3.13 Let X,..., X be i.i.d. r.v.'s from the Gamma distribution with parameters a known and...
Let X be a R.V. with a gamma distribution and the following parameters (X~(α, 1)). What is the pdf and the cdf of Y = X/β, where β > 0 . What is the name of this type of distribution?
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 -...
3.10 Let , X, be 1.1.d. r.v.'s with mean and variance Ơ2, both unknown. Then for any known constants c, , c., consider the linear estimate of μ defined by: (i) Identify the condition that the G's must satisfy, so that u' is an unbiased estimate of . (ii) Show that the sample mean X is the unbiased linear estimate of u with the smallest variance 1-1 (among all unbiased linear estimates of H). Hint. For part (ii), one has...
5.7 Let X, X, be independent r.v.'s from the u(e -a, o+ b) distribution, where a and b are (known) positive constants and θ Ω M. Determine the moment estimate θ of θ, and compute its expectation and variance.
5. Consider the gamma distribution and recall that its mean and variance are μ-αβ and σ2-032, respectively. Assume a is known. Let X1, . . , X,, ~ X where X ~ f(x; α, β). is strict. your findings to verify the additivity property in(3) = n1(3). you computed V(An). Relate V(ßn) and In(3). interval to estimate T (a) Compute the Fisher information I(8) of A (why?) and examine whether the Cramer-Rao inequality (b) Find the score of the sample...
3.18 Let the r.v. X has the Geometric p.d.f. (i) Show that X is both sufficient and complete. U(X )-1 (ii) Show that the estimate U defined by: estimate of 6 if X-1, and U(X) -0 if X 2 2, is an unbiased (iii) Conclude that U is the UNU estimate of θ and also an entirely unreasonable estimate.
Let X1, ..., X., be i.i.d random variables N(u, 02) where u is known parameter and o2 is the unknown parameter. Let y() = 02. (i) Find the CRLB for yo?). (ii) Recall that S2 is an unbiased estimator for o2. Compare the Var(S2) to that of the CRLB for
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
*** SOLVE 8 *** -7. Let X,, X,.. be a sequence of i.i.d. r.v.'s from NCO, 62. Using the MLE ah lEMIOO construct three 1-α asymptotic crs for θ Hint: Use the fact that the sample variance is asymptotically normal Solve the previous problem with Beta(1,0) in place of N(O,e2). -8.
U means Uniform distribution 2. Let X be a r.v. distributed as U(α, β). Show that its ch. f. and m.g.f.x and Mr, respectively, are given by and IM x it(β-a) , t(B-a) ii) By differentiating (ax, show that E(X)-(α + β) / 2 and T 2 (X)-(α-β)2 / 12.