## #StackBounty: #normal-distribution #multivariate-normal-distribution #linear-algebra #complex-numbers Density of a degenerate complex n…

### Bounty: 50

I’m looking for the properties of a degenerate complex normal distribution. Wikipedia has a section on degeneracy in the real multivariate normal case here, stating that the density (in a subspace of $$mathbb R^k$$ where the distribution is supported) looks like:

$$f(mathbf{x}) = (text{det}^{*}(2piSigma))^{-1/2} e^{-1/2 (x-mu)^TSigma^+(x-mu)}$$

… where $$text{det}^*$$ is a pseudo-determinant (product of non-zero eigenvalues) and $$Sigma^+$$ is the pseudo-inverse.

I have a couple of questions relating to whether or not such a formulation exists for the complex case.

Question 1: If $$Z sim mathcal{C}N(mu, Gamma, C)$$ where the support is $$mathbb C^2$$ (for example), then multiplication of $$Z$$ with a matrix $$Ain mathbb{C}_{3,2}$$ would yeild: $$AZ sim mathcal{C}N(Amu, AGamma A^H, ACA^T)$$ where $$AZ in mathbb{C}^3$$ and could, in general, be degenerate.

Assuming that $$A$$ is unknown, how would one find the subspace of $$mathbb{C}^{3}$$ where the density would have a support, and in this subspace, what would the density be? Would be be similar to the real case wherein the determinants and inverses are replaced by their pseudo-counterparts?

Question 2: On the wikipedia article, it is shown that the density of a complex normal can be written as:

$$f(z) = tfrac{sqrt{detleft(overline{P^{-1}}-R^{ast} P^{-1}Rright)det(P^{-1})}}{pi^n}, e^{ -(z-mu)^astoverline{P^{-1}}(z-mu) + operatorname{Re}left((z-mu)^intercal R^intercaloverline{P^{-1}}(z-mu)right)}$$

…where $$P, R$$ are functions of $$Gamma, C$$. I assume that this density is only valid if, for example, $$Gamma$$ is invertible. In a specific problem setting, I’ve encountered a likelihood that can be written in this form, but $$detleft(overline{P^{-1}}-R^{ast} P^{-1}Rright) = 0$$. Does this imply that the density is degenerate? Or is it the case that my likelihood isn’t a likelihood at all?

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## #StackBounty: #hypothesis-testing #self-study #normal-distribution #t-test #likelihood-ratio Likelihood Ratio Test Equivalent with \$t\$ …

### Bounty: 50

$$newcommand{szdp}{!left(#1right)} newcommand{szdb}{!left[#1right]}$$
Problem Statement: Suppose that independent random samples of sizes $$n_1$$ and $$n_2$$
are to be selected from normal populations with means $$mu_1$$ and $$mu_2,$$
respectively, and common variance $$sigma^2.$$ For testing $$H_0:mu_1=mu_2$$
versus $$H_a:mu_1-mu_2>0$$ ($$sigma^2$$ unknown), show that the likelihood
ratio test reduces to the two-sample $$t$$ test presented in Section 10.8.

Note: This is Exercise 10.94 from Mathematical Statistics with Applications, 5th. Ed., by Wackerly, Mendenhall, and Scheaffer.

