3x2 for 0 < x < θ and zero otherwise. With the parameter θ > 0. We wish to Consider the pdf,f(x) estimate θ us...
Suppose thatX1,...Xn are IID with pdf f(x;θ) = 1 /2θ if -θ<x<θ otherwise =0 (a) Find an unbiased estimator of θ. You must prove that your estimator is unbiased. (b) Find the variance of the estimator in (a).
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0 >0 is unknown (a) Find a 1-a confidence interval for 0 by pivoting the cdf of X(n) = max{X1, ... , Xn}. (b) Show that the confidence interval in (a) can also be obtained using a pivotal quantity
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0...
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0 >0 is unknown (a) Find a 1-a confidence interval for 0 by pivoting the cdf of X(n) = max{X1, ... , Xn}. (b) Show that the confidence interval in (a) can also be obtained using a pivotal quantity
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0...
1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where θ is an unknown parameter, θ > 0. Let X, be the sample mean and X(n)max(Xi, X2,,Xn). (i) Show that θ¡n-(1 + )Xn ls an unbiased estimator of θ. Find its mean square error and check whether θ¡r, is consistent for θ. (i) Show that nX(n) is a consistent estimator of o (ii) Assume n > 1 and find MSE's of 02n, and compare...
Let X1, X2, ..., Xn be iid with
pdf f(x|θ) = θ*x(θ-1). a) Find the Maximum Likelihood
Estimator of θ, and b) show that its variance converges to
0 as n approaches infinity.
I have no problem with part a, finding the MLE of θ. However,
I'm having some trouble with finding the variance.
The professor walked us through part b generally, but I need
help with univariate transformation for sigma(-ln(xi))
(see picture below - the professor used Y=sigma(-ln(x)), and...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
Suppose X1, X2, . . . , Xn are iid with pdf f(x|θ) = θx^(θ−1) I(0 ≤ x ≤ 1), θ > 0. (a) Is − log(X1) unbiased for θ^(−1)? (b) Find a better estimator than log(X1) in the sense of with smaller MSE. (c) Is your estimator in part (b) UMVUE? Explain.
3. Let Xi,... , X,n be a random sample from a population with pdf 0, otherwise, where θ > 0. a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ (c) Find the pdf of θ in (b).
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calculate the ML estimator of θ. 6. Consider the pdf g(y|α) = c(1 + αy2 ), −1 < y < 1. (a) Show that g(y|α) is a pdf when c = 3 6 + 2α . (b) Calculate E(Y ) and E(Y 2 ). Referencing your calculations, explain why M1 can’t be...
If X, X2,..., Xn constitute a random sample from the population with pdf ffx) 0 elsewhere a) ind the E(X) and hence show that X is a biased estimator of 0. What is the bias? b)What estimator based on X would be an unbiased estimator of 0? Why? nen( y1-0) y, > c Given g(y,)- show that Yı= min ( X1, X2, Х. ) is a consistent 0 otherwise estimator of the parameter 0 d) Obtain the mean of Y,....