fundamental statistics learning note(16)

LR-test continued

Continued:
If LR test statistics $lambda$ has an unknown distribution, we use large-sample approximate LR test.
Reject $H_O$ if $-2loglambda geq (1-alpha)$ quantile of $chi_p^2$, where $p$ is difference in # free parameters(between $Omega_0 and Omega_0 cup Omega_1$)

Ex. $x_1,dots,x_n overset{text{iid}}sim N(mu_x,sigma_x^2), y_1,dots,y_n overset{text{iid}}sim N(mu_y,sigma_y^2)$, what is the value of p for the approximate LR test?
$$begin{matrix}
H_O:mu_x=mu_y=0, sigma_x^2=sigma_y^2=1text{ vs } H_A:otherwise && p = 4-0 \
H_O:mu_x=mu_y=0 text{ vs } H_A:otherwise && p = 4-2 \
H_O:mu_x=mu_y text{ vs } H_A:mu_xneqmu_y && p = 4-3
end{matrix}$$

Ex. $x_1,dots,x_n overset{text{iid}}sim Pois(theta)$. Derive approximate LR test for $H_O:theta=theta_0$ vs $H_A:theta neq theta_0$(when n is large)

$$begin{equation}begin{split}
L(theta|x_1,dots,x_n)& = prod limits_{i=1}^n e^{-theta}frac{theta^{x_i}}{x_i!} \
& = e^{-ntheta}frac{theta^{sum x_i}}{prod x_i!}
end{split}end{equation}$$
Then LR test $lambda$ is:
$$begin{equation}begin{split}
lambda&= frac{maxlimits_{theta=theta_0}L(theta|x_1,dots,x_n)}{underbrace{maxlimits_{theta>0}L(theta|x_1,dots,x_n)}_{text{MLE is }hat theta = bar x}} \
&=frac{e^{-ntheta_0}theta_0^{sum x_i}/prod x_i!}{e^{-nbar x}bar x^{sum x_i}/prod x_i!} \
& = underbrace{e^{-ntheta_0+nbar x}(frac{theta_0}{bar x})^{nbar x}}_{text{is actually a value given data}}
end{split}end{equation}$$
Though we can’t easily tell the distribution of $lambda$, so approximate $-2loglambda overset{text{approximate}}sim chi_1^2$.
Then approximate LR test: reject $H_O$ if $-2loglambda geq (1-alpha)text{ quantile of }chi_1^2$

p-values

Defnition: Let $W$ be a test statistics for $H_O: theta in Omega_0$.
Data: $ widetilde{X} = X_1,dots,X_n$ and observed value $tilde{x} = x_1,dots,x_n$.
If $W$ is larger when $H_O$ is false: p-value=$maxlimits_{thetainOmega_0}P(W(widetilde{X})geq W(tilde{x})|theta)$
If $W$ is smaller when $H_O$ is false: p-value=$maxlimits_{thetainOmega_0}P(W(widetilde{X})leq W(tilde{x})|theta)$
Note: $widetilde{X}$ is random variable and $tilde{x}$ is observed value.

E.X. $x_1,dots,x_n overset{text{iid}}sim Bern(p). H_O: pleq0.4text{ vs }H_A: pgeq 0.4$.
$W=sum limits_{i=1}^4 X_i$ is test statistics.
Suppose data: $x_1=x_2=x_3=1, x_4=0$, what is the p-value?
$E(W)=4p$, $W$ is larger when $H_O$ is false.
$$Rightarrow Wsim Bin(4,p)$$
$$begin{equation}begin{split}
text{p-value}&=maxlimits_{pleq 0.4}[p(W=3)+P(W=4)] \
&=maxlimits_{pleq 0.4}[{4 choose 3}p^3(1-p)^1+{4choose4}p^4(1-p)^0] \
&=4(0.4)^3(0.6)+0.4^4=0.1792
end{split}end{equation}$$
Note: when $p=0.4$, this makes data most likely in the parameter space ($pleq 0.4$).