Let X1,...,Xn be a large random sample from a distribution

Suppose that a sample comprises the 15 observed values in Table 10.41. Calculate the values of (a) the sample mean, (b) the trimmed means for k = 1, 2, 3, and 4, (c) the sample median, and (d) the M-estimator with k = 1.5 and equal to the sample median absolute deviation divided by 0.6745. Table 10.41 Data for Exercise 1 23.0 21.5 63.0 22.5 2.1 22.1 22.4 2.2 21.7 21.7 22.2 22.9 21.3 21.8 22.1

Suppose that a sample comprises the 14 observed values in Table 10.42. Calculate the values of (a) the sample mean, (b) the trimmed means for k = 1, 2, 3, and 4, (c) the sample median, and (d) the M-estimator with k = 1.5 and equal to the sample median absolute deviation divided by 0.6745. Table 10.42 Data for Exercise 2 1.24 0.36 0.23 0.24 1.78 2.00 0.11 0.69 0.24 0.10 0.03 0.00 2.40 0.1

Suppose that a random sample of n = 100 observations is taken from the normal distribution with unknown mean and known variance 1, and let .5, n denote the sample median. Determine (approximately) the value of Pr(| .5, n | 0.1).

Suppose that a random sample of n = 100 observations is taken from the Cauchy distribution centered at an unknown point , and let .5, n denote the sample median. Determine (approximately) the value of Pr(| .5, n | 0.1

Let f (x) denote the p.d.f. of the contaminated normal distribution given in Eq. (10.7.1) with = 1/2, 2 = 1, and g being the p.d.f. of a normal distribution with mean and variance 4. Suppose that 100 observations are selected at random from a distribution for which the p.d.f. is f (x). Determine the M.S.E. of the sample mean and (approximately) the M.S.E. of the sample median.

Use the data in Table 10.6 on page 640. We want an estimate of the median of the logarithms of sulfur dioxide. Find (a) the sample mean, (b) the trimmed means for k = 1, 2, 3, and 4, (c) the sample median, and (d) the Mestimator with k = 1.5 and equal to the sample median absolute deviation divided by 0.6745.

Suppose that X1,...,Xn are i.i.d. with a distribution that has the p.d.f. in Eq. (10.7.2). Let Xn = 1 n n i=1 Xi. a. Prove that E(Xn) = . b. Prove that Var(Xn) = (1 + 99)2/n.

If Fig. 10.8 were extended all the way to = 1, the variance of the sample median would rise above the variance of the sample mean. Indeed, the ratio of the two variances would be the same at = 1 as it is at = 0. Explain why this should be true. 9

Assume that X1,...,Xn form a random sample from the distribution with p.d.f. given in Eq. (10.7.5). Prove that the M.L.E. of is the sample median. (Hint: Let X have c.d.f. equal to the sample c.d.f. of X1,...,Xn. Then apply Theorem 4.5.3

Let X1,...,Xn be i.i.d. with the p.d.f. in Eq. (10.7.5). Assume that is known. Let be between two of the observed values x1,...,xn. Prove that the derivative of the logarithm of the likelihood at equals 1/ times the difference between the number of observations greater than and the number of observations less than .

Let X be a random variable with a continuous distribution such that the interquartile range (IQR) is . Prove that the IQR of aX + b is a for all a > 0 and all b.

Let X be a random variable with a continuous distribution such that the median absolute deviation is . Prove that the median absolute deviation of aX + b is a for all a > 0 and all b.

Find the median absolute deviation of the Cauchy distribution.

Let X have the exponential distribution with parameter . Prove that the median absolute deviation of X is smaller than one-half of the IQR. (You can do this without actually calculating the median absolute deviation.)

Let X have a normal distribution with standard deviation . a. Prove that the IQR is 21(0.75). b. Prove that the median absolute deviation is 1(0.75).

Darwin (1876, p. 16) reported the results of an experiment in which he grew 15 pairs of Zea mays (corn) plants. Each pair consisted of a self-fertilized and a crossfertilized plant that were grown in the same pot. The numbers below are the differences between heights (in eighths of an inch) of the two plants in each pair (cross-fertilized minus self-fertilized). 49, 67, 8, 16, 6, 23, 28, 41, 14, 29, 56, 24, 75, 60, 48 Find the (a) the sample mean, (b) the trimmed means for k = 1, 2, 3, and 4, (c) the sample median, and (d) the Mestimator with k = 1.5 and equal to

Let X1,...,Xn be a large random sample from a distribution with p.d.f. f . Assume that f is symmetric about the median of the distribution. Find the large-sample distribution of the sample IQR.