4. Find the Fisher Information and the Cramer-Rao lower bound for the variance of an unbiased...
5. Find the Fisher Information and the Cramer-Rao lower bound for the variance of an unbiased estimator of θ given a random sample . . . , xn from the density f(x:0) where < x < oo and-oo < θ < 00 You may use WolframAlpha.com to evaluate a complicated integral that might arise.
5. Find the Fisher Information and the Cramer-Rao lower bound for the variance of an unbiased estimator of θ given a random sample X1, , Xn from the density )=n-li + (r-0)21 where-oo < z < oo and-oo < θ < oo. You may use WolframAlpha.com to evaluate a complicated integral that might arise.
4. Find the Fisher Information and the Cramer-Rao lower bound for the variance of an unbiased estimator of θ given a random sample . , xn from the density r3 -z/θ where x > 0 and f(x:0-6 94e θ > 0.
5. Find the Fisher Information and the Cramer-Rao lower bound for the variance of an unbiased estimator of θ given a random sample X1, , Xn from the density [I + (z-0)21 where _ oo 〈 x 〈 x and You may use WolframAlpha.com to evaluate a complicated integral that might arise.
please use as many steps as possible. 5. Find the Cramer-Rao lower bound for the variance of unbiased estimators of 8 based on a random sample of size n from a distribution with pdf f(1:0) = (1 + (1 - 0)2) for - 00 < < 00
Let X,, X,,...X be a random sample of size n from a normal distribution with parameters a. Derive the Cramer-Rao lower bound matrix for an unbiased estimator of the vector of parameters (μ, σ2). b. Using the Cramer-Rao lower bound prove that the sample mean X is the minimum variance unbiased estimator of u Is the maximum likelihood estimator of σ--σ-->|··( X,-X ) unbiased? c. Let X,, X,,...X be a random sample of size n from a normal distribution with...
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
Let X1, . . . , Xn be a random sample from a population X with p.d.f fθ(x) = θ xθ−1 , for 0 < x < 1 0, otherwise, where θ > 1 is parameter. Find the MLE of 1/θ. If it is an unbiased estimator of 1/θ, compare its variance with the Cramer-Rao lower bound.
B2. (a) Suppose θ is an unknown parameter which is to be estimated from a single measurement X, distributed according to some probability density function f(r0). The Fisher information I(0) is defined by de Show that, under some suitable regularity conditions, the variance of any unbi- ased estimator θ of θ is then bounded by the reciprocal of the Fisher information Var | θ 1(8) Note that the suitable regularity conditions, which are not specified here, allow the interchange of...
difficult…… 2and4 thanks Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....