Question

In: Statistics and Probability

Let X1,...,Xn ∼ Geo(θ). (a) Find a 90% asymptotic confidence interval for θ. (b) Find a...

  1. Let X1,...,Xn ∼ Geo(θ).

    1. (a) Find a 90% asymptotic confidence interval for θ.

    2. (b) Find a 99% asymptotic lower confidence intervals for φ = 1/θ, the expected number of trials until the first success.

Solutions

Expert Solution

quite a long calculation. The student is advised to go through minutely and in case find it difficult to understand, comment.


Related Solutions

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?
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 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 ∼ 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 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 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 N(θ, σ2 ), with one-sided hypotheses H0...
Let X1, . . . , Xn ∼ iid N(θ, σ2 ), with one-sided hypotheses H0 : θ ≤ θ0 vs H1 : θ > θ0. (a) If σ^2 is known, we can use the UMP size-α test. Find the formula for the P-value of this test.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT