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I am trying to solve this complicated Generalised Likelihood Ratio Test problem for 2 samples as part of my independent study course. However, the book that I am using does not talk about 2 samples at all; all the examples provided are 1 sample questions.

The question:

Assume that $X_1, ...,X_m$ is a random sample from $theta_1e^{-theta_1x}I_{(0,infty)}(x)$ and $Y_1, ...,Y_n$ is a random sample from $theta_2e^{-theta_2y}I_{(0,infty)}(y)$. If we assume the samples are independent. What is the GLR for testing
$H_0:theta_1=theta_2$ vs. $H_0:theta_1netheta_2$?

So far I just got to $$Lambda=frac{L(theta,theta)}{L(theta_1,theta_2)}=frac{theta^{m+n}exp[-thetasum x_i+sum y_i]}{theta_1^mexp[-theta_1sum x_i]theta_2^n exp[-theta_2sum y_i]}$$

However, I don't know what I should do for the next step. They say it should be an F test but I just can't see it.

I appreciate your time and help!

The likelihood function given the sample $mathbf x=(x_1,ldots,x_m,y_1,ldots,y_n)$ is

$$L(theta_1,theta_2)=theta_1^mtheta_2^nexpleft(-theta_1sum_{i=1}^m x_i-theta_2sum_{i=1}^n y_iright)mathbf1_{x_1,ldots,x_m,y_1,ldots,y_n>0}quad,,theta_1,theta_2>0$$

Unrestricted MLE of $(theta_1,theta_2)$ is $$(hattheta_1,hattheta_2)=left(frac{1}{bar x},frac{1}{bar y}right)$$

Restricted MLE of $(theta_1,theta_2)$ when $theta_1=theta_2=theta$ (say) is

$$hattheta=frac{m+n}{mbar x+nbar y}$$

So the LR test statistic for testing $H_0$ is

$$Lambda(mathbf x)=frac{sup_{theta_1=theta_2}L(theta_1,theta_2)}{sup_{theta_1,theta_2}L(theta_1,theta_2)}=frac{L(hattheta,hattheta)}{L(hattheta_1,hattheta_2)}$$

Substituting the values of $hattheta_1,hattheta_2,hattheta$, the terms in the exponent of $e$ vanish, and I get

$$Lambda(mathbf x)=bar x^mbar y^nleft(frac{m+n}{mbar x+nbar y}right)^{m+n}$$

The above can be rewritten to get a 'nice' form:

begin{align}
Lambda(mathbf x)&=underbrace{text{constant}}_{>0}left(frac{mbar x}{mbar x+nbar y}right)^mleft(frac{nbar y}{mbar x+nbar y}right)^n
\&=text{constant}cdot,t^m(1-t)^nqquad,text{ where }t=frac{mbar x}{mbar x+nbar y}
\&=g(t)quad,,text{say}
end{align}

Now you have to study the nature of the function $g$, keeping in mind that we reject $H_0$ when $Lambda(mathbf x)<c$ for some $c$ subject to a level restriction.

As for why this should be an $F$ test, note that

$$X_istackrel{text{ i.i.d }}simtext{Exp with mean }1/theta_1implies 2theta_1 X_istackrel{text{ i.i.d }}simtext{Exp with mean }2equiv chi^2_2$$

So summing over independent chi-square variables, $$2mtheta_1overline Xsimchi^2_{2m}$$

Similarly, $$2ntheta_2overline Ysimchi^2_{2n}$$

And since both samples are independent, our test statistic is

$$frac{theta_1overline X}{theta_2overline Y}sim F_{2m,2n}$$