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id 3. Let X1, X2, ..., X 1 N(0,03) and Y1, 72,..., Ym N(02,03) independently. Denote...
Let X1, . . . , Xn be i.i.d. from N(µ1, σ2 ), and Y1, . . . , Ym be i.i.d. from N(µ2, σ2 ). If the two samples are independent, find the maximum likelihood estimates for µ1, µ2, and the common variance σ 2 .
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
Given two independent random samples X1, ..., Xn and Y1, ..., Ym with normal dis- tributions N(Hz, o?) and N(Hy, oz), determine a generalized likelihood ratio test for Ho : Mix - My = 0 versus H : plz – My 70 at a given significance level a (01, 0y unknown but equal).
Let X1,X2,...,Xn denote a random sample from the Rayleigh distribution given by f(x) = (2x θ)e−x2 θ x > 0; 0, elsewhere with unknown parameter θ > 0. (A) Find the maximum likelihood estimator ˆ θ of θ. (B) If we observer the values x1 = 0.5, x2 = 1.3, and x3 = 1.7, find the maximum likelihood estimate of θ.
15. Let X1, . . . , Xn be id from pmf p(z; θ)-(1-0)"-10; ;z=1,2, 3, ,and 0 < θ < 1. (a) Find the maximum likelihood estimator of θ (b) Find the maximum likelihood estimate of θ using the observed sample of 5,8,11.
[20 marks] Let xi, . . . , Xn be a random sample drawn independently from a one-parameter curved normal distribution which has density -oo 〈 x 〈 oo, θ > 0, 2πθ nx, and r2 - enote T-1 Tn (d) [3 marks] Find the maximum likelihood estimator θ2 of. (You do not need to perform the second derivative test.) (e) 3 marks Find the Fisher information T( (f) [3 marks] Is θ2 an MVUE of θ? Justify your answer....
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.
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8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Write down the expression of the likelihood function We assume that min(x1, . . . ,xm) 〈 n and max(x1, . . . ,xm) 〉 0 5 marks) Find , and give all possible solutions to the equation dL dL -...
Consider a random sample X1, ..., Xn from a normal distribution with known mean 0 and unknown variance 0 = 02 (a) Write the likelihood and log-likelihood function (b) Derive the maximum likelihood estimator for 6 (c) Show that the Fisher information matrix is I(O) = 2014 (d) What is the variance of the maximum likelihood estimator for @? Does it attain the Cramer-Rao lower bound? (e) Suppose that you are testing 0 = 1 versus the alternative 0 #...