Regression estimation for bivariate normal distributions

1973 ◽  
Vol 25 (1) ◽  
pp. 335-344 ◽  
Author(s):  
Chien-Pai Han
Keyword(s):  
Regression Estimation ◽  
Normal Distributions ◽  
Bivariate Normal

scholarly journals Bivariate Distributions Underlying Responses to Ordinal Variables

Psych ◽  
2021 ◽  
Vol 3 (4) ◽  
pp. 562-578
Author(s):  
Laura Kolbe ◽  
Frans Oort ◽  
Suzanne Jak
Keyword(s):  
Correlation Coefficient ◽  
Latent Variables ◽  
Ordinal Data ◽  
Bivariate Normal Distribution ◽  
Polychoric Correlation ◽  
Bivariate Distributions ◽  
Normal Distributions ◽  
Ordinal Variables ◽  
Bivariate Normal ◽  
Bivariate Normality

The association between two ordinal variables can be expressed with a polychoric correlation coefficient. This coefficient is conventionally based on the assumption that responses to ordinal variables are generated by two underlying continuous latent variables with a bivariate normal distribution. When the underlying bivariate normality assumption is violated, the estimated polychoric correlation coefficient may be biased. In such a case, we may consider other distributions. In this paper, we aimed to provide an illustration of fitting various bivariate distributions to empirical ordinal data and examining how estimates of the polychoric correlation may vary under different distributional assumptions. Results suggested that the bivariate normal and skew-normal distributions rarely hold in the empirical datasets. In contrast, mixtures of bivariate normal distributions were often not rejected.

scholarly journals Dependence between path-length and size in random digital trees

2017 ◽  
Vol 54 (4) ◽  
pp. 1125-1143
Author(s):  
Michael Fuchs ◽  
Hsien-Kuei Hwang
Keyword(s):  
Path Length ◽  
Normal Limit ◽  
Limit Laws ◽  
Normal Distributions ◽  
Digital Trees ◽  
Bivariate Normal ◽  
Asymmetric Case ◽  
Periodic Fluctuations ◽  
Type Of Behavior ◽  
Small Periodic

AbstractWe study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

scholarly journals Goodman and Kruskal’s Gamma Coefficient for Ordinalized Bivariate Normal Distributions

Psychometrika ◽  
2020 ◽  
Author(s):  
Alessandro Barbiero ◽  
Asmerilda Hitaj
Keyword(s):  
Normal Distribution ◽  
Rank Correlation ◽  
Real Data ◽  
Random Variable ◽  
Bivariate Normal Distribution ◽  
Normal Distributions ◽  
Bivariate Normal ◽  
Ordinal Variable ◽  
Before And After ◽  
Random Components

Abstract We consider a bivariate normal distribution with linear correlation $$\rho $$ ρ whose random components are discretized according to two assigned sets of thresholds. On the resulting bivariate ordinal random variable, one can compute Goodman and Kruskal’s gamma coefficient, $$\gamma $$ γ , which is a common measure of ordinal association. Given the known analytical monotonic relationship between Pearson’s $$\rho $$ ρ and Kendall’s rank correlation $$\tau $$ τ for the bivariate normal distribution, and since in the continuous case, Kendall’s $$\tau $$ τ coincides with Goodman and Kruskal’s $$\gamma $$ γ , the change of this association measure before and after discretization is worth studying. We consider several experimental settings obtained by varying the two sets of thresholds, or, equivalently, the marginal distributions of the final ordinal variables. This study, confirming previous findings, shows how the gamma coefficient is always larger in absolute value than Kendall’s rank correlation; this discrepancy lessens when the number of categories increases or, given the same number of categories, when using equally probable categories. Based on these results, a proposal is suggested to build a bivariate ordinal variable with assigned margins and Goodman and Kruskal’s $$\gamma $$ γ by ordinalizing a bivariate normal distribution. Illustrative examples employing artificial and real data are provided.

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