Let f(x; θ) = 1 θ x 1−θ θ for 0 < x < 1, 0 < θ < ∞.
(1) Show that ˆθ = − 1 n Pn i=1 log(Xi) is the MLE of θ. (2) Show
that this MLE is unbiased.
Exactly 6.4-8. Let f(x; θ)-29 for 0 < z < 1,0 < θ < oo. (1) Show that this MIE is unbiased the MLEofe. 1-0 (2) Show that this MLE is unbiased.
X1, X2, . . . , Xn i.i.d. ∼ N (µ, σ2 ). Assume µ is known; show
that ˆθ = 1 n Pn i=1(Xi− µ) 2 is the MLE for σ 2 and show that it
is unbiased.
Exactly 6.4-2. Xi, X2, . . . , xn i d. N(μ, μ)2 is the MLE for σ2 and show that it is unbiased. r'). Assume μ is known; show that θ- n Ση! (X,-
C3.)Let f(z,0) (1/θ)2(1-0)/0, 0 < x < 1,0 < θ < oo. . Find the maximurn likelihood estimator of θ. Show that the maximurn likelihood estimator is unbiased to θ
Let X1, X2,..., Xn be a r.s. from f(x) = 0x0-1, for 0 < x <1,0 < a < 0o. (a) Find the MLE of 0. (b) Let T = -log X. Find the pdf of T. (c) Find the pdf of Y = DIT: (i.e., distribution of Y = - , log Xi). (d) Find E(). (e) Find E( ). (f) Show that the variance of 0 MLE → as n → 00. (g) Find the MME of 0.
Let Xi, X2, Xn be ar ensity function f(r; θ) = (1/2)e-Iz-이,-oo < x < 00,-00 < θ < oo. Find the d MLE θ
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood function, L(θ), and the log-likelihood function, `(θ). (b) Find the maximum likelihood estimator of θ, ˆθ. (c) Is ˆθ unbiased? (d) What is the distribution of X? Find the moment estimator of θ, ˜θ.
2. Let Xi, X2, . Xn be a random sample from a distribution with the probability density function f(x; θ-829-1, 0 < x < 1,0 < θ < oo. Find the MLE θ
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...
6.2.1 2. Recall that θ--r/ Σ (θ, 1 ) distribution. Also, W - i-1 log Xi has the gamma distribution Г(n, 1/ ) -1 log X, is the mle of θ for a beta (a) Show that 2θW has a X2(2n) distribution. (b) Using part (a), find ci and c2 so that (6.2.35) for 0 < α < 1 . Next, obtain a (1-a) 100% confidence interval for θ.
density function f(x; θ)-829-1, 0 < x < 1, 0 < θ < oo. Find the MLE θ