distribution of the ratio of jointly normal variables

We derive the probability density of the ratio of components of the bivariate normal distribution with arbitrary parameters. The density is a product of two factors, the first is a Cauchy density, the second a very complicated function. We show that the distribution under study does not possess an expected value or other moments of higher order. Our particular interest is focused on the shape of the density. We introduce a shape parameter and show that according to its sign the densities are classified into three main groups. As an example, we derive the distribution of the ratio Z = − Bm−1 /(mBm ) for a polynomial regression of order m. For m=1, Z is the estimator for the zero of a linear regression, for m = 2 , an estimator for the abscissa of the extreme of a quadratic regression, and for m = 3 , an estimator for the abscissa of the inflection point of a cubic regression.

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A note on the generalization of bivariate normal quadrant probabilities to bivariate elliptical distributions

10.31234/osf.io/j4vm3 ◽

2018 ◽

Author(s):

Oscar Lorenzo Olvera Astivia

Keyword(s):

Normal Distribution ◽

Bivariate Normal Distribution ◽

Elliptical Distributions ◽

Bivariate Normal ◽

Geometric Argument

I present a geometric argument to show that the quadrant probability for the bivariate normal distribution can be generalized to the case of all elliptical distributions.

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PSIII-5 Comparison of Bivariate Machine Learning and Linear Model for Genomic Prediction with Different Heritability, QTL and SNP Panel Scenarios

Journal of Animal Science ◽

10.1093/jas/skaa278.424 ◽

2020 ◽

Vol 98 (Supplement_4) ◽

pp. 230-231

Author(s):

Sunday O Peters ◽

Mahmut Sinecan ◽

Kadir Kizilkaya ◽

Milt Thomas

Keyword(s):

Neural Network ◽

Artificial Neural Network ◽

Normal Distribution ◽

Genomic Prediction ◽

Network Models ◽

Bivariate Normal Distribution ◽

Snp Markers ◽

Neural Network Models ◽

Bivariate Normal ◽

Artificial Neural

Abstract This simulation study used actual SNP genotypes on the first chromosome of Brangus beef cattle to simulate 0.50 genetically correlated two traits with heritabilities of 0.25 and 0.50 determined either by 50, 100, 250 or 500 QTL and then aimed to compare the accuracies of genomic prediction from bivariate linear and artificial neural network with 1 to 10 neurons models based on G genomic relationship matrix. QTL effects of 50, 100, 250 and 500 SNPs from the 3361 SNPs of 719 animals were sampled from a bivariate normal distribution. In each QTL scenario, the breeding values (Σgijβj) of animal i for two traits were generated by using genotype (gij) of animal i at QTL j and the effects (βj) of QTL j from a bivariate normal distribution. Phenotypic values of animal i for traits were generated by adding residuals from a bivariate normal distribution to the breeding values of animal i. Genomic predictions for traits were carried out by bivariate Feed Forward MultiLayer Perceptron ANN-1–10 neurons and linear (GBLUP) models. Three sets of SNP panels were used for genomic prediction: only QTL genotypes (Panel1), all SNP markers, including the QTL (Panel2), and all SNP markers, excluding the QTL (Panel3). Correlations from 10-fold cross validation for traits were used to assess predictive ability of bivariate linear (GBLUP) and artificial neural network models based on 4 QTL scenarios with 3 Panels of SNP panels. Table 1 shows that the trait with high heritability (0.50) resulted in higher correlation than the trait with low heritability (0.25) in bivariate linear (GBLUP) and artificial neural network models. However, bivariate linear (GBLUP) model produced higher correlation than bivariate neural network. Panel1 performed the best correlations for all QTL scenarios, then Panel2 including QTL and SNP markers resulted in better prediction than Panel3.

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