{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lec5 - Lecture 5 Let us give one more example of MLE...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 5 Let us give one more example of MLE. Example 3. The uniform distribution U [0 , θ ] on the interval [0 , θ ] has p.d.f. f ( x | θ ) = 1 θ , ≤ x ≤ θ, , otherwise The likelihood function ϕ ( θ ) = n Y i =1 f ( X i | θ ) = 1 θ n I ( X 1 , . . . , X n ∈ [0 , θ ]) = 1 θ n I (max( X 1 , . . . , X n ) ≤ θ ) . Here the indicator function I ( A ) equals to 1 if A happens and 0 otherwise. What we wrote is that the product of p.d.f. f ( X i | θ ) will be equal to 0 if at least one of the factors is 0 and this will happen if at least one of X i s will fall outside of the interval [0 , θ ] which is the same as the maximum among them exceeds θ. In other words, ϕ ( θ ) = 0 if θ < max( X 1 , . . . , X n ) , and ϕ ( θ ) = 1 θ n if θ ≥ max( X 1 , . . . , X n ) . Therefore, looking at the figure 5.1 we see that ˆ θ = max( X 1 , . . . , X n ) is the MLE. 5.1 Consistency of MLE. Why the MLE ˆ θ converges to the unkown parameter θ ? This is not immediately obvious and in this section we will give a sketch of why this happens. 17 LECTURE 5. 18 ϕ(θ29 θ max(X1, ..., Xn) Figure 5.1: Maximize over θ First of all, MLE ˆ θ is a maximizer of L n θ = 1 n n X i =1 log f ( X i | θ ) which is just a log-likelihood function normalized by 1 n (of course, this does not affect the maximization). L n ( θ ) depends on data. Let us consider a function l ( X | θ ) = log f ( X | θ ) and define L ( θ ) = θ l ( X | θ ) , where we recall that θ is the true uknown parameter of the sample X 1 , . . . , X n . By the law of large numbers, for any...
View Full Document

{[ snackBarMessage ]}

Page1 / 7

lec5 - Lecture 5 Let us give one more example of MLE...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online