What is the sufficient condition for the consistency and normality of MLE?
I read several such regular conditions for the consistency and normality of MLE.
The latest one I have read state is one the by Ferguson "a course in the large sample" Sufficient Condition 1 Let $X_1, X_2, \dots$ be i.i.d with density $f(x|\theta)$ (with respect t0 dv), and let $\theta_0$ denoate the true value of the parameter. If
(1) $\Theta$ is an open subset of $R^k$
(2)second partial derivatives of $f(x|\theta)$ with respect to $\theta$ exist and are continuous for all x and may be passed under the integral sign in $\int f(x|\theta)dv(x)$
(3)There exists a function K(x) such that $E_{\theta_0}K(X)<\infty$ and each $\frac{d^2\theta f(x|\theta)}{d\theta^2 }$ is bounded in absolute value by K(x) uniformly in some neigborhood of $\theta_0$
(4)$I(\theta_0)=-E_{\theta_0}\frac{d\theta f(x|\theta)}{d\theta}$ is positive definite
(5) $f(x|\theta)=f(x|\theta_0)$ a.e.dv, thus $\theta=\theta_0$
Then there exists a strongly consistent sequence of $\hat{\theta}$ of roots of the likelihood equation such that
$\sqrt{n}(\hat{\theta}_n-{\theta}_0)\rightarrow ^{d} N(0,I(\theta_0)^{-1})$ Can not make sure the condition (3)?
The details of sufficient conditions are never stated in some elementary statistics courses.
Some books may state that
For consistency, some books may state that $(1)X=(X_1,X_2,\cdots,X_n)$follows $ f(x|\theta)$ i.i.d with respect to u with some common support
(2)The true parameter value $\theta_0$ is an interior point of the parameter space
Similar situation also happens in asymptotic normality:
They told me the MLE can be asymptotic normality or can be not? And then they just show some examples.
I am not so good at real analysis and mathematics. So it is not easy for me to understand the proof in Ferguson. (Especially inconsistent part, it use Uniform Law of Large Number, and based on some semiconscious and compact assumption)
Can anyone give much more simple read proof? And also a simple sufficient statement for asymptotic normality? I know some advanced probability and measure theory. Though I am a beginner.
I try to find a much clear statement for this consistency of MLE. One note from Stanford looks like that (see page 2) https://web.stanford.edu/class/archive/stats/stats200/stats200.1172/Lecture14.pdf Sufficient Condition 2 (1)all pdfs /pmfs $f(x|\theta)$ in the model have the same support,
(2) $\theta_0$ is an interior point(not one the boundary)
(3)The loglikelihood $l(\theta)$ is differentiable in $\theta$
(4)$\hat{\theta}$ is the unique value of $\theta \in \Omega$ that solves the equation $0=l'(\theta)$
My question: (1) It seems that in the sencond item of first sufficient condition:second partial derivatives of $f(x|\theta)$ with respect to $\theta$ exist and are continuous for all x and may be passed under the integral sign in $\int f(x|\theta)dv(x)$
This statement seems very important. Though I never find it in Sufficient Condition 2.
(2) It can be shown that Let $X_1,\cdots, X_n$ follow $U(0,\theta)$.
The MLE is $\hat{\theta}=X_{(n)}$. Though, $\hat{\theta}$ is asymptotic exp(1). Which items in sufficient conditions are not satisfied, both in sufficient conditions one and two?
Many thanks!