Hi all,

I would appreciate some help comprehending a logical step in the proof below about the consistency of MLE. It comes directly from Introduction to Mathematical Statistics by Hogg and Craig and it is slightly different than the standard intuitive one that makes use of the Weak Law of Large Numbers.

So here goes:

Assume that \( \hat{\theta_n} \) solves the estimating equation \( \frac{\partial l(\theta)}{\partial \theta}=0 \). We also assume the usual regularity conditions. Denote \( \theta_0 \) the true parameter which by assumption is an interior point of some set \( \Omega \). Then \( \hat{\theta_n} \xrightarrow{P} \theta_0 \)

Let \( \mathbf{X}=(x_1,x_2, \ldots, {x_n}) \), the vector of observations. Since \( \theta_0 \) is an interior point in \( \Omega \) , \( (\theta_0 -a, \theta_0 +a) \subset \Omega \) for some \( a>0 \). Define \( S_n \) to be the event:

\(S_n= \{ \mathbf {X} : l(\theta_0 ; \mathbf{X}) > l(\theta_0 -a ; \mathbf{X}) \}

\cap \{ \mathbf{X}: l(\theta_0; \mathbf{X}) > l( \theta_0 +a ;\mathbf{X}) \} \)

But on \( S_n \) , \( l (\theta ) \) has a local maximum. \( \hat{\theta_n} \) such that \( \theta_0-a<\hat{\theta_n}<\theta_0+a \) and \( l^{\prime} \left(\hat{\theta_n} \right)=0 \) .

That is:

\(S_n \subset \{ \mathbf{X}: | \hat{ \theta_{n} } \left( \mathbf{X} \right) -\theta_{0} | < a \} \cap \{ \mathbf{X}: l^{ \prime} \left( \hat{\theta_n} \left( \mathbf{X} \right) \right) =0 \} \)

It is precisely at this point that I find their proof a little obscure. How come they consider \( S_n \) a subset of that othet set?Their explanation is unclear.

Of course the proof is not complete at this point but if I have this clarified, I can take it from there. Thank you in advance .

I would appreciate some help comprehending a logical step in the proof below about the consistency of MLE. It comes directly from Introduction to Mathematical Statistics by Hogg and Craig and it is slightly different than the standard intuitive one that makes use of the Weak Law of Large Numbers.

So here goes:

Assume that \( \hat{\theta_n} \) solves the estimating equation \( \frac{\partial l(\theta)}{\partial \theta}=0 \). We also assume the usual regularity conditions. Denote \( \theta_0 \) the true parameter which by assumption is an interior point of some set \( \Omega \). Then \( \hat{\theta_n} \xrightarrow{P} \theta_0 \)

**Proof**Let \( \mathbf{X}=(x_1,x_2, \ldots, {x_n}) \), the vector of observations. Since \( \theta_0 \) is an interior point in \( \Omega \) , \( (\theta_0 -a, \theta_0 +a) \subset \Omega \) for some \( a>0 \). Define \( S_n \) to be the event:

\(S_n= \{ \mathbf {X} : l(\theta_0 ; \mathbf{X}) > l(\theta_0 -a ; \mathbf{X}) \}

\cap \{ \mathbf{X}: l(\theta_0; \mathbf{X}) > l( \theta_0 +a ;\mathbf{X}) \} \)

But on \( S_n \) , \( l (\theta ) \) has a local maximum. \( \hat{\theta_n} \) such that \( \theta_0-a<\hat{\theta_n}<\theta_0+a \) and \( l^{\prime} \left(\hat{\theta_n} \right)=0 \) .

That is:

\(S_n \subset \{ \mathbf{X}: | \hat{ \theta_{n} } \left( \mathbf{X} \right) -\theta_{0} | < a \} \cap \{ \mathbf{X}: l^{ \prime} \left( \hat{\theta_n} \left( \mathbf{X} \right) \right) =0 \} \)

It is precisely at this point that I find their proof a little obscure. How come they consider \( S_n \) a subset of that othet set?Their explanation is unclear.

Of course the proof is not complete at this point but if I have this clarified, I can take it from there. Thank you in advance .

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