Problem Set # 2 November 1, 1999 Economics 504 Robin Sickles

Your completed problem set is due November 15.

1. If θ̂ is an estimator of a parameter θ, its bias is given by b=E(θ̂)-θ. Show that E[(θ̂ − θ)^{2}] = var(θ̂)+b^{2}.

2. Show that the mean of a random sample of size n is a minimum variance unbiased estimator of the parameter λ of a Poisson population.

3. Show that the mean of a random sample of size n from an exponential population is a consistent estimator of its parameter θ.

4. Suppose that we use the largest value of a random sample of size n (namely, the order statistics y_{n}) to estimate the paremeter θ of the population

f(x; θ) = \binom{\frac{1}{θ} for 0 < x < θ}{0 elsewheere}

Check whether this estimator is (a) unbiased and (b) consistent.

5. If x_{1}, x_{2}, ..., x_{n} are the values of a random sample of size n from a population having the density

f(x; θ) = \binom{2(θ − x)/θ^{2} for 0 < x < θ}{0 elsewhere}

find an estimator for θ by the method of moments.

6. If x_{1}, x_{2}, ..., x_{n} are the values of a random sample of size n from a Poisson population with the parameter λ, find an estimate of λ using

(a) the method of moments;

(b) the method of maximum likelihood.

7. Given a random sample of size n from a population having the density

f(x; θ) = \binom{(θ + 1)x^{θ} for 0 < x < 1}{0 elswhere}

find an estimator of θ using

(a) the method of moments; (b) the method of maximum likelihood

8. Given a random sample of size n from a population having the density

f(x : δ, φ) = \binom{\frac{1}{φ}e^{− \frac{x−δ}{φ}} for x > δ}{0 elsewhere}

where δ ∈ (−∞, +∞) and φ ∈ (0, +∞),find simultaneous maximum likelihood estimates of δ, φ.

9. Let x_{1}, x_{2}, ..., x_{n} be iid Uniform [α − β, α + β], where β > 0,and let θ = (α, β).

(a) Find a minimal sufficient statistic τ for θ.

(b) Find the ML estimator of θ. (Hint: Graph the region for θ such that the joint density p(x,θ) > 0.

10. Let x_{1}, x_{2}, ..., x_{n} be a random sample from a normal distribution with mean μ and variance σ^{2}.

(a) Obtain the Cramer-Rao lower bound.

(b) See whether \overline{x} and s^{2}attain the lower bound.

(c) Show that \overline{x} and s^{2}are jointly sufficient for μ and σ^{2}.

11. Measurements of the blood pressure of 25 elderly women have a mean of \overline{x} = 140mm of mercury. If these data can be looked upon as a random sample from a normal population with σ = 10mm of mercury, construct a 95% confidence interval for the population mean μ.

12. The length of the skulls of 10 fossil skeletons of an extinct species of birds has a mean of 5.68cm and a standard deviation of 0.29 cm. Assuming that such measurements are normally distributed, find a 95% confidence interval for the mean length of the skulls of this species of birds.

13. The following are the heat-producing capacities of coal from two mines (in millions of calories per ton):

Mine A: 8, 500 8, 330 8, 480 7, 960 8030
Mine B: 7, 710 7, 890 7, 920 8, 270 7, 860

Assuming that the data constitute independent random samples from normal populations with equal variances, construct a 99%confidence interval form the differences between the true average heat-producing capacities of coal from the two mines.

14. Given a random sample of size n from a normal population with μ = 0, use the Neyman-Pearson lemma to construct the most powerful critical region of size α to test the null hypothesis σ = σ_{0} against the alternative σ = σ_{1} > σ_{0}.

15. A random sample of size 64 is to be used to test the null hypohtesis that the mean of a normal population with the variance tσ^{2} = 256 is less than or equal to 40 against the alternative hypothesis that it is greater than 40. If the null hypothesis is to be rejected if and only if the mean of the random sample exceeds 43, find

(a) the probabilities of type I errors when μ = 37, 38, 39, 40;

(b) the probabilities of type II errors when μ = 41, 42, 43, 44, 45, 46, 47, and 48.

Also plot the graph of the power function.

16. A random sample of size n is to be used to test the null hypothesis that the parameter θ of an exponential population equals θ_{0} against the alternative that it does not equal θ_{0}.

(a) Find an espression for the likelihood ratio statistic.

(b) Use the result of part (a) to show that the critical region of the likelihood ratio test can be written as

\overline{x} · e^{−\overline{x}/θ_{0}} ≤ K.

17. When we test a simple null hypothesis against a compositive alternative, a critical region is said to be unbiased if the corresponding power function takes on its minimum value at the value of the parameter assumed under the null hypothesis. In other words, a critical region is unbaised if the probability of rejecting the null hypothesis is least when the null hypohthesis is true. Given a single observation of the random variable x having the density

f(x; θ) = \binom{1 + θ^{2}(\frac{1}{2} − x) for 0 < x < 1}{0 elsewhere}

where −1 ≤ θ ≤ 1, show that the critical region x ≤ α provides an unbiased and uniformaly most powerful critical region of size α for testing the null hypothesis θ = 0 against the alternative hypothesis θ ≠ 0.