*Bounty: 50*

I recently encountered the three MLE-based tests **Wald test, Likelihood ratio test** and **Lagrange Multiplier test**. Although it seemed at first like the usual hypothesis testing I already know from statistics, I got in some trouble with regard to the actual application. In an attempt to fully understand all three I got my hands on some practice problems. Since all tests are built on MLE, asymptotically equivalent and the points I don’t get are similar for all three, I will just ask my question to all jointly here. Hence the post will be longer, apologies. If I should pose them separately, I will, of course, do so on request.

First, the Wald test. Take for example 100 realizations of a normally distributed random variable with mean $mu = 0.36$ and $sigma^2 = 4$. Define $c_1(mu) = mu$. How would you conduct a Wald test for $H_0: c(mu) = c_1(mu) – 0.8 = 0$ at the 5% significance level practically? Also, what I don’t understand is why one usually defines another function $c_1(mu)$?

What I understood here, is that one tests the restriction $$c(mu) = 0$$ and the closer the value of the test statistic as well as the value of $c(mu)$ is to zero the more valid the restriction. Since it is normally distributed one can set $mu$ as the MLE estimate for the mean. But how does it go from here exactly? I calculate $$ c(0.36) = 0.36 – 0.8 = -0.44$$

and, according to my materials,

$$W = -0.44^{-1}(-0.44)$$

But what is $Var(-0.44)$? And what is the underlying distribution from which I get the p-value?

Second, the likelihood ratio test. Take again a sample of 100 observations. But this time from the poisson distribution. The sample mean here is $mu = 1.7$. Therefore, the MLE estimate of $lambda$ is also $hat{lambda} = 1.7$. This time consider $$c(lambda) = lambda^2 – 3lambda + 2$$ How to test for $c(lambda) = 0$ at the 5% level? Here, I understand even less how to get along with so little information since I thought that one would need to evaluate the log-likelihood function and then decide based on the difference between the log-likelihood value of the restricted and unrestricted MLE estimate? And again, which distribution does the statistic (and thus the p-value) follow?

Finally, the Lagrange Multiplier test. I thought here as well that I would need the log-likelihood function since I have to insert the restricted estimate in its derivative, don’t I? Take the same distribution as before but with the function $$c(lambda) = frac{1}{lambda^2} – 0.1$$ What is the restricted MLE estimate that I insert in the log-likelihood derivative? Is it $frac{1}{1.7^2} – 0.1$? How do I go about it without having the actual sample and the log-likelihood function at my disposal?

Get this bounty!!!