Maximum Likelihood Estimator: Examples and Applications, Study notes of Mathematical Statistics

Download Maximum Likelihood Estimator: Examples and Applications and more Study notes Mathematical Statistics in PDF only on Docsity! TA: Yuan Jiang Email: jiangy@stat.wisc.edu STAT 710: Discussion #7 February 19, 2008 1 Maximum Likelihood Estimator Example 1. Let (X1, . . . , Xn) be a random sample from the uniform distri- bution on (θ, θ + |θ|). Find the MLE of θ when (i) θ ∈ (0,∞); (ii) θ ∈ (−∞, 0); (iii) θ ∈ R, θ 6= 0. Example 2. Let (X1, . . . , Xn) be a random sample from the Weibull distri- bution with Lebesgue density αθ−1xα−1e−x α/θI(0,∞)(x), where α > 0 and θ > 0 are unknown. Show that the likelihood equation are equivalent to h(α) = n−1 ∑n i=1 log Xi and θ = n −1 ∑n i=1 X α i , where h(α) = ( ∑n i=1 X α i ) −1 ∑n i=1 X α i log Xi− α−1, and that the likelihood equations have a unique solution. Example 3. Let (X1, . . . , Xn), n ≥ 2 be a random sample from a distribution having Lebesgue density fθ,j, where θ > 0, j = 1, 2, fθ,1 is the density of N(0, θ2) and fθ,2(x) = (2θ) −1e−|x|/θ. (i) Obtain an MLE of (θ, j). (ii) Show whether the MLE of j in part (i) is consistent. (iii) Show that the MLE of θ is consistent and derive its nondegenerated asymptotic distribution. Office: 1275A MSC 1 Phone: 262-1577