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1. Consider a random sample of size n from a population with probability density function: х...
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...
Suppose X1, X2, , xn is an iid sample from fx(x10)-θe_&z1 (a) For n 2 2, show that (x > 0), where θ > 0 . n- is the uniformly minimum variance unbiased estimator (UMVUE) of θ (b) Calculate varo(0). Comment, in particular, on the n 2 case. (c) Show that vars(0) does not attain the Cramer-Rao Lower Bound (CRLB) on the variance of all unbiased estimators of T(9-0 (d) For this part only, suppose that n 1, 11T(X) is...
Suppose X1, X2, ..., Xn is an iid sample from fx(r ja-θ(1-z)0-11(0 1), where x θ>0. (a) Find the method of moments (MOM) estimator of θ. (b) Find the maximum likelihood estimator (MLE) of θ (c) Find the MLE of Po(X 1/2) d) Is there a function of θ, say T 0), for which there exists an unbiased estimator whose variance attains the Cramér-Rao Lower Bound? If so, find it and identify the corresponding estimator. If not, show why not.
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.
- Suppose a random sample of size n is taken from the following distribution with a known positive parameter a. f(x;0,-) = a20 V 27797z exp 0; ; 0<x<00,0< < 0,0 < 8 < 00 elsewhere For this distruttore, the formats for mye or and x-a are respectively, Myo (1) = exp v{(1 - V1 –24*70)} for 1 < 2112 and exp{}(-VT - 2/0)} My-- (1) for 1 < ✓1 - 2t/0 2 Find the maximum likelihood estimators, 0 and...
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.
Advanced Statistics, I need help with (c) and (d) 2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
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.
QUESTION8 Let Y,,Y2, ..., Yn denote a random sample of size n from a population whose density is given by (a) Find the maximum likelihood estimator of θ given α is known. (b) Is the maximum likelihood estimator unbiased? (c) is a consistent estimator of θ? (d) Compute the Cramer-Rao lower bound for V(). Interpret the result. (e) Find the maximum likelihood estimator of α given θ is known.