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business
modern mathematical statistics with applications
Mathematical Statistics For Economics And Business 1st Edition Ron C. Mittelhammer - Solutions
a. You are random sampling from a Gamma population distribution, and you are testing Ho: ex :5 2 versus Ha : ex> 2.
d. Calculate and interpret the p value for the test.
c. Is the test you used in (a) a UMP test? Is it a UMPU test? Is it an unbiased test? Is it a consistent test?
b. Plot the power function for this test. Interpret the power function from the standpoint of both the pharmaceutical company and the consuming public.
a. Test the null hypothesis Ho : P 2: .50 using as close to a .lD-size test as you can.
2. A pharmaceutical company is analyzing the effectiveness of a new drug that it claims can stimulate hair growth in balding men. For the purposes of an advertising campaign, the marketing department would like to be able to claim that the drug will be effective for at least 50 percent of the
c. Plot the power function of this testing procedure.Interpret the implications of the power function from the standpoint of both the department store and the television manufacturer.
b. Calculate the probabilities of committing type II errors when k 2: 3.
a. Calculate the probabilities of committing type I errors when k = 0, 1, or 2.
1. A shipment of 20 projection screen television sets is at the receiving dock of a large department store. The department store has a policy of not accepting shipments that contain more than lD percent defective merchandise.The receiving clerk is instructed to have a quality inspection done on two
c. For the general case where ~ is unknown, discuss how you would define an operational version of the GMM estimator, and discuss its relationship with the estimator in (a).
b. Discuss conditions under which the estimator you have defined in (a) is consistent, asymptotically normal, and asymptotically efficient.
a. Assuming temporarily that ~ were known, identify the sample moment conditions that would be used to define the GMM estimator. Solve the moment conditions to provide an explicit functional representation of the estimator (called the generalized least squares estimator in the literature).
d. Estimating a and b based on a random sample from a continuous uniform population distribution.*18. (Generalized least squares estimator) Consider the linear model Y = x{3 + E in which all of the classical assumptions apply, except that Eee' = ~ =1= (121. Consider a GMM estimator of the parameter
c. Estimating p based on a random sample from a geometric population distribution.
b. Estimating a and fJ based on a random sample from a Beta population distribution.
a. Estimating J..L and (12 based on a random sample from a normal population distribution.
17. Define MOM estimators in each of the following cases. Are estimators consistent? Are they asymptotically normal?
c. Estimating fJ in the population distribution fIx; fJ) = ~ exp [ -;2] 1,0.col(X)'d. Estimating the parameter p in a negative binomial population distribution, where r is known.
b. Estimating the unknown parameters in the mean vector JL and covariance matrix :E based on a random sample from a bivariate normal distribution.
a. Estimating the p/s based on a random sample from a multinomial population distribution.
16. Define MLEs for the following problems:
'd. Are the estimators in (a) MVUES?
c. Are the estimators in (a) unbiased? consistent?(Hint: Emax(X) = ern/In + 111 and E(max(XIJ2 =:8 2[n/(n + 2)).)
b. Use the MLEs you defined above to generate ML estimates of the respective quantities of interest.
a. Define a MLE for e and for the expected number of minutes past the scheduled departure time that a jet will leave the terminal. Are these MLEs functions of minimal sufficient statistics?
15. The number of minutes past the scheduled departure time that jets with no mechanical problems leave the terminal in an overcrowded airport in the northeast are iid outcomes from a uniform population distribution of the form f(z; 8) = 8-1110.81(z). A random sample of 1,000 departures is to be
d. Define a MLE for the standard deviation of the daily number of wrong connections. Is the MLE consistent?Is it asymptotically normal? Generate a ML estimate of the standard deviation.
c. If E~~ Xi = 4,973, what is the ML estimate of the expected number of wrong connections? If each wrong connection costs the company $0.70, define a MLE for the expected daily cost of wrong connections, and generate a ML estimate of this cost.
b. Is the MLE the MVUE for A? Is it consistent?Asymptotically normal? Asymptotically efficient?
a. Define the MLE of A, the expected number of wrong connections per day.
14. A regional telephone company is analyzing the number of telephone calls that are connected to wrong numbers at its telephone exchange. It collects the number of wrong telephone connections on each of 200 days and treats the observations as the outcome of a random sample of size 200 from a
d. Provide MLE estimates and MVUE estimates of both the expected number of trials needed for the first light and the probability that the lighter lights on any given trial.508 Chapter 8 Point Estimation Methods
c. Define the MLE for the probability that the lighter lights on any given trial. *Is the estimator the MVUE? Is it consistent? Is it asymptotically normal?Is it asymptotically efficient? (Hint: Can you show that t(X) = (n-1)/UL7=1 XiI-I) has an expectation equal to p?)
b. Define the MLE for the expected number of trials needed for a lighter to light. Is the estimator the MVUE? Is it consistent? Is it asymptotically normal?Is it asymptotically efficient?
a. Define an appropriate statistical model for the 10,000 outcomes of how many trials were required for each lighter to light.
