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?)

Expert Solution

Above solution is correct with full explaination so please rate me high.

orchestra answered 2 years ago

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

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 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.

R simulation: Let X1, . . . , Xn be i.i.d. random variables from a uniform...

R simulation: Let X1, . . . , Xn be i.i.d. random variables from a uniform distribution on [0, 2]. Generate and plot 10 paths of sample means from n = 1 to n = 40 in one figure for each case. Give some comments to empirically check the Law of Large Numbers. (a) When n is large, X1 + · · · + Xn/n converges to E[Xi]. (b) When n is large, X1^2+ · · · + Xn^2/n converges to...

Let X1,X2,...,Xn be i.i.d. Gamma random variables with parameters α and λ. The likelihood function is...

Let X1,X2,...,Xn be i.i.d. Gamma random variables with parameters α and λ. The likelihood function is diﬃcult to diﬀerentiate because of the gamma function. Rather than ﬁnding the maximum likelihood estimators, what are the method of moments estimators of both parameters α and λ?

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).

6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1) random variables. Let M =...

6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1) random variables. Let M = max(X1,...,Xn). (a) Find the density function of M. (b) Find E[M] and V[M].

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 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 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|...

Question