3.18 Let the r.v. X has the Geometric p.d.f. (i) Show that X is both sufficient...
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...
3.13 Let X,..., X be i.i.d. r.v.'s from the Gamma distribution with parameters a known and β θ eQ (0,0) unknown. (i) Determine the Fisher information I(e). U = U (X, , ,X" ) = ' (ii) Show that the estimate ηα 1.1 is unbiased and calculate its variance.
2.4 Consider the independent r.v.'s x,..,.X with the Weibull p.d.f. (i) Show that- is the MLE of θ.
3.3 Let X, ., X, be a random sample of size n from the U(0, e) distribution, Be Ω (0, o), and let Yz be the largest order statistic of the X,'s. Then (i) Employ formula (29) in Chapter 6 in order to obtain the p.d.f. of Y,. (ii) Use part (i) in order to construct an unbiased estimate of θ depending only on (iii) By Example 6 here (with α-0 and A-0) in conjunction with Theorem 3, show that...
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...
f (x, θ)-θ(1 _ θ)-1 2.7 Let X be a r.v. having th Then show that X is sufficient for θ , X-1, 2, , θ e Ω (0,1)
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
362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0), x = 0, 1, 2, . .. , zero elsewhere, where 0 0s I (a) Find the m. l.e., 6, of 0. X, is a complete sufficient statistic for 0. (b) Show that (c) Determine the unbiased minimum variance estimator of 0.
362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0),...
362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0), x = 0, 1, 2, . .. , zero elsewhere, where 0 0s I (a) Find the m. l.e., 6, of 0. X, is a complete sufficient statistic for 0. (b) Show that (c) Determine the unbiased minimum variance estimator of 0.
362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0),...
2.3 Let X be a r.v. describing the lifetime of a certain equipment, and suppose that the p.d.f. of X is f (ii) We know (see Exercise 2.1) that the MLE of θ, based on a random sample of size n from the above pd.f., is θ = 1/ X. Then determine the MLE of g(9).