Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ ...
difficult…… 2and4 thanks Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
1.(c) 2.(a),(b) 5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
Suppose that Xi, X2, ..., Xn is an iid sample from where θ > 0. (a) Show that is a complete and sufficient statistic for σ (b) Prove that Y1-X11 follows an exponential distribution with mean σ (c) Find the uniformly minimum variance unbiased estimator (UMVUE) of T(o-o", where r is a fixed constant larger than 0.
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ, ie. I. What is the Maximum Likelihood estimate θ of θ? 2. Show that the maximum likelihood estimator of θ is unbiased. 3. We're looking to cstimate the variance θ (1-9) of Xi . x being the empirical average 2(1-2). Check that T is not unli ator propose an unbiased estimator of θ(1-0).
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...
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...
Show that the family of distributions GAM , к): > 0, к > 0 belongs to the regular exponential class, and use this information to find com plete sufficient statistics based on a random sample Xi,... , X, Let Xi,... , Xn be a random sample from a Bernoulli distribution, Xi BIN(1,p):0<p1. EX)-Σ x, n(n -1) Verify that T- is an unbiased estimator of p Then show that T is a UMVUE of p2 Show that the family of distributions...
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...
Ex (5) Let X = (Xi, X2, ,X") be a random sample with size n taken from population has e-부) a) 71 2 X is an unbiased estimator of τ (θ)-2(J+ b) T-X is a consistent estimator of τ (9) (J+ β fx(x ; θ) , β < x <。。.Show that 2)