scholarly journals Two component modified Lilliefors test for normality

Equilibrium ◽  
2021 ◽  
Vol 16 (2) ◽  
pp. 429-455
Author(s):  
Piotr Sulewski
Keyword(s):  
Similarity Measure ◽  
Goodness Of Fit ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Absolute Difference ◽  
Test Statistics ◽  
Excess Kurtosis ◽  
Two Component ◽  
Lilliefors Test ◽  
Test For Normality

Research background: Commonly known and used parametric tests e.g. Student, Behrens? Fisher, Snedecor, Bartlett, Cochran, Hartley tests are applicable when there is an evidence that samples come from the Normal general population. What makes things worse is that testers are not fully aware in what degree of abnormality distorts results of parametric tests listed above and suchlike. So, it is no exaggeration to say that testing for normality (goodness-of-fit testing, GoFT) is a gate to proper parametric statistical reasoning. It seems that the gate opens too easily. In other words, most popular goodness-of-fit tests are weaker than statisticians want them to be. Purpose of the article: The main purpose of this paper is to put forward the GoFT that is, in particular circumstances, more powerful than GoFTs used until now. The other goals are to define a similarity measure between an alternative distribution and the normal one and to calculate the power of normality tests for a big set of alternatives. And, of course, to interest statisticians in using the GoFTs in their practice. Method: There are two ways to make GoFT more powerful: extensive and intensive one. The extensive method consists in drawing large samples. The intensive method consists in extracting more information from mall samples. In order to make the test method intensive, the test statistics, as distinct from all existing GoFTs, has two components. The first component (denoted by ?) is a classic Kolmogorov / Lilliefors test statistics i.e. the greatest absolute difference between theoretical and empirical cumulative distribution functions. The second component is the order statistics (r) at which the ?_max^((r) ) locate itself. Of course ?_max^((r) ) is the conditional random variable with (r) being the condition. Large scale Monte Carlo simulations provided data sufficient to in-depth study of properties of distributions of ?_max^((r) ) random variable. Findings & value-added: Simulation study shows that the Two Component Modified Lilliefors test for normality is the most powerful for some type of alternatives, especially for the symmetrical, unimodal and bimodal distributions with positive excess kurtosis, for symmetrical and unimodal distributions with negative excess kurtosis and small sample sizes. Due to the values of skewness and excess kurtosis, and the defined similarity measure between the ND and an alternative, alternative distributions are close to the normal distribution. Numerous examples of real data show the usefulness of the proposed GoFT.

scholarly journals Goodness-of-fit tests for the beta Gompertz distribution

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 69-81
Author(s):  
Hanaa Abu-Zinadah ◽  
Asmaa Binkhamis
Keyword(s):  
Goodness Of Fit ◽  
Real Data ◽  
Likelihood Method ◽  
Test Statistics ◽  
Gompertz Distribution ◽  
Goodness Of Fit Tests ◽  
Lilliefors Test ◽  
Kolmogorov Smirnov ◽  
Anderson Darling ◽  
Von Mises

This article studied the goodness-of-fit tests for the beta Gompertz distribution with four parameters based on a complete sample. The parameters were estimated by the maximum likelihood method. Critical values were found by Monte Carlo simulation for the modified Kolmogorov-Smirnov, Anderson-Darling, Cramer-von Mises, and Lilliefors test statistics. The power of these test statistics founded the optimal alternative distribution. Real data applications were used as examples for the goodness of fit tests.

scholarly journals Goodness-of-fit tests for the beta Gompertz distribution

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 69-81
Author(s):  
Hanaa Abu-Zinadah ◽  
Asmaa Binkhamis
Keyword(s):  
Goodness Of Fit ◽  
Real Data ◽  
Likelihood Method ◽  
Test Statistics ◽  
Gompertz Distribution ◽  
Goodness Of Fit Tests ◽  
Lilliefors Test ◽  
Kolmogorov Smirnov ◽  
Anderson Darling ◽  
Von Mises

This article studied the goodness-of-fit tests for the beta Gompertz distribution with four parameters based on a complete sample. The parameters were estimated by the maximum likelihood method. Critical values were found by Monte Carlo simulation for the modified Kolmogorov-Smirnov, Anderson-Darling, Cramer-von Mises, and Lilliefors test statistics. The power of these test statistics founded the optimal alternative distribution. Real data applications were used as examples for the goodness of fit tests.

Bootstrap Analysis

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Goodness Of Fit ◽  
Null Distribution ◽  
Real Data ◽  
Confidence Regions ◽  
Confidence Bands ◽  
Cumulative Distribution ◽  
Attractive Alternative ◽  
Bootstrap Analysis ◽  
Parametric Bootstrapping ◽  
Anderson Darling

Parametric bootstrapping (BS) provides an attractive alternative, both theoretically and numerically, to asymptotic theory for estimating sampling distributions. This chapter summarizes its use not only for calculating confidence intervals for estimated parameters and functions of parameters, but also to obtain log-likelihood-based confidence regions from which confidence bands for cumulative distribution and regression functions can be obtained. All such BS calculations are very easy to implement. Details are also given for calculating critical values of EDF statistics used in goodness-of-fit (GoF) tests, such as the Anderson-Darling A2 statistic whose null distribution is otherwise difficult to obtain, as it varies with different null hypotheses. A simple proof is given showing that the parametric BS is probabilistically exact for location-scale models. A formal regression lack-of-fit test employing parametric BS is given that can be used even when the regression data has no replications. Two real data examples are given.

