6. Let X1...., X, be a random sample from Unif(0,0). 1 a. Find E(4X1X3). b. Find...
1. Let X1, X2,...,x. be a random sample from the unif(0,0) distribution (a) Find an unbiased estimatior of O based on the sample mean X (b) Find an unbiased estimator of based on the sample maximum X (c) Which estimator is better in terms of variance?
Let X1, ..., Xn be a random sample from a distribution with pdf 2πσχ (a) If σ and μ are both unknown, find a minimal sufficient statistic T. (b) If σ is known and μ is unknown, is T from last part a sufficient statistic? Is it a minimal sufficient statistic? Prove your answer. (c) Let V (II1 X)/m, what is the distribution of V? Are V andindependently distributed? Let X1, ..., Xn be a random sample from a distribution...
1. (20 points) Let X1....X be a random sample from a uniform distribution over [0,0]. (a) (4 points) Find the maximum likelihood estimator (MLE) 0 MLE for 0. (b) (3 points) Is the MLE ONLE unbiased for 0? If yes, prove it: If not, construct an unbiased estimator 0, based on the MLE. (c) (4 points) Find the method of moment estimator (MME) OM ME for 8. (d) (3 points) Is the MME OMME tnbiased for 6? If yes, prove...
Let X1, . . . , Xn be a random sample from a population with density 8. Let Xi,... ,Xn be a random sample from a population with density 17 J 2.rg2 , if 0<、〈릉 0 , if otherwise ( a) Find the maximum likelihood estimator (MLE) of θ . (b) Find a sufficient statistic for θ (c) Is the above MLE a minimal sufficient statistic? Explain fully.
Let X1, ... ,X, be a sample of iid N(0,0) random variables with © = R. a) Show that T = - X-1 Xş is a pivotal quantity. d) Determine an exact (1 – a) x 100% confidence interval for SD(X) = V0 based on T.
2.a. Let X1, X2, ..., X., be a random sample from a distribution with p.d.f. (39) f( 0) = (1 - 1) if 0 < x <1 elsewhere ( 1 2.) = where 8 > 0. Find a sufficient statistic for 0. Justify your answer! Hint: (2(1-)). b. Let X1, X2,..., X, be a random sample from a distribution with p.d.f. (1:0) = 22/ if 0 < I< elsewhere where 8 >0. Find a sufficient statistic for 8. Justify your...
Consider X1,X2, , Xn be an iid random sample fron Unif(0.0). Let θ = (끄+1) Y where Y = max(X1, x. . . . , X.). It can be easily shown that the cdf of Y is h(y) = Prp.SH-()" 1. Prove that Y is a biased estimator of θ and write down the expression of the bias 2. Prove that θ is an unbiased estimator of θ. 3. Determine and write down the cdf of 0 4. Discuss why...
Let X1, . . . , Xn be a random sample from the discrete uniform distribution on 1, 2, . . . , θ. Using the definition of sufficient statistic, show that X(n) is a sufficient statistic for θ.
Let X1,X2Xn be a random sample from a uniform distribution on the interval (0,0) (a) Show that the density function of Xcp-minXXXn) is given by n-1 72 0 otherwise (b) Use (a) to calculate E[Xcu]. Calculate the bias, B(6). Find a function of Xo) that is an unbiased estimator of 0
1. Let X1, X2,...,X, be a random sample from each of the distributions having the to lowing pdfs or pmfs: (a) f(x; 0) = 6"e-/x!, r = 0,1,2, ..., 0< < oo, zero elsewhere, where f(0,0) = 1. (b) f(2; 6) = 0.00-110,1)(2), 0 <O< 0o. (c) f(x; 6) = (1/0)e-1/10,00) (2), 0 <$<. (d) f(x; 0) = e-(2-) 110,00) (2), - < < . • For each case, find the ML estimator ômle of 0; • For each case,...