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).

Expert Solution

Page 1page 2page 3page 4

orchestra answered 2 years ago

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. 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.

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.

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

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 ∼ iid Beta(θ, 1). (a) Find the UMP test...

Let X1, . . . , Xn ∼ iid Beta(θ, 1). (a) Find the UMP test for H0 : θ ≥ θ0 vs H1 : θ < θ0. (b) Find the corresponding Wald test. (c) How do these tests compare and which would you prefer?

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, 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.

6. Let X1; : : : ;Xn be i.i.d. samples from Uniform(0;theta ). (a) Find cn...

6. Let X1; : : : ;Xn be i.i.d. samples from Uniform(0;theta ). (a) Find cn such that Theta1 = cn min(X1; : : : ;Xn) is an unbiased estimator of theta (b) It is easy to show that theta2 = 2(X-bar) is also an unbiased estimator of theta(you do not need to show this). Compare theta1 and theta2. Which is a better estimator of theta? Specify your criteria.

Let X1, X2, · · · , Xn (n ≥ 30) be i.i.d observations from N(µ1,...

Let X1, X2, · · · , Xn (n ≥ 30) be i.i.d observations from N(µ1, σ12 ) and Y1, Y2, · · · , Yn be i.i.d observations from N(µ2, σ22 ). Also assume that X's and Y's are independent. Suppose that µ1, µ2, σ12 , σ22 are unknown. Find an approximate 95% confidence interval for µ1µ2.

Question