scholarly journals Nonparametric Competitors to the Two-Way ANOVA

1994 ◽  
Vol 19 (3) ◽  
pp. 237-273 ◽  
Author(s):  
Larry E. Toothaker ◽  
De Newman
Keyword(s):  
Normal Distribution ◽  
Normal Distributions ◽  
Power Simulation ◽  
Rank Transform ◽  
Small N ◽  
Mixed Normal Distribution

The ANOVA F and several nonparametric competitors for two-way designs were compared for empirical α and power. Simulation of 2 × 2, 2 × 4, and 4 × 4 designs was done with cell sizes of 5 and 10 when sampling from normal, exponential, and mixed normal distributions. Conservatism of both α and power in the presence of other nonnull effects was seen in the tests due to Puri and Sen (1985) and, to a lesser degree, in the rank transform tests ( Conover & Iman, 1981 ). Tests by McSweeney (1967) and Hettmansperger (1984) had liberal α for some designs and distributions, especially for small n. The ANOVA F suffers from conservative α and power for the mixed normal distribution, but it is generally recommended.

scholarly journals Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

2021 ◽  
Vol 53 (1) ◽  
pp. 162-188
Author(s):  
Krzysztof Bartoszek ◽  
Torkel Erhardsson
Keyword(s):  
Normal Distribution ◽  
Phenotypic Trait ◽  
Phenotypic Traits ◽  
Normal Distributions ◽  
Average Value ◽  
Conditional Moments ◽  
Mixture Of Normal Distributions ◽  
Explicit Bounds ◽  
Mixtures Of Normal Distributions ◽  
Parameter Values

AbstractExplicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

NON-NORMAL STATISTICAL TOLERANCE ANALYSIS USING ANALYTICAL CONVOLUTION METHOD

2008 ◽  
Vol 07 (01) ◽  
pp. 127-130 ◽  
Author(s):  
S. G. LIU ◽  
P. WANG ◽  
Z. G. LI
Keyword(s):  
Normal Distribution ◽  
Closed Loop ◽  
Tolerance Analysis ◽  
Accurate Method ◽  
Probability Density Functions ◽  
Triangular Distribution ◽  
Normal Distributions ◽  
Statistical Tolerance ◽  
Statistical Tolerance Analysis ◽  
Convolution Method

In statistical tolerance analysis, it is usually assumed that the statistical tolerance is normally distributed. But in practice, there are many non-normal distributions, such as uniform distribution, triangular distribution, etc. The simple way to analyze non-normal distributions is to approximately represent it with normal distribution, but the accuracy is low. Monte-Carlo simulation can analyze non-normal distributions with higher accuracy, but is time consuming. Convolution method is an accurate method to analyze statistical tolerance, but there are few reported works about it because of the difficulty. In this paper, analytical convolution is used to analyze non-normal distribution, and the probability density functions of closed loop component are obtained. Comparing with other methods, convolution method is accurate and faster.

scholarly journals MIXED NORMAL INFERENCE ON MULTICOINTEGRATION

2010 ◽  
Vol 26 (5) ◽  
pp. 1565-1576 ◽  
Author(s):  
H. Peter Boswijk
Keyword(s):  
Brownian Motion ◽  
Normal Distribution ◽  
Asymptotic Distribution ◽  
Present Note ◽  
Hypothesis Test ◽  
Null Distribution ◽  
Test Statistics ◽  
Parameter Estimator ◽  
Likelihood Analysis ◽  
Mixed Normal Distribution

Asymptotic likelihood analysis of cointegration in I (2) models (see Johansen, 1997, 2006; Boswijk, 2000; Paruolo, 2000) has shown that inference on most parameters is mixed normal, implying hypothesis test statistics with an asymptotic χ2 null distribution. The asymptotic distribution of the multicointegration parameter estimator so far has been characterized by a Brownian motion functional, which has been conjectured to have a mixed normal distribution, based on simulations. The present note proves this conjecture.

scholarly journals Normal Distribution on the Rotation Group So(3)

1997 ◽  
Vol 29 (3-4) ◽  
pp. 201-233 ◽  
Author(s):  
Dmitry I. Nikolayev ◽  
Tatjana I. Savyolov
Keyword(s):  
Normal Distribution ◽  
Orientation Distribution ◽  
Rotation Group ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Normal Distributions ◽  
The Central Limit Theorem ◽  
Special Cases ◽  
Wrapped Normal ◽  
Circular Normal Distribution

We study the normal distribution on the rotation group SO(3). If we take as the normal distribution on the rotation group the distribution defined by the central limit theorem in Parthasarathy (1964) rather than the distribution with density analogous to the normal distribution in Eucledian space, then its density will be different from the usual (1/2πσ) exp⁡(−(x−m)2/2σ2) one. Nevertheless, many properties of this distribution will be analogous to the normal distribution in the Eucledian space. It is possible to obtain explicit expressions for density of normal distribution only for special cases. One of these cases is the circular normal distribution.The connection of the circular normal distribution SO(3) group with the fundamental solution of the corresponding diffusion equation is shown. It is proved that convolution of two circular normal distributions is again a distribution of the same type. Some projections of the normal distribution are obtained. These projections coincide with a wrapped normal distribution on the unit circle and with the Perrin distribution on the two-dimensional sphere. In the general case, the normal distribution on SO(3) can be found numerically. Some algorithms for numerical computations are given. These investigations were motivated by the orientation distribution function reproduction problem described in the Appendix.

scholarly journals A Probability Distribution for a Time-varying Quantity

1966 ◽  
Vol 19 (1) ◽  
pp. 119-122 ◽  
Author(s):  
D. A. Lloyd
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Gaussian Distribution ◽  
Time Varying ◽  
Normal Distributions

In many practical situations in the field of navigation, it has been noticed that the probability distribution of measured errors has a shape which has considerable departures from that of the normal distribution. These departures are particularly noticeable in the ‘tails’ of the distribution of practical cases, which are often higher than those of the corresponding normal distributions (see, for example, ‘Is the gaussian distribution normal’, W/Cdr. E. W. Anderson. This Journal, 18, 65).

