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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...
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- Spring '09