For the conditions of Exercise 6, find the M.L.E. of =

Suppose that X1,...,Xn form a random sample from a distribution with the p.d.f. given in Exercise 10 of Sec. 7.5. Find the M.L.E. of e1/

Suppose that X1,...,Xn form a random sample from a Poisson distribution for which the mean is unknown. Determine the M.L.E. of the standard deviation of the distribution

Suppose that X1,...,Xn form a random sample from an exponential distribution for which the value of the parameter is unknown. Determine the M.L.E. of the median of the distribution

Suppose that the lifetime of a certain type of lamp has an exponential distribution for which the value of the parameter is unknown. A random sample of n lamps of this type are tested for a period of T hours and the number X of lamps that fail during this period is observed, but the times at which the failures occurred are not noted. Determine the M.L.E. of based on the observed value of X.

Suppose that X1,...,Xn form a random sample from the uniform distribution on the interval [a, b], where both endpoints a and b are unknown. Find the M.L.E. of the mean of the distribution

Suppose that X1,...,Xn form a random sample from a normal distribution for which both the mean and the variance are unknown. Find the M.L.E. of the 0.95 quantile of the distribution, that is, of the point such that Pr(X < ) = 0.95.

For the conditions of Exercise 6, find the M.L.E. of = Pr(X > 2).

Suppose that X1,...,Xn form a random sample from a gamma distribution for which the p.d.f. is given by Eq. (7.6.2). Find the M.L.E. of  ()/ ()

Suppose that X1,...,Xn form a random sample from a gamma distribution for which both parameters and are unknown. Find the M.L.E. of /.

Suppose that X1,...,Xn form a random sample from a beta distribution for which both parameters and are unknown. Show that the M.L.E.s of and satisfy the following equation:  () ()  () () = 1 n n i=1 log Xi 1

Suppose that X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, ], where the value of is unknown. Show that the sequence of M.L.E.s of is a consistent sequence.

Suppose that X1,...,Xn form a random sample from an exponential distribution for which the value of the parameter is unknown. Show that the sequence of M.L.E.s of is a consistent sequence.

Suppose that X1,...,Xn form a random sample from a distribution for which the p.d.f. is as specified in Exercise 9 of Section 7.5. Show that the sequence of M.L.E.s of is a consistent sequence

Suppose that a scientist desires to estimate the proportion p of monarch butterflies that have a special type of marking on their wings. a. Suppose that he captures monarch butterflies one at a time until he has found five that have this special marking. If he must capture a total of 43 butterflies, what is the M.L.E. of p? b. Suppose that at the end of a day the scientist had captured 58 monarch butterflies and had found only three with the special marking. What is the M.L.E. of p?

Suppose that 21 observations are taken at random from an exponential distribution for which the mean is unknown ( > 0), the average of 20 of these observations is 6, and although the exact value of the other observation could not be determined, it was known to be greater than 15. Determine the M.L.E. of .

Suppose that each of two statisticians A and B must estimate a certain parameter whose value is unknown ( > 0). Statistician A can observe the value of a random variable X, which has the gamma distribution with parameters and , where = 3 and = ; statistician B can observe the value of a random variable Y , which has the Poisson distribution with mean 2. Suppose that the value observed by statistician A is X = 2 and the value observed by statistician B is Y = 3. Show that the likelihood functions determined by these observed values are proportional, and find the common value of the M.L.E. of obtained by each statistician

Suppose that each of two statisticians A and B must estimate a certain parameter p whose value is unknown (0

Prove that the method of moments estimator for the parameter of a Bernoulli distribution is the M.L.E.

Prove that the method of moments estimator for the parameter of an exponential distribution is the M.L.E.

Prove that the method of moments estimator of the mean of a Poisson distribution is the M.L.E.

Prove that the method of moments estimators of the mean and variance of a normal distribution are also the M.L.E.s.

Let X1,...,Xn be a random sample from the uniform distribution on the interval [0, ]. a. Find the method of moments estimator of . b. Show that the method of moments estimator is not the M.L.E

Suppose that X1,...,Xn form a random sample from the beta distribution with parameters and . Let = (, ) be the vector parameter. a. Find the method of moments estimator for . b. Show that the method of moments estimator is not the M.L.E.

Suppose that the two-dimensional vectors (X1, Y1), (X2, Y2), . . . , (Xn, Yn) form a random sample from a bivariate normal distribution for which the means of X and Y , the variances of X and Y , and the correlation between X and Y are unknown. Show that the M.L.E.s of these five parameters are as follows: 1 = Xn and 2 = Y n, 52 1 = 1 n n i=1 (Xi Xn) 2 and 52 2 = 1 n n i=1 (Yi Y n) 2, = n i=1(Xi Xn)(Yi Y n) #n i=1(Xi Xn)2 $1/2 #n i=1(Yi Y n)2 $1/2 . Hint: First, rewrite the joint p.d.f. of each pair (Xi, Yi) as the product of the marginal p.d.f. of Xi and the conditional p.d.f. of Yi given Xi. Second, transform the parameters to 1, 2 1 and = 2 21 1 , = 2 1 , 2 2.1 = (1 2)2 2 . Third, maximize the likelihood function as a function of the new parameters. Finally, apply the invariance property of M.L.E.s to find the M.L.E.s of the original parameters. The above transformat

Consider again the situation described in Exercise 24. This time, suppose that, for reasons unrelated to the values of the parameters, we cannot observe the values of Ynk+1,...,Yn. That is, we will be able to observe all of X1,...,Xn and Y1,...,Ynk, but not the last k Y values. Using the hint given in Exercise 24, find the M.L.E.s of 1, 2, 2 1 , 2 2 , and .