If we have a hypothesis that is composite, i.e. $H: \theta\in \Theta_\text{(a set)}$ (instead of a hypothesis that is simple, i.e. $H: \theta=\theta^*_\text{(a number)}$), then we have likelihood $L(H)$ of this hypothesis defined as supremum of $\{f(\mathbf{x}|\theta), \theta\in\Theta\}$, i.e. $L(H)=\sup_\Theta f(\mathbf{x}|\theta)=\sup_\Theta (\prod_if(x_i|\theta))$.
My question is $L(H)$ is only one of values of $f(\mathbf{x}|\theta)$ for all allowed values of $\theta$, (though the most notable one), and I think it can't well represent the likelihood for all $\theta$, so why we use such 'unrepresentative' definitoin, is it really reasonable?
A related issue is that if we define likelihood of hypothesis this way and if we use the ratio of $L(H_0)$ and $L(H_a)$ ($H_a$ is the alternative hypothesis) to judge which hypothesis is more possible, then the corresponding $\Theta_0$ and $\Theta_a$ can have very different size. And so, for example, $L(H_0)$ is o number picked up from a large set of $\{f(\mathbf{x}|\theta), \theta\in\Theta_0\}$, and $L(H_a)$ is o number picked up from a small set of $\{f(\mathbf{x}|\theta), \theta\in\Theta_a\}$, and it seems to me that this big asymmetry doesn't make $L(H_0)/L(H_a)$ a good indicator of comparison of likelihood of the two hypotheses.
Why don't we use something like $\frac{\int_\Theta f(\mathbf{x}|\theta)d\theta}{|\Theta|}$ (where $|\Theta|$ is the size of $|\Theta|$) to define likelihood $L(H)$ of hypothesis?
Updated:
I think the LRT should be understood this way, first we calculate MLE of $\theta$, and it will be in $\Theta_0$ (or $\Theta_a$), the corresponding likelihood is $\sup_{\Theta_0} f(\mathbf{x}|\theta)$, but this is not a good indicator that we should accept null/alternative hypothesis. So we find a way to measure how much MLE in $\Theta_0$ and not in the other set. The way we do so is to calculate the max likelihood for $\theta \in \Theta_a$), and compare it with the max likelihood for $\theta \in \Theta_0$ (actually it’s for all $\theta$).
This makes sense. LRT actually measures how much the division $\theta*$ between $\Theta_0$ and $\Theta_a$ (This value is not test static, but possibly $z_{\alpha/2}$ or alike) is far from MLE. And it uses ratio of likelihood at (instead of distance between) MLE and $\theta*$ to measure it. And so it makes sense that we ignore all but the two points that give max likelihood for null and alternative hypothesis. I guess we may also use integral of distribution curve of $(-\infty, \theta*)$ and $(\theta*, \infty)$. (One of which contains MLE.) The sizes of $\Theta_0$ and $\Theta_a$ don’t matter since we just need to consider two points, MLE and $\theta*$. Also, both sizes can be infinity, and even if one has larger size and takes ‘advantage’ because of that, that’s an indicator that that one hypothesis is more probably correct, in other words, here we needn’t ‘fair game’, since what we measure is [essentially how one hypothesis is more at advantages than the other, or] a distance, and the difference in sizes actually reflects the distance.
But the question is 1. that I’m not sure how one sets division between $\Theta_0$ and $\Theta_a$, in the definition it is predetermined, which I feel is too arbitrary; for example we can set it to be MLE and in that case we will always get the conclusion that the two hypotheses are equally probably correct, which is problematic. 2. Why we use likelihood, not distance nor integral to measure how far MLE is from the division $\theta*$.
My thought about question 1:
$\Theta_0, \Theta_a$ are predetermined, since the hypotheses are usually, for example, whether Distribution 1 (parameter unknown) equals Distribution 2 (parameter known) or not, this is equal to saying $\theta=\theta_0$ or not, and so naturally $\Theta_0=\{\theta\}, \Theta_a= \mathbb{R}-\{\theta\} $ (the two don’t have to complement each other.) The point is that since the hypotheses specify some aspects of the distribution, so they limit the range of the values of the parameter.
