MAST90082 Mathematical Statistics 2022
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Semester 1 Mock Exam, 2022
MAST90082 Mathematical Statistics
Question 1 (30 marks)
Let X1, . . . , Xn be a random sample from the Pareto distribution with pdf
f (x|θ) =
where θ > 1 is unknown.
(a) Find the Method of Moments Estimator (MME) of θ .
(b) Show that T = log Xi is complete and sufficient for θ .
(c) Find E9(T ← 1 ), where T = log Xi . Hint: the pdf of Gamma(r, λ) is f(x|r, λ) = ,
where r is the shape parameter, λ is the scale parameter, and Γ(r) = 0尸 xr ← 1 e ←z dx is the gamma function.
Find the UMVUE of θ .
(e) Find a uniformly most powerful (UMP) test of size α for testing
H0 : θ < θ0 versus H1 : θ > θ0 ,
where θ0 > 0 is a fixed real number. (Use quantiles of chi-square distributions to express the test)
(f) Find a confidence interval for θ with confidence coefficient 1 _ α by pivoting a random variable based on T = log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use an equal-tailed confidence interval)
(g) Find a confidence interval for θ with confidence coefficient 1 _ α by pivoting the cdf of X(1) = min{X1, . . . , Xn}. (Use an equal-tailed confidence interval)
Question 2 (20 marks)
Let X1, . . . , Xn be a random sample from a population with pdf
f (x|θ, λ) =
where θ e 戊 and λ > 0 are both unknown.
(a) Prove that (Xi _ X(1)) and X(1) are independent for any (θ, λ). You may assume that X(1) is complete and sufficient for θ for any fixed λ .
Find a sufficient statistic for (θ, λ) and prove its sufficiency.
(c) Find the maximum likelihood estimator (MLE) of (θ, λ). Hint: the first step would be maximizing the likelihood function for any fixed λ
(d) Find a likelihood ratio test (LRT) of size α for testing
H0 : λ = λ0 versus H1 : λ λ0 ,
where λ0 > 0 is a fixed real number. (Use quantiles of chi-square distributions to express the test) Hint: you may use the fact that (Xi _ X(1)) ~ Gamma(n _ 1, λ)
Question 3 (6 marks)
Mention whether the following statements are true or false. Prove it if you think it is true, otherwise provide a counterexample.
(i) Every function of a sufficient statistic is sufficient.
(ii) Every sufficient statistic is complete.
(iii) Unbiased estimator of any parameter is unique.
Question 4 (6 marks)
Let X1, . . . , Xn be a random sample from Uniform[_θ, θ], θ > 0. Find the Maximum Likelihood Estimator (MLE) of θ .
Question 5 (6 marks)
A statistic is ancillary for a parameter θ if its distribution does not depend on θ. Let X1, . . . , Xn be i.i.d. from a scale parameter family with cdf F (x/σ), σ > 0 and density 口(1)f (z口), where f (.) is not a function of σ. Let X(1), . . . , X(n) be the order statistics. Show that X(n)/X(1) is ancillary for σ .
Question 6 (6 marks)
Let X1, . . . , Xn be a random sample from Poisson(λ), λ > 0. Find the uniformly mini- mum variance unbiased estimator (UMVUE) of exp(aλ), where a is a fixed real number. You may assume T = Xi is complete and sufficient for λ . Hint: if Y1 , Y2 are inde- pendent and Yi ~ Poisson(λi), then Y1+Y2 ~ Poisson(λ1 +λ2 ). Also, ifX ~ Poisson(λ), then EA(bx ) = e(b ← 1)A, where b is a fixed real number.
Question 7 (6 marks)
Let X1, . . . , Xn be a random sample from a population with pdf
f (x|θ) =
where θ > 0. Find a most powerful test (MPT) for testing
H0 : θ = θ0 versus H1 : θ = θ 1 ,
where θ0 and θ 1 are fixed real numbers satisfy 0 < θ0 < θ 1 .
2022-06-13