where, X(1) = minimum ( X1, X2, .........Xn)
Is the maximum likelihood estimator of theta.
Let X1, X2, ..., Xn be iid random variables from a Uniform(-0,0) distribution, where 8 >...
6.1.10. Let X1, X2..... Xn be a random sample from a N(0,0%) distribution, where o? is fixed but-X <O<O. (a) Show that the mle ofis X. (b) If is restricted by 0 < < oc, show that the mie of 8 is 8 = max{0,X}.
7.6.4. Let X1, X2,... , Xn be a random sample from a uniform (0,) distribution. Continuing with Example 7.6.2, find the MVUEs for the following functions of (a) g(0)-?2, i.e., the variance of the distribution (b) g(0)- , i.e., the pdf of the distribution C) or t real, g(9)- , î.?., the mgf of the distribution. Example 7.6.2. Suppose X1, X2,... , Xn are iid random variables with the com- mon uniform (0,0) distribution. Let Yn - max{X1, X2,... ,...
3. Let X1, ..., Xn, ... be iid random variables from the shifted exponential distribution: Se-(2-0) f( x0) = л VI (a) Find the MLE for 0. (b) Find the MLE for ø= EX. (c) Find the MOM estimator for 0.
Let X1, X2, ..., Xn be iid random variables with a "Rayleigh” density having the following pdf: 22 -12 10 f(x) = e x > 0 > 0 0 пе a) (3 points) Find a sufficient estimator for 0 using the Factorization Theorem. b) (3 points) Find a method of moments estimator for 0. Small help: E(X1) = V c) (7 points) What is the MLE of 02 + 0 – 10 ? d) (7 points) For a fact, 21–1...
3. Let X1, X2, . . . , Xn be a random sample from a distribution with the probability density function f(x; θ) (1/02)Te-x/θ. O < _T < OO, 0 < θ < 00 . Find the MLE θ
Suppose X1, X2, , Xn is an iid sample from a uniform distribution over (θ, θΗθ!), where (a) Find the method of moments estimator of θ (b) Find the maximum likelihood estimator (MLE) of θ. (c) Is the MLE of θ a consistent estimator of θ? Explain.
Problem 8: 5 points] Let Xi,.,.Xn be IID from a Uniform distribution on (-0,0) where 0 0 is an unknown parameter (a) Find a minimal sufficient statistic T. (b) Define Show that T and V 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.
5. Let X1, X2, ..., Xn be a random sample from a distribution with pdf of f(x) = (@+1)xº,0<x<1. a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for @ ?
5. Let X1, X2,. , Xn be a random sample from a distribution with pdf of f(x) (0+1)x,0< x<1 a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for 0?