Suppose X1,··· ,Xn are i.i.d. with pdf if 0 < x < 1 and 0 otherwise.
(a) Construct the MP test for the hypothesis v.s. with α=0.05.
(b) Derive the power function of the test in (a).
Suppose X1,··· ,Xn are i.i.d. with pdf if 0 < x < 1 and 0 otherwise....
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 < x < 1, > 0 a) Find a sufficient statistic for . Is the statistic minimal sufficient? b) Find the MLE for and verify that it is a function of the statistic in a) c) Find IX() and hence give the CRLB for an unbiased estimator of . pdf means probability distribution function We were unable to transcribe this...
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) = for x>0 (a) Use method of moments to find estimators for µ and µ^2 . (b) What is the log likelihood as a function of µ after observing X1 = x1, . . . , Xn = xn? (c) Find the MLEs for µ and µ^2 . Are they the same as those you find in part (a)? (d) According to the Central Limit...
Let X1, X2, ..., Xn be a random sample from X which has pdf depending on a parameter and (i) (ii) where < x < . In both these two cases a) write down the log-likelihood function and find a 1-dimensional sufficient statistic for b) find the score function and the maximum likelihood estimator of c) find the observed information and evaluate the Fisher information at = 1. f(20) We were unable to transcribe this image((z(0 – 2) - )dxəz(47)...
Suppose n independent, identically distributed observations are drawn from an exponential () distribution, with pdf given by f(x,)=, 0 < x < . The data are x1, x2, .. , xn Construct a likelihood ratio hypothesis test of Ho : vs H1: (where and are known constants, with ), where the critical value is taken to be a constant c We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Suppose X1, X2, .., Xn is an iid sample from where >0. (a) Derive the size α likelihood ratio test (LRT) for Ho : θ-Bo versus H : θ θο. Derive the power function of the LRT (b) Suppose that n 10, Derive the most powerful (MP) level α-0.10 test of Ho : θ 1 versus Hi: 0-2. Calculate the power of your test
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
A random variable X has probability density function f(x)=(a-1)x^(-a),for x>=1. (a) For independent observations x1,...,xn show that the log-likelihood is given by, l(a;x1,...,xn)=nlog(a-1)-a (b) Hence derive an expression for the maximum likelihood estimate for ↵. (c) Suppose we observe data such that n = 6 and 6 i=1 log(xi) = 12. Show that the associated maximum likelihood estimate for ↵ is given by aˆ ↵ =1 .5. logri We were unable to transcribe this image
Suppose that X1, X2,..., Xn are iid from where a 1 is a known constant and θ > 0 is an unknown parameter. (a) Show that the likelihood ratio rejection region for testing Ho : θ θο versus H : θ > θο can be written in terms of X(n), the maximum order statistic. (b) Derive the power function of the test in part (a). (c) Derive the most powerful (MP) level α test of Ho : θ-5 versus H1...
3. Let X1,..Xn be a sample with joint pdf (or pmf) f(x,0), 0 e 0 c R. Suppose that {f(x, 0) 0 e 0} has monotone likelihood ratio (MLR) in T(X,). Consider test function if T(xn)> c if T(xn) c if T(x)<c 0 E [0,1 and c 2 0 are constants. Prove that the power function of ¢(X,) is where non-decreasing in 0 3. Let X1,..Xn be a sample with joint pdf (or pmf) f(x,0), 0 e 0 c R....
let x1.........xn be independent where xi is normally distributed with unknown mean u and unknown variance 0 find the UMP test for testing =0 against 0 when it is assumed that is known.=1 We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image