STAT608 Mathematical Statistics II
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STAT608 Mathematical Statistics II Final Review
1. Let X1 , X2 , . . . , Xn be a random sample from the uniform distribution on (0, θ) where θ > 0. (a) Find E(X1 ) and E(X1(2)).
(b) Find the method of moments estimator, θMME , for θ . Unbiased? Consistent? (c) Find the maximum likelihood estimator, θMLE , for θ . Unbiased? Consistent? (d) Show that T = θMLE is a sufficient statistic.
(e) Show whether or not T = θMLE is complete.
(f) Find UMVUE (Uniform Minimum Variance Unbiased Estimator) using T = θMLE in (d) and (e).
2. Let X1 , X2 , . . . , Xn be an i.i.d. sample from Bernoulli(p) distribution.
(a) Find the maximum likelihood estimator (MLE) of the probability of success p, pˆMLE .
(b) Find the asymptotic distribution of ^n(pˆMLE 一 p).
(c) Let θ = loge ╱ 、. Find the MLE of θ , θˆMLE .
(d) Find the asymptotic distribution of ^n(θˆMLE 一 θ). State how this distribution behaves in terms of n and p.
(e) Obtain an approximate 95% confidence interval for θ for a fixed large n.
(f) Perform the Likelihood Ratio Test (LRT) for H0 : p ≤ p0 vs. H1 : p > p0 and show that the LRT will reject H0 if Xi > b for some constant b.
3. Let X1 , X2 , . . . , Xn be a random sample from a Normal(0, σ2 ). Perform the most powerful test (Neyman-Pearson Lemma) of H0 : σ = σ0 vs. H1 : σ = σ 1 where σ0 < σ 1 .
(a) Show that the test will reject H0 if Xi(2) > c for some constant c.
(b) Determine the constant c explicitly for a given value of α, the size of Type I Error.
4. Find a pivotal quantity.
(a) Let X1 , X2 , . . . , Xn be a random sample from a location-scale family: f (). Show that q = is a pivotal quantity where s is the sample standard deviation.
(b) Let X ~ Beta(θ, 1). Find a pivotal quantity and find a 100(1 一 α) confidence interval using the pivotal quantity.
5. Let X1 , X2 , . . . , Xn be a random sample from a Normal(µ, 1) distribution:
f (x|µ, 1) = exp , 一(x 2(一) µ)2 }.
(a) Prove that
exp , 一 i1 (xi 一 µ)2 } = exp , 一 (n 一 1)s2 一 n( 一 µ)2 }
where s2 = (xi 一 )2 .
(b) Let T = Xi . Show that T is sufficient using the likelihood of µ .
(c) Find a minimal sufficient statistic, S. Is S complete?
(d) Let µ has a Normal (θ, τ2 ) prior distribution. Derive the posterior distribution of µ . (e) Find the posterior mean of µ .
(f) Find a 95% equal-tail posterior interval for µ .
6. Let X1 , X2 , . . . , Xn be a random sample from an Exponential distribution:
f (x|θ) = θe −θx , x ≥ 0 and θ > 0.
(a) Compute the method of moments estimator for θ .
(b) Find the likelihood as a function of the observed data X1 , X2 , . . . , Xn and the parameter θ . (c) Find the maximum likelihood estimator (MLE) for θ , θˆMLE .
(d) Find the bias of θˆMLE .
(e) Find the Fisher information to find the estimated SE of θˆMLE .
(f) Show whether θˆMLE is consistent. Explain.
(g) Find the asymptotic distribution of (θMLE 一 θ).
(h) Find a (1 一 α) asymptotic confidence interval for θ .
Let ϕ = g(θ) = log(θ). Use the Delta Method.
(i) Find the maximum likelihood estimator for ϕ , MLE .
(j) Find the estimated standard error of MLE .
(k) Find an approximated 95% percent confidence interval.
7. Let X1 , X2 , . . . , Xn be a random sample from an Exponential distribution as in Problem 6.
Consider a prior distribution for the parameter θ as an exponential distribution, π(θ|τ ) = τ e −τθ , θ > 0 and τ > 0.
(a) Find the posterior distribution of θ . Show that it is a weighted average of the prior mean for θ and θˆMLE .
(b) Find the posterior mean and variance.
(c) Describe how to construct a 90% Bayes credible interval (equal-tail posterior interval) for θ .
2023-05-24