Question

Let X1, X2,..., Xn be a random sample from a population with probability density function f(x) = theta(1-x)^(theta-1), where 0<x<1, where theta is a positive unknown parameter

a) Find the method of moments estimator of theta

b) Find the maximum likelihood estimator of theta

c) Show that the log likelihood function is maximized at theta(hat)

Answer #1

a)

b)

c)

6. Let X1, X2, ..., Xn be a random sample of a random variable X
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where θ > -1. Obtain,
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b) Maximum Likelihood Estimator (MLE) of parameter θ.
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c) Is each of the estimators of a) and b) a sufficient
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. a. Justify the claim that Y = X1 + X2 + ... + Xn is a complete
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b. Compute...

Let X1, X2, · · · , Xn be a random sample from the distribution,
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a) Obtain the maximum likelihood estimator of θ, ˆθ.
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Let B > 0 and let X1 , X2 , … , Xn be a random sample from
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(i) Obtain the maximum likelihood estimator for B, β ˆ .
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Problem 1. The Cauchy distribution with scale 1 has following
density function
f(x) = 1 / π [1 + (x − η)^2 ] , −∞ < x < ∞.
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l(η; x1, x2, ..., xn) = log L(η; x1, x2, ..., xn) =
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Let X1, X2, . . . Xn be iid
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