13. Your company markets a disposable butane lighter called "surelight." In your product advertising, you use the slogan "lights on the first try-every time!" As a quality check, you intend to examine a random sample of 10,000 lighters from the assembly line and observe for each lighter the number
c. The disturbance terms are related as St = PSt-1 + Vt , where the Vt's are iid with EVt = 0 and varWt) = a2 V t, and ipi < 1. The least-squares estimator is both BLUE and consistent.
b. The s;'s are homoskedastic and independent with ESi = 8 =p 0 V i. Also, xf3 = /3IL + /32Z, where L is an (n xl) column vector of l's, and Z is an (n xl)column vector of explanatory variable values. Then if fJ is the least-squares estimator of /3, fi2 is the BLUE of /32,
a. Let the random (n xl) vector Y represent a random sample from some composite experiment, where Ee = [0], and Eee' = a 21. Suppose the x-matrix has full column rank, but that the first and second columns of x are nearly linearly dependent and, as a 507 result, the determinant of x'x is near zero,
12. In each case, indicate whether the statement regarding the relationship Y = xf3 + e is true or false, and justify your answer.
b. Is the least-squares estimator unbiased? BLUE?Asymptotically unbiased?c. Letting x = [; ~~], 1 Xn(a symmetric, posltlve definite matrix) as n -+00, would it follow that the least-squares estimator is consistent and asymptotically normally distributed?Why or why not?d. Letting _ [In.YI Y. - :
a. Transform the model into GLM form. What parameters or functions of parameters are being estimated by the least-squares estimator applied to the transformed model?
Assuming the model specification is correct, answer the following questions:
11. The following statistical model is postulated for representing the relationship between real aggregate disposable income and real aggregate expenditure on nondurable goods:Yj = exp(pi + {32Xj + Ej)Problems where Yi = real aggregate expenditure on nondurables in period i measured in billions of
d. Yj = X2;!( 1 + ex,dP+ti)
c. Yj = {3 0 nki= IXjP; i +Ej
b. Yj = {3o + PIXj + P2xl + Ej
a. Yj = exp(L:7=1 Xij{3; + Ed
j. Estimate the probability that your MVUE estimator of the price elasticity of demand will generate an estimate that is within ±.15 of the true price elasticity. You may use estimates of unknown parameters A •• hiJ A A A k. Is EYt = e1hpt itl, where ({31·,{32,{33) is the BLUE estimator of
i. Present conditions under which the MVUE for a2 would be a consistent estimator.
Justify that the estimator is, in fact, the MVUE. Generate an MVUE estimate of a2 •
h. Define the MVUE for a2
g. Present conditions under which the MVUE of(In{3l, 132, {33) would be: (I) a consistent estimator, and (2) an asymptotically normally distributed estimator.
f. Define the probability distribution of the MVUE for({32 + {33). What is the probability distribution of the MVUE for (In {31, {32, {33)?
e. What is the MVUE for ({32 + {33), i.e., what is the MVUE for the degree of homogeneity of the demand function in terms of relative prices and real income?Justify that your estimator is the MVUE for ({32 +{33).Generate an MVUE estimate of (132 + {33).
d. Define the MVUE for the vector (In {31, {32, {33). Justify that your estimator is in fact the MVUE. Generate a MVUE estimate for the vector.
c. Define the BLUE estimator of (In {31, {32, {33)' Generate a BLUE estimate for this vector.
b. Define complete (and minimal) sufficient statistics for the parameters (,8, a).
Can the random sample (YI, ... , Yn) be interpreted as a random sample from a population distribution? Why or why not?
a. Present the statistical model in a form that is consistent with the general linear model framework(variables should be measured in a way that coincides with the way they are used in the GLM specification).
Yt = per capita textile consumption in year t, represented as an index with base year 1925;Pt = retail price of clothing divided by a general cost-of-living index in year t,represented as an index with base year 1925, it = real income per capita in year t,defined as the total money income of
during the period 1923-1939, using a general linear 506 Chapter 8 Point Estimation Methods model framework. In particular, he specified the relationship between per-capita textile consumption, real price of textiles, and per-capita real income as t = 1923, ... , 1939, where
9. Henri Theil, a famous economist/econometrician, analyzed the demand for textiles in the Netherlands
If not, can you characterize the problem conditions under which each estimator would be superior in terms of MSE?
d. Compare the mean square errors of the leastsquares estimator and the estimator ~'. Is one estimator superior in MSE to the other V /3 and a2 ?
c. Is the estimator a consistent estimator of /3? Justify your answer, being explicit about any assumptions you have made about the behavior of the Xi values.
b. Derive an expression for the variance of this estimator.
a. Is the estimator unbiased? If the estimator is not unbiased, derive an expression for the bias.
