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3. Suppose that X1, . . . ,Xn IID X ∼ unif [0, θo] with θo > 0.
a) Obtain both the maximum likelihood estimate, θn = θn(X), and (thus) the
minimum relative likelihood, λn(θ,X) as functions of X = (X1, . . . ,Xn).
b) For U1, . . . ,Un IID U ∼ unif [0, 1] determine the asymptotic distribution
of Zn = n(1 − U(n)) where U(n) = max(U1, . . . ,Un) and use this result
to obtain the asymptotic distribution of θn.
c) Using the so-called Δ-method, use the result in b) to obtain the asymptotic
distribution of ln λn.
4. Suppose that X ∼ laplace(θo) with probability density function, f(x, θ) = 2−1e−|x−θ|
for some unique θ = θo ∈ R. [helpful: Z = |X−θ| ∼ exp (1) ]
a) For sample size n = 1, obtain the maximum likelihood estimate, θ = θ(X), and the
minimum relative likelihood, λ(θ,X), and thus determine aθ and the form of the
minimum relative likelihood confidence set C(x) for θo for any level 0 < β < 1.
b) For n = 1, determine the score function, S(θ) = S(θ,X), the fisher information,
I(θ) = varθS(θ), the mean value, −EθDS(θ), and thus (or otherwise) obtain
both
∞
−∞
∂f (x, θ)/∂θ dx and
∞
−∞
∂2f(x, θ)/∂θ2 dx.
[ hint: |a| = −aI(a < 0) + aI(a ≥ 0) ]
2. Native Americans
Paper details:
Choose either the anti slavery movement or the movements against the Native Americans. Discuss how your choice of topic evolved to the point where a change was deemed necessary by society and the government.
Describe societal and cultural reaction to the movement you have chosen. Be sure to use plenty of quotes and specific examples to support your points. Must follow the guidelines for writing a paper that is attached. Use the lecture packet for sources.
cite the sources also. Must have a introduction, body and conclusions. must have a total of 10 paragraphs minimum
