispersed real data sets are considered to illustrate a superior performance of the proposed model.

scholarly journals Water Particles Monitoring in the Atacama Desert: SPC Approach Based on Proportional Data

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 154
Author(s):  
Anderson Fonseca ◽  
Paulo Henrique Ferreira ◽  
Diego Carvalho do Nascimento ◽  
Rosemeire Fiaccone ◽  
Christopher Ulloa-Correa ◽  
...  
Keyword(s):  
Control Chart ◽  
Average Run Length ◽  
Atacama Desert ◽  
Real Data ◽  
Interval Data ◽  
Total Quality ◽  
Statistical Process ◽  
Monte Carlo Simulation Study ◽  
Run Length ◽  
Study Results

Statistical monitoring tools are well established in the literature, creating organizational cultures such as Six Sigma or Total Quality Management. Nevertheless, most of this literature is based on the normality assumption, e.g., based on the law of large numbers, and brings limitations towards truncated processes as open questions in this field. This work was motivated by the register of elements related to the water particles monitoring (relative humidity), an important source of moisture for the Copiapó watershed, and the Atacama region of Chile (the Atacama Desert), and presenting high asymmetry for rates and proportions data. This paper proposes a new control chart for interval data about rates and proportions (symbolic interval data) when they are not results of a Bernoulli process. The unit-Lindley distribution has many interesting properties, such as having only one parameter, from which we develop the unit-Lindley chart for both classical and symbolic data. The performance of the proposed control chart is analyzed using the average run length (ARL), median run length (MRL), and standard deviation of the run length (SDRL) metrics calculated through an extensive Monte Carlo simulation study. Results from the real data applications reveal the tool’s potential to be adopted to estimate the control limits in a Statistical Process Control (SPC) framework.

scholarly journals The Gamma Generalized Pareto Distribution with Applications in Survival Analysis

2017 ◽  
Vol 6 (3) ◽  
pp. 141 ◽  
Author(s):  
Thiago A. N. De Andrade ◽  
Luz Milena Zea Fernandez ◽  
Frank Gomes-Silva ◽  
Gauss M. Cordeiro
Keyword(s):  
Monte Carlo Simulation ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Real Data ◽  
Model Parameters ◽  
Monte Carlo Simulation Study ◽  
Lorenz Curves ◽  
Generalized Pareto ◽  
Mathematical Properties ◽  
Pareto Model

We study a three-parameter model named the gamma generalized Pareto distribution. This distribution extends the generalized Pareto model, which has many applications in areas such as insurance, reliability, finance and many others. We derive some of its characterizations and mathematical properties including explicit expressions for the density and quantile functions, ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating function, R\'enyi entropy and order statistics. We discuss the estimation of the model parameters by maximum likelihood. A small Monte Carlo simulation study and two applications to real data are presented. We hope that this distribution may be useful for modeling survival and reliability data.

Detecting common breaks in the means of high dimensional cross-dependent panels

2021 ◽  
Author(s):  
Lajos Horváth ◽  
Zhenya Liu ◽  
Gregory Rice ◽  
Yuqian Zhao
Keyword(s):  
Panel Data ◽  
Common Factors ◽  
Real Data ◽  
Change Points ◽  
High Dimensional ◽  
Asymptotic Results ◽  
Cross Sectional ◽  
Data Set ◽  
Monte Carlo Simulation Study ◽  
Cross Sectional Dependence

Abstract The problem of detecting change points in the mean of high dimensional panel data with potentially strong cross–sectional dependence is considered. Under the assumption that the cross–sectional dependence is captured by an unknown number of common factors, a new CUSUM type statistic is proposed. We derive its asymptotic properties under three scenarios depending on to what extent the common factors are asymptotically dominant. With panel data consisting of N cross sectional time series of length T, the asymptotic results hold under the mild assumption that min {N, T} → ∞, with an otherwise arbitrary relationship between N and T, allowing the results to apply to most panel data examples. Bootstrap procedures are proposed to approximate the sampling distribution of the test statistics. A Monte Carlo simulation study showed that our test outperforms several other existing tests in finite samples in a number of cases, particularly when N is much larger than T. The practical application of the proposed results are demonstrated with real data applications to detecting and estimating change points in the high dimensional FRED-MD macroeconomic data set.

scholarly journals Parametric and Semiparametric Estimations of Bivariate Truncated Type I Generalized Logistic Models driven from Copulas

2017 ◽  
Vol 7 (1) ◽  
pp. 72 ◽  
Author(s):  
Lamya A Baharith
Keyword(s):  
Logistic Models ◽  
Real Data ◽  
Information Criterion ◽  
Logistic Distribution ◽  
Type I ◽  
Copula Functions ◽  
Data Set ◽  
Monte Carlo Simulation Study ◽  
Generalized Logistic Distribution ◽  
Weibull Models

Truncated type I generalized logistic distribution has been used in a variety of applications. In this article, a new bivariate truncated type I generalized logistic (BTTGL) distributional models driven from three different copula functions are introduced. A study of some properties is illustrated. Parametric and semiparametric methods are used to estimate the parameters of the BTTGL models. Maximum likelihood and inference function for margin estimates of the BTTGL parameters are compared with semiparametric estimates using real data set. Further, a comparison between BTTGL, bivariate generalized exponential and bivariate exponentiated Weibull models is conducted using Akaike information criterion and the maximized log-likelihood. Extensive Monte Carlo simulation study is carried out for different values of the parameters and different sample sizes to compare the performance of parametric and semiparametric estimators based on relative mean square error.

scholarly journals Interpolation of groundwater quality parameters with some values below the detection limit

2011 ◽  
Vol 15 (9) ◽  
pp. 2763-2775 ◽  
Author(s):  
A. Bárdossy
Keyword(s):  
Detection Limit ◽  
Maximum Likelihood ◽  
Groundwater Quality ◽  
Likelihood Estimation ◽  
Quality Parameters ◽  
Indicator Kriging ◽  
West Germany ◽  
Marginal Distributions ◽  
Copula Models ◽  
Measuring Device

Abstract. For many environmental variables, measurements cannot deliver exact observation values as their concentration is below the sensitivity of the measuring device (detection limit). These observations provide useful information but cannot be treated in the same manner as the other measurements. In this paper a methodology for the spatial interpolation of these values is described. The method is based on spatial copulas. Here two copula models – the Gaussian and a non-Gaussian v-copula are used. First a mixed maximum likelihood approach is used to estimate the marginal distributions of the parameters. After removal of the marginal distributions the next step is the maximum likelihood estimation of the parameters of the spatial dependence including taking those values below the detection limit into account. Interpolation using copulas yields full conditional distributions for the unobserved sites and can be used to estimate confidence intervals, and provides a good basis for spatial simulation. The methodology is demonstrated on three different groundwater quality parameters, i.e. arsenic, chloride and deethylatrazin, measured at more than 2000 locations in South-West Germany. The chloride values are artificially censored at different levels in order to evaluate the procedures on a complete dataset by progressive decimation. Interpolation results are evaluated using a cross validation approach. The method is compared with ordinary kriging and indicator kriging. The uncertainty measures of the different approaches are also compared.

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