Likelihood Ratio test

I have a pdf where I would like to test for the following hypothesis: H0: θ1=θ1,0; Ha: θ1>θ1,0, and f(x) =(1/θ1) exp((-x-θ2)/θ1), if x>θ2, and 0 otherwise, with θ2 unknown.

I constructed the likelihood function: L(x|θ1,θ2) = (1/θ1)^n*exp(∑(i=1 to n)((-xi-θ2)/θ1) and the log likelihood: -n*log(θ1)+∑((-xi-θ2)/θ1) under H0, θ1 = θ1, to find θ2, I set the derivative of l(x|θ1,θ2) = 0 with respect to θ2, but since the derivative doesn't yield to a solution, I realized that the maximizer for the likelihood function is the x(1), which is the minimum of xi, so θhat2 = x(1). Under Ha, I set the derivative of l(x|θ1,θ2) = 0 with respect to θ1, and found that θhat1 = (∑(xi+θ2))/n. But when I construct the likelihood ratio, things get so complicated and I can't follow. Can anyone suggest a way to tackle this part of the problem?


Active Member
Sure, but when you run your LR on real data you will get actual numbers, numbers that you will be able to put into optimization routines... Or is this a homework problem from a statistics course?


TS Contributor
There should be have some simplification in the likelihood ratio. Please show the work.

Also, try to use math tag to present the idea. In the likelihood function, do you mean

\( \exp\left\{-\frac{x-\theta_2} {\theta_1}\right\} \)


\( \exp\left\{-\frac{x+\theta_2} {\theta_1}\right\} \)

You seeming doing the latter case, but the former case should be the more common location-scale form.
Sorry I don't know how to use the math tag (couldn't find it in the reply box), but the correct form is the first one:
exp(-(x- θ2)/θ1), where x>θ2
As I mentioned, I have calculated the estimation for θ1 to be θhat1 = (∑(xi-θ2))/n, and θhat2 = x(1), where x(1) is the minimum of xi. When I plug these estimations into the likelihood ratio, the expression is huge so I suspect that I am doing something wrong.