My Work So Far: We have the likelihood as
$$L(mu_1, mu_2,sigma^2)= szdp{frac{1}{sqrt{2pi}}}^{!!(n_1+n_2)} szdp{frac{1}{sigma^2}}^{!!(n_1+n_2)/2} expszdb{-frac{1}{2sigma^2}szdp{sum_{i=1}^{n_1}(x_i-mu_1)^2 +sum_{i=1}^{n_2}(y_i-mu_2)^2}}.$$
To compute $$Lbig(hatOmega_0big),$$ we need to find the MLE for $$sigma^2:$$
begin{align} hatsigma^2&=frac{1}{n_1+n_2}szdp{sum_{i=1}^{n_1}(x_i-mu_1)^2 +sum_{i=1}^{n_2}(y_i-mu_2)^2}. end{align}
This is the MLE for $$sigma^2$$ regardless of what $$mu_1$$ and $$mu_2$$ are.
Thus, under $$H_0,$$ we have that
$$hatsigma_0^2=frac{1}{n_1+n_2}szdp{sum_{i=1}^{n_1}(x_i-mu_0)^2 +sum_{i=1}^{n_2}(y_i-mu_0)^2},$$
and the unrestricted case is
$$hatsigma^2=frac{1}{n_1+n_2}szdp{sum_{i=1}^{n_1}(x_i-overline{x})^2 +sum_{i=1}^{n_2}(y_i-overline{y})^2}.$$
Under $$H_0,;mu_1=mu_2=mu_0,$$ so that
begin{align} Lbig(hatOmega_0big) &=szdp{frac{1}{sqrt{2pi}}}^{!!(n_1+n_2)} szdp{frac{1}{hatsigma_0^2}}^{!!(n_1+n_2)/2} expszdb{-frac{n_1+n_2}{2}}\ Lbig(hatOmegabig) &=szdp{frac{1}{sqrt{2pi}}}^{!!(n_1+n_2)} szdp{frac{1}{hatsigma^2}}^{!!(n_1+n_2)/2} expszdb{-frac{n_1+n_2}{2}}, end{align}
and the likelihood ratio is given by
begin{align} lambda &=frac{Lbig(hatOmega_0big)}{Lbig(hatOmegabig)}\ &=szdp{frac{hatsigma^2}{hatsigma_0^2}}^{!!(n_1+n_2)/2}\ &=szdp{frac{displaystylesum_{i=1}^{n_1}(x_i-overline{x})^2 +sum_{i=1}^{n_2}(y_i-overline{y})^2} {displaystylesum_{i=1}^{n_1}(x_i-mu_0)^2 +sum_{i=1}^{n_2}(y_i-mu_0)^2}}^{!!(n_1+n_2)/2}. end{align}
It follows that the rejection region, $$lambdale k,$$ is equivalent to
begin{align} frac{displaystylesum_{i=1}^{n_1}(x_i-overline{x})^2 +sum_{i=1}^{n_2}(y_i-overline{y})^2} {displaystylesum_{i=1}^{n_1}(x_i-mu_0)^2 +sum_{i=1}^{n_2}(y_i-mu_0)^2}&frac{1}{k’}-1=k”\ frac{n_1(overline{x}-mu_0)^2+n_2(overline{y}-mu_0)^2} {displaystyle(n_1-1)S_1^2+(n_2-1)S_2^2}&>k”\ frac{n_1(overline{x}-mu_0)^2+n_2(overline{y}-mu_0)^2} {displaystyledfrac{(n_1-1)S_1^2+(n_2-1)S_2^2} {n_1+n_2-2}}&>k”(n_1+n_2-2)\ frac{n_1(overline{x}-mu_0)^2+n_2(overline{y}-mu_0)^2} {S_p^2}&>k”(n_1+n_2-2). end{align}
Here
$$S_p^2=frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}.$$

My Question: The goal is to get this expression somehow to look like
$$t=frac{overline{x}-overline{y}}{S_psqrt{1/n_1+1/n_2}}>t_{alpha}.$$
But I don’t see how I can convert my expression, with the same sign for $$overline{x}$$ and $$overline{y},$$ to the desired formula with its opposite signs. What am I missing?

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## #StackBounty: #normal-distribution #circular-statistics Expected ratio of x'Ax and x'AAx on a unit sphere?

### Bounty: 50

Suppose $$A$$ is symmetric positive definite matrix. Is there a nice expression for the first moment of the following quantity?

$$frac{x^TAx}{x^TA^2 x}$$

Where $$x$$ is distributed as $$text{Normal}(0,I_n)$$. This is the ratio of two quadratic forms evaluated on the surface of the sphere. When $$A$$ has eigenvalues $$langle 1, frac{1}{2}rangle$$, this expectation is equal to $$frac{4}{3}$$

Get this bounty!!!