8. A business consultant to the ACME Textile Co.suggests that the estimator f/ = (x'x +ktlx'y might be useful to consider as an alternative to the leastsquares estimator of /3 in the preceding problem (the estimator ~. is a special case of the so-called "ridge regression"estimator in the statistics
Use the estimator you derived in (b) to generate an estimate of /3.
d. From a sample of size n = 100, the sample outcome resulted in 100 LXiYi = 92,017 and i=I nLX; = 897,235.i=I
c. Presuming that you could increase the sample size without bound, is the estimator you derived in (b) a consistent estimator of /3? Is it asymptotically normally distributed? Justify your answer, being explicit about any assumptions you have made about the behavior of the Xi values and/or the Si
b. Derive the functional form of the least-squares estimator of the proportionality factor, /3. Is the 505 least-squares estimator BLUE in this case? Is it the MVUE of /3?
a. Should the random sample {Y1, ... ,Yn} be interpreted as a random sample from some population distribution, or should it be interpreted as a random sample generated from a composite experiment?(Note: It cannot be expected that the aptitude scores will all be the same.)
you may also assume that ESi = 0 and EST = a2, V i. Suppose you had the outcome, {YI, Y2, ... , Yn}, of a random sample of average production rates for n employees, together with their associated scores, {XI,X2, ... ,Xn}, on the aptitude test.
You may assume that the s/s are independent, and
7. The personnel department of the ACME Textile Co.administers an aptitude test to all prospective assemblyline employees. The average number of garments per hour that an employee can produce is approximately proportional to the score received on the aptitude test.In particular, the relationship is
k. Calculate the maximum likelihood estimate of the probability that there will be greater than or equal to 97% correct cash transactions on a given day.(Hint: Determine the appropriate function of a in this case, and use the invariance principle.)
Calculate the estimate of a using the MVUE of a.j. Calculate the maximum likelihood estimate of q(a) = a/(a + I), the expected proportion of correct transactions.
Calculate the maximum likelihood estimate of a.
h. Is the MLE of q(a) asymptotically normal and asymptotically efficient? If so, define the asymptotic distribution of the MLE estimator.Problems i. The outcome of the sufficient statistic was nLlnxi = -9.725.i=I
g. Is the MLE of q(a) a consistent estimator?
f. Define the MLE of q(a) = alIa + 1), which is the expected proportion of correct cash transactions.
d. It can be shown (you don't have to) that E (tlnX;) -I = -a/In - 1).(See W. C. Guenther (1967), "A best statistic with variance not equal to the Cramer-Rao lower bound." American Mathematical Monthly, 74, pp. 993-994, or else you can derive the density of('L7=llnX;)-1 and find its
c. Is the MLE of a a consistent estimator?
b. Show that the MLE is a function of the complete sufficient statistic for this problem.
a. Define the maximum likelihood estimator of a.
6. A large commercial bank intends to analyze the accuracy with which its bank tellers process cash transactions.In particular, it desires an estimate of the expected proportion of daily cash transactions that the bank tellers process correctly. It plans to analyze 200 past observations on the
'e. Is the MLE the MVUE of f3? Is the MLE asymptotically normally distributed? Asymptotically efficient?
d. Define the maximum likelihood estimator for the parameter f3. Is the MLE a function of the complete sufficient statistic? Is the MLE a consistent estimator of f3?
c. Does there exist an unbiased estimator of f3 whose variance achieves the Cramer-Rao lower bound?
b. Define a set of minimal sufficient statistics for the reparameterized density function. Are the minimal sufficient statistics complete sufficient statistics?
a. Does the reparameterized family of density functions belong to the exponential class of density functions?
The magazine publishes an index defined by f3 = 1/(J to measure the quality of computer chips, where the closer f3 is to zero, the better the computer chip. The joint density of X is reparameterized so that the density function is parameterized by f3, as nX - f3ne-fJE7~lxi n Ilo,ool(Xi), where f3 >
5. The research department of the Personal Computer Monthly magazine is analyzing the operating life of computer chips that are produced by a major manufacturer located in Silicon Valley. The research staff postulates that a random sample of lifetimes being analyzed adheres to the statistical model
e. Discuss any alterations to the specification of the relationship between food expenditure and disposable income that you feel is appropriate for this problem.
d. What is the probability distribution of /32' the leastsquares estimator of f32? Generate the MVUE estimates of the mean and variance of this distribution.(You might find it useful to know that(Y'Y - y'x(x'x)-l x'y)/(4998) = 25.369055.) Given the assumptions of the model, and using the MVUE
c. The actual survey consisted of 5 000 observations and the following summary of th~ data is available;(X'XJ-I = [ .17577542 -.019177442J-.019177442 .0020946798', _ [10759.646J x y - 98598.324 'y'y = 149965.67, 504 Chapter 8 Point Estimation Methods where the x matrix is a (5, 000 x 2) matrix
b. Is the least-squares estimator unbiased, the BLUE, and/or the MVUE for the parameters or functions of parameters being estimated?
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