Question

In: Statistics and Probability

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.

Solutions

Expert Solution


Related Solutions

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 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
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 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 an i.i.d. sample from a geometric distribution with parameter p. U =...
Let X1,..., Xn be an i.i.d. sample from a geometric distribution with parameter p. U = ( 1, if X1 = 1, 0, if X1 > 1) find a sufficient statistic T for p. find E(U|T)
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, 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.
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,...,Xn ∼ Geo(θ). (a) Find a 90% asymptotic confidence interval for θ. (b) Find a...
Let X1,...,Xn ∼ Geo(θ). (a) Find a 90% asymptotic confidence interval for θ. (b) Find a 99% asymptotic lower confidence intervals for φ = 1/θ, the expected number of trials until the first success.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT