fundatmental statistics learning note(end)

Some Exercises

Let $x_1,dots,x_n overset{text{iid}}sim N(mu_x,1)$, $y_1,dots,y_n overset{text{iid}}sim N(mu_y,1)$, and both are independent.
Derive LR test for $H_O:mu_x=mu_y text{ vs } H_A:mu_xneqmu_y$. <

$$begin{equation}begin{split}
L(mu_x,mu_y|x_1,dots,x_n,y_1,dots,y_n) & = f(x_1,dots,x_n,y_1,dots,y_n|mu_x,mu_y)
& overset{text{indep}}=prod_{i=1}^nf(x_i|mu)prod_{j=1}^m f(y_j|mu_y)
& = prod_{i=1}^{n}frac{1}{sqrt{2pi}}e^{-frac{(x_i-mu_x)^2}{2}}cdot prod_{i=1}^{n}frac{1}{sqrt{2pi}}e^{-frac{(y_i-mu_y)^2}{2}}
end{split}end{equation}$$
$Rightarrow (frac{1}{2pi})^{frac{m+n}{2}}e^{-frac{sum(x_i-mu_x)^2}{2}}e^{-frac{sum(y_i - mu_y)^2}{2}}$

$x_1,dots,x_n,y_1,dots,y_n overset{text{iid}}sim N(bar mu, 1)$. all $x,y$ are iid with same unknown mean, so maximized by $hat mu = frac{sum x_i + sum y_i}{n+m}$ (Overall average)

$$begin{equation}begin{split}
LR & = frac{maxlimits_{mu_x=mu_y}L(mu_x,mu_y|mathbf{x},mathbf{y})}{maxlimits_{mu_x,mu_y}L(mu_x,mu_y|mathbf{x},mathbf{y})}
& = frac{(frac{1}{2pi})^{frac{m+n}{2}}e^{-frac{sum(x_i-hat x)^2}{2}}e^{-frac{sum(y_i - hat y)^2}{2}}}{(frac{1}{2pi})^{frac{m+n}{2}}e^{-frac{sum(x_i-bar x)^2}{2}}e^{-frac{sum(y_i - bar y)^2}{2}}}
end{split}end{equation}$$
where $hat mu_x = bar x, hat mu_y = bar y$.

To get approx LR test, $-2logLR overset{text{approx}}sim chi_{2-1}^2$, reject $H_O$ if $-2logLR > (1-alpha)$ quantile of $chi_1^2$

Let $x$ has PDF $f(x|theta)=frac{2x}{theta^2}, 10$
Set prior $pi(theta) propto frac{1}{theta^2}$, derive posterior PDF pf $theta|x$
$pi(theta|x)propto underbrace{frac{2x}{theta^2}I(theta>x)}_{text{likelihood}}cdot underbrace{frac{1}{theta^2}}_{text{prior}} = frac{2x}{theta^2}I(theta>x)$

$Rightarrow int_x^{infty}frac{2x}{theta^2}dtheta = -frac{2x}{theta^2}|_x^infty = frac{2}{3x^2}$
Hence, the valid PDF is $pi(theta|x) = frac{3x^2}{2}cdot frac{2x}{theta^2}I(theta>x)overset{text{set}}=1$