scholarly journals Goodness-of-Fit Tests for Bivariate Time Series of Counts

Econometrics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 10
Author(s):  
Šárka Hudecová ◽  
Marie Hušková ◽  
Simos G. Meintanis
Keyword(s):  
Goodness Of Fit ◽  
Probability Generating Function ◽  
Parametric Bootstrap ◽  
Real Data ◽  
Data Sets ◽  
Test Statistics ◽  
Finite Sample ◽  
Generalized Poisson ◽  
Goodness Of Fit Tests ◽  
Monte Carlo Experiments

This article considers goodness-of-fit tests for bivariate INAR and bivariate Poisson autoregression models. The test statistics are based on an L2-type distance between two estimators of the probability generating function of the observations: one being entirely nonparametric and the second one being semiparametric computed under the corresponding null hypothesis. The asymptotic distribution of the proposed tests statistics both under the null hypotheses as well as under alternatives is derived and consistency is proved. The case of testing bivariate generalized Poisson autoregression and extension of the methods to dimension higher than two are also discussed. The finite-sample performance of a parametric bootstrap version of the tests is illustrated via a series of Monte Carlo experiments. The article concludes with applications on real data sets and discussion.

A test for fuzzy exponentiality based on Kullback-Leibler information

2021 ◽  
pp. 1-8
Author(s):  
Lingtao Kong
Keyword(s):  
Biological Sciences ◽  
Monte Carlo ◽  
Goodness Of Fit ◽  
Experimental Studies ◽  
Real Data ◽  
Test Statistics ◽  
Test Statistic ◽  
Goodness Of Fit Test ◽  
Higher Power ◽  
Leibler Information

The exponential distribution has been widely used in engineering, social and biological sciences. In this paper, we propose a new goodness-of-fit test for fuzzy exponentiality using α-pessimistic value. The test statistics is established based on Kullback-Leibler information. By using Monte Carlo method, we obtain the empirical critical points of the test statistic at four different significant levels. To evaluate the performance of the proposed test, we compare it with four commonly used tests through some simulations. Experimental studies show that the proposed test has higher power than other tests in most cases. In particular, for the uniform and linear failure rate alternatives, our method has the best performance. A real data example is investigated to show the application of our test.

scholarly journals On the control of the difference between two Brownian motions: a dynamic copula approach

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Thomas Deschatre
Keyword(s):  
Cumulative Distribution Function ◽  
Survival Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Gaussian Copula ◽  
Brownian Motions ◽  
Dynamic Copula ◽  
Copula Approach ◽  
The Difference ◽  
The Right

AbstractWe propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Considering two Brownian motions B1t and B2t, the main result is that the range of possible values for is the same for Markovian pairs and all pairs of Brownian motions, that is with φ being the cumulative distribution function of a standard Gaussian random variable.

3 Power and sample size for ABE in the 2× 2 design Here we give formulae for the calculation of the power of the TOST procedure assuming that there are in total n subjects in the 2× 2 trial. Suppose that the null hypotheses given below are to be tested using a 100(1 − α)% two-sided confidence interval and a power of (1 − β) is required to reject these hypotheses when they are false. H :µ ≤− ln 1.25 H :µ ≥ ln 1.25. Let us define x to be a random variable that has a noncentral t-distribution with df degrees of freedom and noncentrality parameter nc, i.e., x ∼ t(df,nc). The cumulative distribution function of x is defined as CDF(t,df,nc) = Pr(x≤ t). Assume that the power is to be calculated using log(AUC). If σ is the common within-subject variance for T and R for log(AUC), and n/2 subjects are allocated to each of the sequences RT and TR, then 1−β = CDF(t , df,nc )−CDF(t , df,nc ) (7.1) where √ n(log(µ )− log(0.8)) nc = 2σ √ n(log(µ )− log(1.25)) nc = 2σ √ (n− 2)(nc −nc = ) df 2t and t is the 100(1 − α)% point of the central t-distribution on n− 2 degrees of freedom. Some SAS code to calculate the power for an ABE 2 × 2 trial is given below, where the required input variables are α, σ value of the ratio µ and n, the total number of subjects in the trial.

2003 ◽  
pp. 361-361
Keyword(s):  
Distribution Function ◽  
Confidence Interval ◽  
Degrees Of Freedom ◽  
Random Variable ◽  
Noncentrality Parameter ◽  
Cumulative Distribution ◽  
Total N ◽  
The Common ◽  
Input Variables ◽  
T Distribution

scholarly journals On Distribution Characteristics of a Fuzzy Random Variable

2018 ◽  
Vol 47 (2) ◽  
pp. 53-67 ◽  
Author(s):  
Jalal Chachi
Keyword(s):  
Probability Theory ◽  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Fuzzy Random Variable ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Distribution Characteristics ◽  
Fuzzy Random Variables ◽  
Fuzzy Random

In this paper, rst a new notion of fuzzy random variables is introduced. Then, usingclassical techniques in Probability Theory, some aspects and results associated to a randomvariable (including expectation, variance, covariance, correlation coecient, etc.) will beextended to this new environment. Furthermore, within this framework, we can use thetools of general Probability Theory to dene fuzzy cumulative distribution function of afuzzy random variable.

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