Another point to make is that we don’t make judgement about accepting hypotheses or not by choosing (the division of) $\Theta_0, \Theta_a$, we do so by choosing a range of sample $\mathbf{x}$ where we say a hypothesis is accepted, that is, though $\Theta_0, \Theta_a$ are fixed, LRT is still function of the sample (random variables), and we set a range of LRT (like one where LRC is less than a chosen constant c) where we accept a hypothesis.
In other words, first, $\theta*$ is fixed, and so is one of the nominator and the denominator of LRT that corresponds to the range of $\theta$ where where MLE is not. Second, MLE is calculated and it shifts with the sample $\mathbf{x}$. Third, we set c, the range of LRT, or the range of $\mathbf{x}$ where we accept a hypothesis, this is roughly (not exactly; since what we calculate is not MLE but likelihood at MLE and $\theta*$) that we set a range of MLE where we accept a hypothesis. Overall, the range of LRT, and MLE and $\mathbf{x}$ are somehow equivalent, since they are all random variables and functions of the third; and the process is not that we have MLE and then set $\theta*$. In a word, from the discussion below, we have first $\hat \theta_0$ and then $\hat \theta$, a random variable, a function of sample, we can write it as $\hat \theta(\mathbf{x})$ . With $c$ we roughly set an allowable distance $|\hat \theta_0-\hat \theta (\mathbf{x}) |$, and, when MLE is not in $\Theta_0$, exactly $\theta*-\mathrm{MLE}$.
(PS: things can be simplified when we consider simple $H_0: \theta= \theta_0$, in this case LRT$=\frac {f(\mathbf{x}|\theta_0) }{f(\mathbf{x}|\hat\theta) }$, here we needn't consider sets of $\theta$ and supremums, and it is a common hypothesis test.
About change of LRT with the sample $\mathbf{x}$. In this case if the sample just moves as an intact body around the real line, (or equivalently, if the sample stays and $\theta$ moves around the real line) then the denominator doesn't change, and the nominator changes 'geometrically (since the likelihoods of $X_i$'s are multiplied) and exponentially(since the more the sample gets close to the tail of the distribution parameterized by $\theta$'; if besides that, the sample changes its size and observed 'distribution', then it would be more complicated.
But if the sample size $n\to\infty$, then the observed 'distribution' would be very probably the same as the actual distribution of iid $X_i$. And so the mentioned complexity disappears. LRT as a randomn variable (and function of $X_i$'s) will have an easier-to-define 'geometrical and exponential 'distribution, which turns out to be chi-square distribution. This gives a rough description of asymptotic behavior of the static LRT for hypothesis testing.)
My question here is why we need to set a range of LRT, MLE or $\mathbf{x}$ to decide if we accept a hypothesis or not. And it seems when setting the range, we have a preference that null hypothesis should not be easily rejected, why so? Overall, what are factors to consider when we set such a range.
I notice a major error, that is in the denominator it’s not $\Theta_a$ $\quad$ (2) but the set $\Theta_a$ of all possible parameters. $\quad$ (1) But this doesn’t affect seriously my above discussion.
Casella in Section 8.2.1 has a similar discussion about the relation between LRT and MLE, and gives $LRT=\frac {f(\mathbf{x}|\hat \theta_0)} {f(\mathbf{x}|\hat \theta)}$, where $\hat \theta_0$ maximize the likelihood for $\theta\in \Theta_0$. This fact (1) avoids the introduction of $\theta*$ and whether MLE is in $\Theta_0$ or $\Theta_a$. I’m not sure if this is reasonable but it looks neater.
My another question is from this we know LRT is always no more than 1, that is, if MLE is in $\Theta_0$, LRT has its max value and null hypothesis is certainly not rejected. $\quad$ (3) (It is so even when we define LRT with (2).) But is it reasonable? For example, even if $\Theta_0$ has just a single element, $\Theta_a$ can be an interval (on the real line) far or near from $\Theta_0$, and this will affect the value of $\sup_{\Theta_a} f(\mathbf{x}|\theta)$, and it seems that defining LRT with (2) is more reasonable. Also, it is more ‘symmetrical’. So why don’t we do so?
With (3) we can understand that c (or $(-\infty, c)$ or more exactly (0,1)) actually measures how far LRT (we can say, of $\hat \theta_0$) is away from 1, LRT of MLE; this corresponds to how far $\theta*$ (when MLE is not in $\Theta_0$) or $\hat \theta_0$, is from MLE. And this correspondence is very clear in the Example 8.2.2 of normal distribution.
