Review Questions

1. A population is distributed according to the probability function

P (x; r, l) = \{ \begin{matrix}\frac{1}{r}, & x = l, 2l, …, rl, \\ 0, & otherwise, \end{matrix}

where r and l are positive integer unknown parameters. A random sample X_{1}, X_{2}, …, X_{n} is drawn from the population.

(a) Estimate (r, l) by the method of moments.

(b) Find the maximum likelihood estimator of (r, l).

2. A population is distributed (continuously) uniformly U(a, 2a). A random sample X_{1}, X_{2}, …, X_{n} is drawn from the population.

(a) Suggest an unbiased estimator for a in the case n = 1.

(b) For general n, suggest a better unbiased estimator for a employing the sufficient statistic S = ( min_{1≤i≤n} X_{i}, max_{1≤i≤n} X_{i}).

(c) Is the statistic found in (b) of minimal variance out of all unbiased estimators for a? If in your opinion the answer is positive – prove it, otherwise – prove that S is not a complete statistic.

3. Opine on the following statement: A population f(x; θ) with an unknown parameter θ is given. X_{1}, X_{2}, …, X_{n} is a random sample from the population. T_{1} and T_{2} are two estimators for θ. If T_{1} is unbiased and T_{2} is biased, then T_{2} cannot be better than T_{1} (in the sense that |T_{2} − θ| ≤ |T_{1} − θ| for every value of θ and every possible outcome of the sample, with strict inequality in at least one case).

4. A certain population is distributed χ^{2}(r) and a second – N(r, 1), where the parameter r is unknown. A random sample X_{1}, X_{2}, …, X_{n} has been drawn from the first population and a random sample of equal size Y_{1}, Y_{2}, …, Y_{n}, independent of the first, has been drawn from the second population. Assume n to be large.

(a) Find a confidence interval of confidence level 1 − α for r, based on either the statistic T_{1} = ∑_{1≤i≤n}(X_{i} + Y_{i}) or the statistic T_{2} = ∑_{1≤i≤n}(X_{i} − Y_{i}).

(b) Test the hypothesis

\begin{matrix}H_{0} : & r = r_{0}, \\ H_{1} : & r > r_{0}, \end{matrix}

with significance level α employing the same statistic.

(c) Can the other statistic be used to provide an indication regarding the value of r? If yes – explain intuitively how, otherwise – explain why not.

5. A population is distributed according to the density function

f(x; a) = \{ \begin{matrix}\frac{3a^{3}}{x^{4}} , & x ≥ a, \\ 0, & x < a, \end{matrix}

where a > 0 is an unknown parameter. A random sample X_{1}, X_{2}, …, X_{n} is drawn from the population.

(a) Find the method of moments estimator of a.

(b) Find the maximum likelihood estimator of a.

(c) Find a 1-dimensional sufficient statistic for a.

(d) Find an unbiased estimator of \frac{1}{a}.

(e) Improve the estimator found in (d), employing the sufficient statistic found in (c).

(f) Calculate the variance of the estimator found in (e).

(g) Does the estimator attain the lower bound of Cramér-Rao? Explain!

(h) Find a confidence interval of confidence level 1 − α for \frac{1}{a}.

(i) Test the hypothesis

\begin{matrix}H_{0} : & a = a_{0}, \\ H_{1} : & a > a_{0}. \end{matrix}

with significance level α using the statistic max_{1≤i≤n} X_{i}.