Let X1, . . . , Xn ∼ Unif(0, θ). a) Is this family MLR in Y = X(n) (the sample maximum)? (b) Find the UMP size-α test for H0 : θ ≤ θ_0 vs H1 : θ > θ_0.
Let X ∼ Geo(?) with Θ=[0, 1]. a) Show that pdf of the random variable X is in the one-parameter regular exponential family of distributions. b) If X1,…, Xn is a sample of iid Geo(?) random variables with Θ=(0, 1), determine a complete minimal sufficient statistic for ?.
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
Show that minimal sufficient statistic for uniform (θ, θ +1) is not complete F(x/θ) = (1 θ<xi< θ+1, 1…n) (0 otherwise )
Exercise 2.4: Suppose θ is an estimator for θ with probability function Pr 2(n-1)/n and Pr[θ θ+n] 1/n (and no other values of θ are possible); show oj that θ is consistent but that bias (0) as n- → oo. Exercise 2.4: Suppose θ is an estimator for θ with probability function Pr 2(n-1)/n and Pr[θ θ+n] 1/n (and no other values of θ are possible); show oj that θ is consistent but that bias (0) as n- → oo.
Let f(x; θ) = 1 θ x 1−θ θ for 0 < x < 1, 0 < θ < ∞. (1) Show that ˆθ = − 1 n Pn i=1 log(Xi) is the MLE of θ. (2) Show that this MLE is unbiased. Exactly 6.4-8. Let f(x:0)-缸붕 for 0 < x < 1,0 < θ < oo 1 1-0 (1) Show that θ Σ-1 log(X) is the MLE of θ (2) Show that this MLE is unbiased.
Explain how to determine the angle θ when given the y-value, given 0 ≤ θ ≤ 2pi, and where tan θ =-8/15. Include a sketch with your explanation and solve for θ 8 8. Explain how to determine the angle 8 when given the y-value, given 0 so s 27, and where tan 8 Include a sketch with your explanation and solve for 8. [COM: /4*] 15
Let Xi iid∼ N(0, θ) for i = 1, ..., n. a) Find the MLE for θ. Call it b) Is biased? c) Is consistent? d) Find the variance of (e) What is the asymptotic distribution of ?
Let X1, ..., Xn be a sample from a U(0, θ) distribution where θ > 0 is a constant parameter. a) Density function of X(n) , the largest order statistic of X1,..., Xn. b) Mean and variance of X(n) . c) show Yn = sqrt(n)*(θ − X(n) ) converges to 0, in prob. d) What is the distribution of n(θ − X(n)).
Let Xi, , Xn be a sample from U(0,0), θ 0. a. Find the PDF of X(n). b. Use Factorization theorem to show that X(n) is sufficient for θ. C. Use the definition of complete statistic to verify that X(n) is complete for θ.