Heuristic Procedures for Simultaneous Estimation of Several Normal Means

1988 ◽  
Vol 2 (1) ◽  
pp. 95-113 ◽  
Author(s):  
Chi-Hyuck Jun
Keyword(s):  
Normal Distribution ◽  
Simultaneous Estimation ◽  
Shrinkage Estimators ◽  
Squared Error Loss ◽  
Limiting Behavior ◽  
Normal Distributions ◽  
Squared Error ◽  
Normal Means ◽  
Heuristic Procedures ◽  
Estimation Problems

Simultaneous estimation problems that deal with the estimation of several means for normal distributions are considered under the squared-error loss. Heuristic procedures are presented which can be applied to parameter estimation problems for a wide class of distributions specified only by their means and variances. Explicit results are obtained for the heuristic shrinkage estimators in the normal distribution case. Limiting behavior of the relative risk savings of these estimators is studied. The performances of the proposed estimators for normal distribution means are compared with other existing estimators by a computer simulation.

scholarly journals On Properties of the Bimodal Skew-Normal Distribution and an Application

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 703
Author(s):  
David Elal-Olivero ◽  
Juan F. Olivares-Pacheco ◽  
Osvaldo Venegas ◽  
Heleno Bolfarine ◽  
Héctor W. Gómez
Keyword(s):  
Maximum Likelihood ◽  
Normal Distribution ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Skew Normal Distribution ◽  
Data Set ◽  
Normal Distributions ◽  
Stochastic Representations ◽  
Skew Normal

The main object of this paper is to develop an alternative construction for the bimodal skew-normal distribution. The construction is based upon a study of the mixture of skew-normal distributions. We study some basic properties of this family, its stochastic representations and expressions for its moments. Parameters are estimated using the maximum likelihood estimation method. A simulation study is carried out to observe the performance of the maximum likelihood estimators. Finally, we compare the efficiency of the new distribution with other distributions in the literature using a real data set. The study shows that the proposed approach presents satisfactory results.

scholarly journals Prefeasibility Study of Photovoltaic Power Potential Based on a Skew-Normal Distribution

Energies ◽  
2020 ◽  
Vol 13 (3) ◽  
pp. 676
Author(s):  
Shin Young Kim ◽  
Benedikt Sapotta ◽  
Gilsoo Jang ◽  
Yong-Heack Kang ◽  
Hyun-Goo Kim
Keyword(s):  
Normal Distribution ◽  
Solar Irradiance ◽  
Information Criterion ◽  
Energy Performance ◽  
Skew Normal Distribution ◽  
Normal Distributions ◽  
Photovoltaic Power ◽  
Natural Energy ◽  
The Difference ◽  
Skew Normal

Solar energy does not always follow the normal distribution due to the characteristics of natural energy. The system advisor model (SAM), a well-known energy performance analysis program, analyzes exceedance probabilities by dividing solar irradiance into two cases, i.e., when normal distribution is followed, and when normal distribution is not followed. However, it does not provide a mathematical model for data distribution when not following the normal distribution. The present study applied the skew-normal distribution when solar irradiance does not follow the normal distribution, and calculated photovoltaic power potential to compare the result with those using the two existing methods. It determined which distribution was more appropriate between normal and skew-normal distributions using the Jarque–Bera test, and then the corrected Akaike information criterion (AICc). As a result, three places in Korea showed that the skew-normal distribution was more appropriate than the normal distribution during the summer and winter seasons. The AICc relative likelihood between two models was more than 0.3, which showed that the difference between the two models was not extremely high. However, considering that the proportion of uncertainty of solar irradiance in photovoltaic projects was 5% to 17%, more accurate models need to be chosen.

Simulating Complex Supply Chain Relationships Using Copulas

2019 ◽  
pp. 1263-1276
Author(s):  
Krishnamurty Muralidhar ◽  
Rathindra Sarathy
Keyword(s):  
Supply Chain ◽  
Supply Chain Management ◽  
Supply Chains ◽  
Normal Distribution ◽  
Real World ◽  
Marginal Distributions ◽  
Chain Management ◽  
Normal Distributions ◽  
Supply Chain Relationships ◽  
Complex Relationships

Simulation is often used as a tool to analyze and understand complex systems in supply chain management research. Supply chains involve complex relationships between different variables. Hence, it is necessary to simulate related non-normal distributions to simulate these systems. The simulation of related normal distributions is relatively easy and can be found in most simulation texts. However, when the marginal distributions under investigation do not have a normal distribution, it becomes very difficult to generate values from these related distributions. In this study, the authors illustrate a method based on copulas that allows for the generation of related distributions with arbitrary marginals. The procedure suggested in this study is simple and easy to implement. Using this procedure will enable researchers in supply chain management to more effectively simulate complex real-world scenarios resulting in better analysis and understanding of supply chains.

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