A transformation for normal extremes

1982 ◽  
Vol 19 (A) ◽  
pp. 201-206
Author(s):  
H. E. Daniels
Keyword(s):  
Normal Distribution ◽  
Sample Sizes ◽  
Practical Implications ◽  
Transformed Distribution

A transformation is introduced to stabilize the variance of the largest value in a sample from a normal distribution. The transformed distribution is found to approximate closely to the limiting Gumbel form for all sample sizes. The practical implications of the result are discussed.

A transformation for normal extremes

1982 ◽  
Vol 19 (A) ◽  
pp. 201-206
Author(s):  
H. E. Daniels
Keyword(s):  
Normal Distribution ◽  
Sample Sizes ◽  
Practical Implications ◽  
Transformed Distribution

A transformation is introduced to stabilize the variance of the largest value in a sample from a normal distribution. The transformed distribution is found to approximate closely to the limiting Gumbel form for all sample sizes. The practical implications of the result are discussed.

On the Mathematical Basis of Zelen’s Prerandomized Designs

1985 ◽  
Vol 24 (03) ◽  
pp. 120-130 ◽  
Author(s):  
E. Brunner ◽  
N. Neumann
Keyword(s):  
Clinical Trial ◽  
Normal Distribution ◽  
Random Variables ◽  
Small Samples ◽  
Selection Effects ◽  
Sample Sizes ◽  
Mathematical Basis ◽  
Additional Assumptions

SummaryThe mathematical basis of Zelen’s suggestion [4] of pre randomizing patients in a clinical trial and then asking them for their consent is investigated. The first problem is to estimate the therapy and selection effects. In the simple prerandomized design (PRD) this is possible without any problems. Similar observations have been made by Anbar [1] and McHugh [3]. However, for the double PRD additional assumptions are needed in order to render therapy and selection effects estimable. The second problem is to determine the distribution of the statistics. It has to be taken into consideration that the sample sizes are random variables in the PRDs. This is why the distribution of the statistics can only be determined asymptotically, even under the assumption of normal distribution. The behaviour of the statistics for small samples is investigated by means of simulations, where the statistics considered in the present paper are compared with the statistics suggested by Ihm [2]. It turns out that the statistics suggested in [2] may lead to anticonservative decisions, whereas the “canonical statistics” suggested by Zelen [4] and considered in the present paper keep the level quite well or may lead to slightly conservative decisions, if there are considerable selection effects.

Determination of sample sizes for setting β-content tolerance limits controlling both tails of the normal distribution

1984 ◽  
Vol 2 (5) ◽  
pp. 311-314 ◽  
Author(s):  
Youn-Min Chou ◽  
Robert W. Mee
Keyword(s):  
Normal Distribution ◽  
Sample Sizes ◽  
Tolerance Limits

scholarly journals Comparative evaluation of goodness of fit tests for normal distribution using simulation and empirical data

2020 ◽  
Vol 57 (2) ◽  
pp. 237-251
Author(s):  
Achilleas Anastasiou ◽  
Alex Karagrigoriou ◽  
Anastasios Katsileros
Keyword(s):  
Sample Size ◽  
Normal Distribution ◽  
Empirical Data ◽  
Goodness Of Fit ◽  
Small Sample Size ◽  
Small Sample ◽  
Type I ◽  
Sample Sizes ◽  
Omnibus Test ◽  
Wide Range

SummaryThe normal distribution is considered to be one of the most important distributions, with numerous applications in various fields, including the field of agricultural sciences. The purpose of this study is to evaluate the most popular normality tests, comparing the performance in terms of the size (type I error) and the power against a large spectrum of distributions with simulations for various sample sizes and significance levels, as well as through empirical data from agricultural experiments. The simulation results show that the power of all normality tests is low for small sample size, but as the sample size increases, the power increases as well. Also, the results show that the Shapiro–Wilk test is powerful over a wide range of alternative distributions and sample sizes and especially in asymmetric distributions. Moreover the D’Agostino–Pearson Omnibus test is powerful for small sample sizes against symmetric alternative distributions, while the same is true for the Kurtosis test for moderate and large sample sizes.

scholarly journals On the Order Statistics of Standard Normal-Based Power Method Distributions

