Stat 425 Investigation 4 Estimating Maximum Value

Please prepare a report containing your analysis of the following questions. You may work with one other person, submitting one report with both names. Word-processed reports are preferred to hand-written ones. Please integrate relevant computer output into your report. This report is due by 4pm on Thursday, December 2.

Distribution of Sample Maximum:

Let X₁, X₂, …, X_n be a random sample from a continuous uniform distribution on the interval (0, theta), where the parameter theta is a positive real number. Let the random variable Y be the sample maximum; in other words, Y = max{ X₁, X₂, …, X_n}.

(a) Determine the probability density function of Y, as a function of n and theta.

(b) Graph this pdf when theta = 100 and n = 4. Also graph it when theta = 100 and n = 10. Comment on how the pdf’s compare in these two cases.

(c) Determine the expected value of Y, as a function of n and theta.

(d) Determine the variance of Y, as a function of n and theta.

(e) How does the expected value of Y compare to theta? How does this change as the sample size n increases?

(f) Calculate the expected value and variance of Y when theta = 100 and n = 4. Also calculate them when theta = 100 and n = 10. Comment on how the expected value and variance compare in these two cases.

Estimators of Theta:

Consider these definitions:

• A random variable used to estimate the value of a parameter theta is called an estimator.
• If the expected value of an estimator T equals the parameter theta, then T is said to be an unbiased estimator of theta.
• The bias of an estimator T is defined to be the difference between its expected value and the parameter: bias(T) = E(T) – theta.
• The mean squared error of an estimator T is defined to be: MSE(T) = E[(T-theta)^2].

(g) Determine the bias of Y as an estimator of theta. Is the bias positive or negative?

(h) Does the absolute value of the bias increase, decrease, or stay the same as the sample size n increases? What happens to the bias in the limit as n approaches infinity?

Now consider estimators of the form T_c = cY, where Y = max{ X₁, X₂, …, X_n} as above, and c is a constant that could depend on the sample size n.

(i) Determine the value of the constant c, possibly as a function of the sample size n, so that T_c is an unbiased estimator of theta.

(j) Determine the variance of this unbiased estimator.

(k) Report the unbiased estimator and its variance in the n = 4 case. Then do the same for the n = 10 case. How do they compare?

(l) Explain why the MSE of an unbiased estimator is equal to its variance.

It can be shown (see exercise #5 on page 203) that for any estimator T, MSE(T) = [bias(T)]^2 + Var(T).

(m) Use this result to derive an expression for MSE(T_c), as a function of c, n, and theta.

(n) Use calculus to determine the value of c that minimizes MSE(T_c), possibly as a function of n.

(o) Is the estimator of theta that minimizes MSE equal to the unbiased estimator? Explain.

Simulation Analysis:

As a check and follow-up on your theoretical analysis above, you will simulate this situation.

(p) Use Minitab to simulate the taking of 1000 random samples of size n=4 from a continuous uniform distribution on the interval (0,100):

MTB> random 1000 c1-c4;

SUBC> uniform 0 100.

Determine the sample maximum for each of the 1000 samples:

MTB> rmax c1-c4 c11

Produce histograms of these sample maxima, and calculate their mean and standard deviation. Comment on whether these appear to be consistent with your theoretical analysis above.

(q) Use the let command to calculate unbiased estimates and store them in c12. Produce histograms of these estimates, and calculate their mean and standard deviation. Is their mean close to the actual value of theta? Is their standard deviation close to what your answer to (j) would have predicted? Explain.

(r) Now use the let command to calculate minimum MSE estimates and store them in c13. Produce histograms of these estimates, and calculate their mean and standard deviation.

(s) Calculate the squared error for each of your estimates:

MTB> let c14=(c12-100)**2

MTB> let c15=(c13-100)**2

Report the mean of these squared errors for the unbiased estimates and for the minimum MSE estimates. Which is smaller?

Other Estimators:

(t) Now consider estimators of the form kW, where W = min{ X₁, X₂, …, X_n}. Determine the value of k for which kW is an unbiased estimator of theta. Then determine the variance of this unbiased estimator. How does it compare to the variance of the unbiased estimator based on the sample maximum?

(u) Now consider estimators of the form aX-bar, where X-bar is the sample mean of the X_i’s. Determine the value of a for which aX-bar is an unbiased estimator of theta. Then determine the variance of this unbiased estimator. How does it compare to the variance of the unbiased estimators based on the sample maximum and on the sample minimum?

(v) Use simulation to investigate the behavior of these two unbiased estimators. Produce histograms and descriptive statistics to analyze them. Continue to use a sample size of n=4 and to let theta=100 as in the simulations above.