Let X1. . . . Xn be i.i.d f(x; θ) = θ(1 − θ)^x x =...

Let X1. . . . Xn be i.i.d f(x; θ) = θ(1 − θ)^x x = 0.. Is there a function of θ for which there exists an unbiased estimator of θ whose variance achieves the CRLB? If so, find it

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orchestra answered 1 year ago

Let X1, ..., Xn be i.i.d random variables with the density function f(x|θ) = e^(θ−x) ,...

Let X1, ..., Xn be i.i.d random variables with the density function f(x|θ) = e^(θ−x) , θ ≤ x. a. Find the Method of Moment estimate of θ b. The MLE of θ (Hint: Think carefully before taking derivative, do we have to take derivative?)

Let X1, . . . , Xn i.i.d. Uniform(θ, θ + 1). Show that: ˆθ1 =...

Let X1, . . . , Xn i.i.d. Uniform(θ, θ + 1). Show that: ˆθ1 = X¯ − 1 2 and ˆθ2 = X(n) − n n + 1 are both consistent estimators for θ.

Let X1,...,Xn be i.i.d. N(θ,1), where θ ∈ R is the unknown parameter. (a) Find an...

Let X1,...,Xn be i.i.d. N(θ,1), where θ ∈ R is the unknown parameter. (a) Find an unbiased estimator of θ^2 based on (Xn)^2. (b) Calculate it’s variance and compare it with the Cram ́er-Rao lower bound.

Suppose X1; : : : ; Xn is i.i.d Exponential distribution with density f(xjθ) = (1/θ)...

Suppose X1; : : : ; Xn is i.i.d Exponential distribution with density f(xjθ) = (1/θ) * e(-x/θ); 0 ≤ x < 1; θ > 0: (a) Find the UMVUE (the best unbiased estimator) of θ. (b) What is the Cramer-Rao lower bound of all unbiased estimator of all unbiased estimator of θ. Does the estimator from (a) attain the lower bound? Justify your answer. (c) What is the Cramer-Rao lower bound of all unbiased estimator of θ^2? 3 (d)...

Let X1, . . . , Xn be i.i.d. samples from Uniform(0, θ). Show that for...

Let X1, . . . , Xn be i.i.d. samples from Uniform(0, θ). Show that for any α ∈ (0, 1), there is a cn,α, such that [max(X1,...,Xn),cn,α max(X1,...,Xn)] is a 1−α confidence interval of θ.

For a fixed θ>0, let X1,X2,…,Xn be i.i.d., each with the beta (1,θ) density. i) Find...

For a fixed θ>0, let X1,X2,…,Xn be i.i.d., each with the beta (1,θ) density. i) Find θ^ that is the maximum likelihood estimate of θ. ii) Let X have the beta (1,θ) density. Find the density of −log⁡(1−X). Recognize this as one of the famous ones and provide its name and parameters. iii) Find f that is the density of the MLE θ^ in part (i).

Let X1, X2, · · · , Xn be iid samples from density: f(x) = {θx^(θ−1),...

Let X1, X2, · · · , Xn be iid samples from density: f(x) = {θx^(θ−1), if 0 ≤ x ≤ 1} 0 otherwise Find the maximum likelihood estimate for θ. Explain, using Strong Law of Large Numbers, that this maximum likelihood estimate is consistent.

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 Ix>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|...

Suppose X1, ..., Xn are i.i.d. from an exponential distribution with mean θ. If we are...

Suppose X1, ..., Xn are i.i.d. from an exponential distribution with mean θ. If we are testing H0 : θ = θ0 vs Ha : θ > θ0. Suppose we reject H0 when ( X¯n/ θ0) > 1 + (1.645/ √n) (a) (10 points) Calculate the power function G(ζ). You may leave your answer in terms of the standard normal cdf Φ(x). (b) (5 points) Is this test consistent?

1. Let X1, X2 be i.i.d with this distribution: f(x) = 3e cx, x ≥ 0...

1. Let X1, X2 be i.i.d with this distribution: f(x) = 3e cx, x ≥ 0 a. Find the value of c b. Recognize this as a famous distribution that we’ve learned in class. Using your knowledge of this distribution, find the t such that P(X1 > t) = 0.98. c. Let M = max(X1, X2). Find P(M < 10)

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