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Expectation and variance of bivariate skew normal distribution

Azzalini and Della Valle (Biometrika, 1996) have defined a bivariate skew Normal distribution, with density $$f_2(x,y;\theta)=2\phi_2(x,y;\omega)\Phi(\alpha_x x+\alpha_y y)$$ where $\Phi$ is the ...
Xi'an's user avatar
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6 votes

How is that possible that simple arithmetic mean works well even for strongly skewed distribution?

What is your definition about "good"? I assume you want to say the bias of sample mean is zero, it is consistent, if so, in 1947, Hsu and Robbins proved that the arithmetic mean converges ...
Tuobang Li's user avatar
3 votes

Mean Squared Error for point estimation

$$ \text{MSE}=(\text{bias})^2+\text{var}. $$ An unbiased estimator has $\text{bias}=0$. $$\begin{align} \text{MSE}&=0^2+\text{var}\\\implies \text{MSE}&=\text{var}.\end{align} $$
Dave's user avatar
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1 vote

Using Law of iterated expectations, I want to calculate mean of Y, E(Y)

If you want to perform these calculations with R, you can use the package rje to get the conditional distributions, and then the package ...
Stéphane Laurent's user avatar
1 vote
Accepted

Using Law of iterated expectations, I want to calculate mean of Y, E(Y)

Use the definition of conditional expectation of Y given X to verify your answers $$ E(Y|X = 0) = \sum_{y} y \cdot P(Y=y|X=0) = 0 \cdot \frac{f(0,0)}{f_{X}(0)} + 1 \cdot \frac{f(0,1)}{f_{X}(0)} + 2 \...
ADAM's user avatar
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