7. Let X1, · · · , Xn be i.i.d. with the density p(x, θ) = θ k (1 − θ) 1−k I{x = 0, 1}
(a) Find the ML estimator of θ.
(b) Is it unbiased ?
(c) Compute its MSE
7. Let X1, · · · , Xn be i.i.d. with the density p(x, θ) = θ k (1 − θ) 1−k I{x = 0, 1} (a) Find the ML estimator of θ. (...
7. Let X1 , Xn be i.i.d. with the density p(r,0) = a*(1 - 0)1-k1{x = 0,1} (a) Find the ML estimator of 0 (b) Is it unbiased? (c) Compute its MSE 7. Let X1 , Xn be i.i.d. with the density p(r,0) = a*(1 - 0)1-k1{x = 0,1} (a) Find the ML estimator of 0 (b) Is it unbiased? (c) Compute its MSE
pr(1-0)1-k1(a 0, 1} Let Xi, . . . ,X, be i.id. with the density p(x, θ) (a) Find the ML estimator of . b) Is it unbiased? (c) Compute its MSE pr(1-0)1-k1(a 0, 1} Let Xi, . . . ,X, be i.id. with the density p(x, θ) (a) Find the ML estimator of . b) Is it unbiased? (c) Compute its MSE
2) 6. Let Xi, , xn be i.i.d. Ņ(μ, σ (a) Find the sample analogue estimator of θ (b) Find the ML estimator of θ. 2) 6. Let Xi, , xn be i.i.d. Ņ(μ, σ (a) Find the sample analogue estimator of θ (b) Find the ML estimator of θ.
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ E Θ : (0.00) (a) Calculate the distribution function Fo of Xi (b) Let x1, .., xn be a realization of X1, Xn. What is the log-likelihood- function for the parameter θ? (c) Calculate the maximum-likelihood-estimator θ(x1, , xn) for the unknown parameter θ
Let X1. . . . Xn be i.i.d Uniform over the interval (θ, θ + 1].Show that X(1)+X(n) )/2- 1/2 is also an unbiased estimator of θ, whereX(1) is the minimum order statistic and X(n) is the maximum order statistic. If X - 1/2 is also an unbiased estimator of θ which of the two estimators would you prefer to use.
Additional Question i.i.d. ˆ Fix θ > 0 and let X1,...,Xn ∼ Unif[0,θ]. We saw in class that the MLE of θ, θMLE = max(X1, . . . , Xn), is biased. I give two other estimators of θ, which can be made unbiased by appropriate choice of constants C1, C2: ADDITIONAL QUESTION Fix θ 0 and let Xi, . . . , Xn iid. Unifl0.0]. We saw in class that the MLE of θ, θΜ1E- max(Xi,..., Xn), is biased....
Suppose X1, X2, . . . , Xn are iid with pdf f(x|θ) = θx^(θ−1) I(0 ≤ x ≤ 1), θ > 0. (a) Is − log(X1) unbiased for θ^(−1)? (b) Find a better estimator than log(X1) in the sense of with smaller MSE. (c) Is your estimator in part (b) UMVUE? Explain.
Additional Question Fix θ > 0 and let X1, . . . , Xn i.i.d. ∼ Unif[0, θ]. We saw in class that the MLE of θ, ˆθMLE = max(X1, . . . , Xn), is biased. I give two other estimators of θ, which can be made unbiased by appropriate choice of constants C1, C2: ˆθ1 = C1 max(X1, . . . , Xn) and ˆθ2 = C2Σxi We have two questions: (1) Find values of C1, C2 for...
Let X1, X2, ..., Xn be a random sample with probability density function a) Is ˜θ unbiased for θ? Explain. b) Is ˜θ consistent for θ? Explain. c) Find the limiting distribution of √ n( ˜θ − θ). need only C,D, and E Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
6. Let X1, , Xn be i.i.d. N(u,a2) (a) Find the sample analogue estimator of 0. (b) Find the ML estimator of 0