Let X1, ..., Xn be iid with pdf f(x; θ) = (1/ x√ 2π) e(-(logx- theta)^2)...

Let X1, ..., Xn be iid with pdf f(x; θ) = (1/ x√ 2π) e^{(-(logx- theta)^2) /2} I_x>0 for θ ∈ R.

(a) (15 points) Find the MLE of θ.

(b) (10 points) If we are testing H₀ : θ = 0 vs Ha : θ != 0. Provide a formula for the likelihood ratio test statistic λ(X).

(c) (5 points) Denote the MLE as ˆθ. Show that λ(X) is can be written as a decreasing function of | ˆθ| or ( | theta hat | )

(d) (Extra 10 points) From the previous question, we know we reject H₀ when λ(X) is too small, i.e., | ˆθ| is too large. If you want to reject H₀ with actual significance level α, what is the cutoff. In other words, find out the cutoff so that you reject H_o with actual significance level α.

Hint: You can do this in one of the two ways. (You don’t need to do both.) The first way is to provide a critical value for λ(X) by taking the formula in (c), relating it to an example in lecture notes in Chapter 4, and arguing convincingly that the conclusion of that example should hold here. The second way is to provide a critical value for | ˆθ| by identifying the distribution of | ˆθ| under H0 and isolating some areas in the left and right tails of that distribution.

Expert Solution

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orchestra answered 2 years ago

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