scholarly journals Identifying Which of J Independent Binomial Distributions Has the Largest Probability of Success

2020 ◽  
Vol 18 (2) ◽  
pp. 2-9
Author(s):  
Rand Wilcox
Keyword(s):  
Binomial Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Pairwise Comparisons ◽  
Type I ◽  
Type I Errors ◽  
Binomial Distributions ◽  
Probability Of Success

Let p1,…, pJ denote the probability of a success for J independent random variables having a binomial distribution and let p(1) ≤ … ≤ p(J) denote these probabilities written in ascending order. The goal is to make a decision about which group has the largest probability of a success, p(J). Let p̂1,…, p̂J denote estimates of p1,…,pJ, respectively. The strategy is to test J − 1 hypotheses comparing the group with the largest estimate to each of the J − 1 remaining groups. For each of these J − 1 hypotheses that are rejected, decide that the group corresponding to the largest estimate has the larger probability of success. This approach has a power advantage over simply performing all pairwise comparisons. However, the more obvious methods for controlling the probability of one more Type I errors perform poorly for the situation at hand. A method for dealing with this is described and illustrated.

Heteroscedastic Methods for Performing All Pairwise Comparisons of Regression Lines Associated With J Independent Groups

Methodology ◽  
2015 ◽  
Vol 11 (3) ◽  
pp. 110-115 ◽  
Author(s):  
Rand R. Wilcox ◽  
Jinxia Ma
Keyword(s):  
Least Squares ◽  
Pairwise Comparisons ◽  
Simple Extension ◽  
Type I ◽  
Type I Errors ◽  
Well Elderly ◽  
Using Data

Abstract. The paper compares methods that allow both within group and between group heteroscedasticity when performing all pairwise comparisons of the least squares lines associated with J independent groups. The methods are based on simple extension of results derived by Johansen (1980) and Welch (1938) in conjunction with the HC3 and HC4 estimators. The probability of one or more Type I errors is controlled using the improvement on the Bonferroni method derived by Hochberg (1988) . Results are illustrated using data from the Well Elderly 2 study, which motivated this paper.

scholarly journals Assessment of the Randomization Test for Binomial Sex-Ratio Distributions in Birds

The Auk ◽  
2003 ◽  
Vol 120 (1) ◽  
pp. 62-68
Author(s):  
John G. Ewen ◽  
Phillip Cassey ◽  
Robert A. R. King
Keyword(s):  
Sex Ratio ◽  
Binomial Distribution ◽  
Type I Error ◽  
Analytical Techniques ◽  
Randomization Test ◽  
Entire Population ◽  
Type I ◽  
Primary Sex Ratio ◽  
Binomial Distributions ◽  
Common Distribution

Abstract We assessed a randomization test frequently used in studies that aim to detect bias in primary sex ratio of avian species. Three different treatments were examined that represent simple but ecologically realistic cases of interest to researchers. The randomization test was successful in reducing Type I error when testing for a significant departure from a single binomial distribution. When brood sizes or sample sizes were low, however, the randomization test lacked power to detect departures from a population of broods with multiple binomial distributions of sons and daughters. We recommend analytical techniques available to researchers that do not require a common distribution of the sexes to broods for an entire population.

Correction of the Student T Statistic for Nonindependence of Sample Observations

1992 ◽  
Vol 75 (3) ◽  
pp. 1011-1020 ◽  
Author(s):  
Donald W. Zimmerman ◽  
Richard H. Williams ◽  
Bruno D. Zumbo
Keyword(s):  
Unbiased Estimate ◽  
Random Variables ◽  
Type I ◽  
Type Ii ◽  
Population Variance ◽  
Type I Errors ◽  
Computer Simulation Study ◽  
The One ◽  
Type Ii Errors ◽  
Independent Identically Distributed

A computer-simulation study examined the one-sample Student t test under violation of the assumption of independent sample observations. The probability of Type I errors increased, and the probability of Type II errors decreased, spuriously elevating the entire power function. The magnitude of the change depended on the correlation between pairs of sample values as well as the number of sample values that were pairwise correlated. A modified t statistic, derived from an unbiased estimate of the population variance that assumed only exchangeable random variables instead of independent, identically distributed random variables, effectively corrected for nonindependence for all degrees of correlation and restored the probability of Type I and Type II errors to their usual values.

Protecting the overall rate of Type I errors for pairwise comparisons with an omnibus test statistic.

1979 ◽  
Vol 86 (4) ◽  
pp. 884-888 ◽  
Author(s):  
Harvey J. Keselman ◽  
Paul A. Games ◽  
Joanne C. Rogan
Keyword(s):  
Pairwise Comparisons ◽  
Type I ◽  
Test Statistic ◽  
Omnibus Test ◽  
Type I Errors

scholarly journals Bayesian Estimation of the Parameters of Discrete Weibull Type (I) Distribution

2020 ◽  
Vol 18 (2) ◽  
pp. 2-13
Author(s):  
Samir Kamel Ashour ◽  
Mohamed Salem Abdelwahab Muiftah
Keyword(s):  
Weibull Distribution ◽  
Bayesian Estimation ◽  
Shape Parameter ◽  
Random Variables ◽  
Independent Random Variables ◽  
Parameter Estimates ◽  
Type I ◽  
Sample Sizes ◽  
Bayesian Estimates ◽  
Distribution Parameters

Bayesian estimation of the continuous Weibull distribution parameters was studied by Ahmad and Ahmad (2013) under the assumption of knowing the shape parameter. Bayesian estimates are considered here of the parameters of the discrete Weibull Type I [DW(I)] distribution and are obtained under two different assumptions: when the shape parameter is known, and when both parameters are independent random variables. A Mathcad program is performed to simulate data from the DW(I) distribution considering different values of the parameters and different sample sizes, and to obtain Bayesian parameter estimates. The resulted estimates are compared to the ML and proportion estimates obtained by Khan et al. (1989).

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

Methodology of Calculation the Terminal Settling Velocity Distribution of Irregular Particles for High Values of the Reynold’s Number/Metodologia Wyliczania Rozkładu Granicznej Prędkości Opadania Ziaren Nieregularnych Dla Wysokich Wartości Liczb Reynoldsa

2014 ◽  
Vol 59 (2) ◽  
pp. 553-562 ◽  
Author(s):  
Agnieszka Surowiak ◽  
Marian Brożek
Keyword(s):  
Velocity Distribution ◽  
Settling Velocity ◽  
Random Variables ◽  
Independent Random Variables ◽  
Calculation Algorithm ◽  
Shape Coefficient ◽  
Geometrical Properties ◽  
Size And Shape ◽  
Physical Density ◽  
Settling Velocity Distribution

Abstract Settling velocity of particles, which is the main parameter of jig separation, is affected by physical (density) and the geometrical properties (size and shape) of particles. The authors worked out a calculation algorithm of particles settling velocity distribution for irregular particles assuming that the density of particles, their size and shape constitute independent random variables of fixed distributions. Applying theorems of probability, concerning distributions function of random variables, the authors present general formula of probability density function of settling velocity irregular particles for the turbulent motion. The distributions of settling velocity of irregular particles were calculated utilizing industrial sample. The measurements were executed and the histograms of distributions of volume and dynamic shape coefficient, were drawn. The separation accuracy was measured by the change of process imperfection of irregular particles in relation to spherical ones, resulting from the distribution of particles settling velocity.

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