Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ 0. În turn, the parameter 0 is best regarded as a random variable with a Pareto(a, b) distribution, that is, bab 0, otherwise, where a 〉 0 and b 〉 0 are known. (a) Turn the "Bayesian crank" to find the posterior distribution of θ. I would probably start by working with a sufficient statistic (b) Find the posterior mean and use this as...
Let XI, X2, , Xn İs a random sample from the probability density function Use factorization theorem to show that X(1) = min(X1 , . . . , Xn) is sufficient for θ Is X(1) minimal sufficient for θ? a. b.
Let Xi, , Xn be a random sample from a n(o, σ*) distribution with pdf given by 2πσ I. Is the distribution family {f(x; σ), σ 0} complete? 2. Is PCH)〈1) the same for all σ ? 3. Find a sufficient statistic for σ. 4. Is the sufficient statistic from (c) also complete!? Let Xi, , Xn be a random sample from a n(o, σ*) distribution with pdf given by 2πσ I. Is the distribution family {f(x; σ), σ 0}...
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find a complete sufficient statistic for 0. d. Find the UMVUE for θ. Make sure you indicate how you know it is the UMVUE. Let Xi , X2,. … X, denote a random sample of size n...
3. Let X1, X2, .., Xn be a sample from the PDF 25 points 2r a. Show that X(n) is a complete-sufficient statistic for θ. b. Show that (3/2)X is unbiased for 0 c. Find the UMVUE for based on Xi, x2, x". Note: your final answer should be written as a computable formnla based on an olbserved sample. Con- sider using order statistics and their conditional distributions
Let Xi, , xn be a randon sannple fron f,(z0)-e-(z-0),0 є (-00,00), z > θ a. Show that X(1) is a complete statistic for θ. Hint: First find the PDF of XI) b. Show that the sample variance S is an ancllar statistic,and use this result to show that Xa) and S2 are independent.
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xi) and X(n)- (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , xn (iid) Uniform(0,0), E(R)-θ . What happens to E® as n increases? Briefly explain in words why this makes sense intuitively.
Please answer the following question and show every step. Thank you. Let Xi,..,Xn be a random sample from a population with pdf 0, x<0, where θ > 0 is unknown. (a) Show that the Gamma(a, b) prior with pdf 0, θ < 0. is a conjugate prior for θ (a > 0 and b > 0 are known constants). (b) Find the Bayes estimator of θ under square error loss. (c) Find the Bayes estimator of (2π-10)1/2 under square error...
Let X1, ..., Xn be a sample from a U(0, θ) distribution where θ > 0 is a constant parameter. a) Density function of X(n) , the largest order statistic of X1,..., Xn. b) Mean and variance of X(n) . c) show Yn = sqrt(n)*(θ − X(n) ) converges to 0, in prob. d) What is the distribution of n(θ − X(n)).
Let X,X,, X, be a random sample of size 3 from a uniform distribution having pdf /(x:0) = θ,0 < x < 0,0 < θ, and let):く,), be the corresponding order statistics. a. Show that 2Y, is an unbiased estimator of 0 and find its variance. b. Y is a sufficient statistic for 8. Determine the mean and variance of Y c. Determine the joint pdf of Y, and Y,, and use it to find the conditional expectation Find the...