## #StackBounty: #normal-distribution #circular-statistics Expected ratio of x'Ax and x'AAx on a unit sphere?

### Bounty: 50

Suppose $$A$$ is symmetric positive definite matrix. Is there a nice expression for the first moment of the following quantity?

$$frac{x^TAx}{x^TA^2 x}$$

Where $$x$$ is distributed as $$text{Normal}(0,I_n)$$. This is the ratio of two quadratic forms evaluated on the surface of the sphere. When $$A$$ has eigenvalues $$langle 1, frac{1}{2}rangle$$, this expectation is equal to $$frac{4}{3}$$

Get this bounty!!!

## #StackBounty: #normal-distribution #circular-statistics Expected ratio of x'Ax and x'AAx on a unit sphere?

### Bounty: 50

Suppose $$A$$ is symmetric positive definite matrix. Is there a nice expression for the first moment of the following quantity?

$$frac{x^TAx}{x^TA^2 x}$$

Where $$x$$ is distributed as $$text{Normal}(0,I_n)$$. This is the ratio of two quadratic forms evaluated on the surface of the sphere. When $$A$$ has eigenvalues $$langle 1, frac{1}{2}rangle$$, this expectation is equal to $$frac{4}{3}$$

Get this bounty!!!

## #StackBounty: #normal-distribution #circular-statistics Expected ratio of x'Ax and x'AAx on a unit sphere?

### Bounty: 50

Suppose $$A$$ is symmetric positive definite matrix. Is there a nice expression for the first moment of the following quantity?

$$frac{x^TAx}{x^TA^2 x}$$

Where $$x$$ is distributed as $$text{Normal}(0,I_n)$$. This is the ratio of two quadratic forms evaluated on the surface of the sphere. When $$A$$ has eigenvalues $$langle 1, frac{1}{2}rangle$$, this expectation is equal to $$frac{4}{3}$$

Get this bounty!!!

## #StackBounty: #normal-distribution #circular-statistics Expected ratio of x'Ax and x'AAx on a unit sphere?

### Bounty: 50

Suppose $$A$$ is symmetric positive definite matrix. Is there a nice expression for the first moment of the following quantity?

$$frac{x^TAx}{x^TA^2 x}$$

Where $$x$$ is distributed as $$text{Normal}(0,I_n)$$. This is the ratio of two quadratic forms evaluated on the surface of the sphere. When $$A$$ has eigenvalues $$langle 1, frac{1}{2}rangle$$, this expectation is equal to $$frac{4}{3}$$

Get this bounty!!!

## #StackBounty: #normal-distribution #circular-statistics Expected ratio of x'Ax and x'AAx on a unit sphere?

### Bounty: 50

Suppose $$A$$ is symmetric positive definite matrix. Is there a nice expression for the first moment of the following quantity?

$$frac{x^TAx}{x^TA^2 x}$$

Where $$x$$ is distributed as $$text{Normal}(0,I_n)$$. This is the ratio of two quadratic forms evaluated on the surface of the sphere. When $$A$$ has eigenvalues $$langle 1, frac{1}{2}rangle$$, this expectation is equal to $$frac{4}{3}$$

Get this bounty!!!

## #StackBounty: #normal-distribution #circular-statistics Expected ratio of x'Ax and x'AAx on a unit sphere?

### Bounty: 50

Suppose $$A$$ is symmetric positive definite matrix. Is there a nice expression for the first moment of the following quantity?

$$frac{x^TAx}{x^TA^2 x}$$

Where $$x$$ is distributed as $$text{Normal}(0,I_n)$$. This is the ratio of two quadratic forms evaluated on the surface of the sphere. When $$A$$ has eigenvalues $$langle 1, frac{1}{2}rangle$$, this expectation is equal to $$frac{4}{3}$$

Get this bounty!!!

## #StackBounty: #normal-distribution #circular-statistics Expected ratio of x'Ax and x'AAx on a unit sphere?

### Bounty: 50

Suppose $$A$$ is symmetric positive definite matrix. Is there a nice expression for the first moment of the following quantity?

$$frac{x^TAx}{x^TA^2 x}$$

Where $$x$$ is distributed as $$text{Normal}(0,I_n)$$. This is the ratio of two quadratic forms evaluated on the surface of the sphere. When $$A$$ has eigenvalues $$langle 1, frac{1}{2}rangle$$, this expectation is equal to $$frac{4}{3}$$

Get this bounty!!!