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.

Expert Solution

orchestra answered 2 years ago

Let X = ( X1, X2, X3, ,,,, Xn ) is iid, f(x, a, b) =...

Let X = ( X1, X2, X3, ,,,, Xn ) is iid, f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b < 1 then, find a two dimensional sufficient statistic for (a, b)

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 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|...

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?

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, . . . , Xn ∼ iid Exp(θ) and consider the test for H0...

Let X1, . . . , Xn ∼ iid Exp(θ) and consider the test for H0 : θ ≥ θ0 vs H1 : θ < θ0. (a) Find the size-α LRT. Express the rejection region in the form of R = {X > c ¯ } where c will depend on a value from the χ 2 2n distribution. (b) Find the appropriate value of c. (c) Find the formula for the P-value of this test. (d) Compare this test...

Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Is this family MLR...

Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Is this family MLR in Y = X(n)? (b) Find the UMP size-α test for H0 : θ ≤ θ0 vs H1 : θ > θ0. (c) Find the UMP size-α test for H0 : θ ≥ θ0 vs H1 : θ < θ0. (d) Letting R1 be the rejection region for the test in part (b) and R2 be the rejection region for the test in part...

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2)...

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2) = ?1 e^(−?1(x−?2)) with S = [?2, ∞) and Θ = ℝ+ × ℝ. Determine a) L(?1, ?2). b) the MLE of ?⃗ = (?1, ?2). c) E(? ̂ 2).

Let X1, X2, . . . , Xn be iid following a uniform distribution over the...

Let X1, X2, . . . , Xn be iid following a uniform distribution over the interval (θ, 2θ) (θ > 0). (a) Find a method of moments estimator of θ. (b) Find the MLE of θ. (c) Find a constant k such that E(k ˆθ) = θ. (d) By using the Rao-Blackwell, which estimators of (a) and (b) can be improved?

Question