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Todd C. Headrick ◽  
Mohan D. Pant
Keyword(s):  
Normal Distribution ◽  
Order Statistics ◽  
Remainder Term ◽  
Standard Normal Distribution ◽  
Power Method ◽  
Sample Sizes ◽  
Elementary Functions ◽  
Standard Normal

This paper derives a procedure for determining the expectations of order statistics associated with the standard normal distribution () and its powers of order three and five ( and ). The procedure is demonstrated for sample sizes of . It is shown that and have expectations of order statistics that are functions of the expectations for and can be expressed in terms of explicit elementary functions for sample sizes of . For sample sizes of the expectations of the order statistics for , , and only require a single remainder term.

scholarly journals Some Estimation for the Parameters and Hazard Function of Kummer Beta Generalized Normal Distribution

2021 ◽  
Vol 27 (127) ◽  
pp. 188-212
Author(s):  
Manal Mahmoud Rashid ◽  
Entsar Arebe Aldoori
Keyword(s):  
Genetic Algorithm ◽  
Maximum Likelihood ◽  
Normal Distribution ◽  
Failure Rate ◽  
Hazard Function ◽  
Sample Sizes ◽  
Distribution Parameters ◽  
The Common ◽  
Generalized Normal Distribution

Transforming the common normal distribution through the generated Kummer Beta model to the Kummer Beta Generalized Normal Distribution (KBGND) had been achieved. Then, estimating the distribution parameters and hazard function using the MLE method, and improving these estimations by employing the genetic algorithm. Simulation is used by assuming a number of models and different sample sizes. The main finding was that the common maximum likelihood (MLE) method is the best in estimating the parameters of the Kummer Beta Generalized Normal Distribution (KBGND) compared to the common maximum likelihood according to Mean Squares Error (MSE) and Mean squares Error Integral (IMSE) criteria in estimating the hazard function. While the practical side showed that the hazard function is increasing, i.e. the increment in staying the teachers in the service, they will be exposed to a greater failure rate as a result of the staying period which decreases in its turn.

Best Linear Unbiased Estimators for Normal Distribution Quantiles for Sample Sizes up to 20

1986 ◽  
Vol 35 (3) ◽  
pp. 327-329 ◽  
Author(s):  
Khatab M. Hassanein ◽  
A. K. Md. Ehsanes Saleh ◽  
Edward F. Brown
Keyword(s):  
Normal Distribution ◽  
Sample Sizes ◽  
Unbiased Estimators ◽  
Best Linear Unbiased ◽  
Best Linear Unbiased Estimators

scholarly journals Sample Sizes Based on Weibull Distribution and Normal Distribution for FRP Tensile Coupon Test

Materials ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 126 ◽  
Author(s):  
Yongxin Yang ◽  
Weijie Li ◽  
Wenshui Tang ◽  
Biao Li ◽  
Dengfeng Zhang
Keyword(s):  
Weibull Distribution ◽  
Sample Size ◽  
Normal Distribution ◽  
Fiber Reinforced Polymer ◽  
Sample Sizes ◽  
Frp Composites ◽  
Reinforced Polymer ◽  
Size Evaluation ◽  
Sample Coefficient Of Variation ◽  
Current Guidelines

Current guidelines stipulate a sample size of five for a tensile coupon test of fiber reinforced polymer (FRP) composites based on the assumption of a normal distribution and a sample coefficient of variation (COV) of 0.058. Increasing studies have validated that a Weibull distribution is more appropriate in characterizing the tensile properties of FRP. However, few efforts have been devoted to sample size evaluation based on a Weibull distribution. It is not clear if the Weibull distribution will result in a more conservative sample size value. In addition, the COV of FRP’s properties can vary from 5% to 15% in practice. In this study, the sample size based on a two-parameter Weibull distribution is compared with that based on a normal distribution. It is revealed that the Weibull distribution results in almost the same sample size as the normal distribution, which means that the sample size based on a normal distribution is applicable. For coupons with COVs varying from 0.05 to 0.20, the sample sizes range from less than 10 to more than 60. The use of only five coupons will lead to a prediction error of material property between 6.2% and 24.8% for COVs varying from 0.05 to 0.20.

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