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modern mathematical statistics with applications

Questions and Answers of Modern Mathematical Statistics With Applications

b. Using the fact that T has a t-distribution with (n - k)degrees of freedom, define a random interval (21, 22)that satisfies P(l'{3 E (ZI,Z2)) = .95whenn-k = 25.(Hint: Define y~ur ZI and Z2
3. Suppose Ylnxll = xlnxkl{3lkxll + elnxll, where e N([Oj, a2 I) and x has full column rank.a. Show that T = l'(i3 - (3)(S2l'(x'X)-ll)l/2'for any conformable l 1= [01, has at-distribution with (n -
e. Regarding Smith's linear model assumptions, are there other distributional assumptions that you would suggest for consideration besides the uniform distribution? Would this change any of your
d. Given that raw milk sells for $10 per hundredweight, define a BLUE for the expected marginal revenue per cow obtained from administering the growth hormone. If the total cost of the hormone
c. In the outcome of a random sample of size 100, Smith's found that y'y = 4,454,656.3 and x'y =[ 1211,709577.607738] ' ES tI' mat e the parameters 0 f t h e lm' ear model using i3 and S2.
b. Is the estimator S2 of a2 (1) unbiased, (2) consistent, and/or (3) asymptotically normal? Justify your answer.(Hint: The Xt'S occur in a repeating sequence of the numbers I, 2, 3, ... , 10. It
a. Is the least-squares estimator of (3 (1) unbiased, (2)BLUE, (3) consistent, and/or (4) asymptotically normal?Justify your answer.
1 I(z;a, b) = b _ /la,bl(Z)with a = -b, and b is some positive number.The XI values are the dosages of the growth hormone, measured in cc units, at levels defined by Xt = t - 10 trunc C ;01) , t =
2. Smith's Dairy is contemplating the profitability of utilizing a new bovine growth hormone for increasing the milk production of its cows. Smith's randomly selects cows and administers given
. Define an MVUE estimator for the covariance matrix of the least-squares estimator. Estimate the covariance matrix of the least-squares estimator.
d. Define the MVUE for the degree of homogeneity of the production function (i.e., define the MVUE for q.(,B) = {J2 + {Ja. Estimate the degree of homogeneity of the production function using the
c. Is the estimator you used to estimate 0'2 (1) unbiased,(2) asymptotically unbiased, (3) BLUE, (4)MVUE, (5) consistent, and/or (6) Gamma distributed?
b. Is the estimator you used in part (a) to estimate q(,B)(1) unbiased, (2) asymptotically unbiased, (3) BLUE,(4) MVUE, (5) consistent, and/or (6) normally distributed?
a. Transform the production function into a form in which parameters can be estimated using the leastsquares estimator. Estimate the parameters of the transformed model.
1. The daily production of electricity generated by a coal-fired power plant operating in the midwest can be represented as Yj = {Jll~m~JeE;where Yj = quantity of electricity produced on day i,
f. Is the MVUE of [ILl, IL2, ar, ail' consistent?g. If an investor invests $500 in each of the two investments, what is the MVUE of her expected dol·lar return on the investment during the
Calculate the MVUE outcome for [ILl, IL2, ar, ail'.
e. A random sample of size 50 has an outcome that is summarized by x = [.048 .077]', si = .5 x 10-3, s~ = .3 X 10-4, and S12 = .2 X 10-4
d. Define the MVUE for the vector [ILl, IL2, ar, ail'.
c. Define the MVUE for E and for diag(E).
b. Define the MVUE for IL.
. Find a minimal, complete (vector) sufficient statistic for N(IL, E).
15. The rates of return per dollar invested in two common stocks over a given investment period can be viewed as the outcome of a bivariate normal distribution N(IL, E). The rates are independent
d. Define an MVUE for the number of defective toys in the shipment, and provide an MVUE estimate of this number.
c. Suppose that the outcome of X was 3. Define a MVUE estimate of the proportion of defectives in the shipment.
b. Define the MVUE for the proportion of defectives in the shipment.
·a. Show that X - fIx; 8) is a minimal, complete sufficient statistic for fIx; 8).
14. An incoming shipment of 1000 toys from a toy manufacturer is received by a large department store for a pre-Christmas sale. The store randomly samples 50 toys from the shipment, without
In a mean square error sense, which estimator would you prefer for estimating 02, t(X), or t*(XJ? Note that J..L~ = r!Or for an exponential PDF.
13. In problem 12(b), above, consider the alternative estimator t*(X) = 52 for estimating 02
d. The random sample (XI,'" ,Xn) is generated from a Bernoulli population distribution. The estimator t(X) = X( 1 - XJ will be used to estimate var(Xi) =pl1 - pJ. (Hint: 2 L7=1 L7>i a = n(n - l)a).
c. The random sample (XI, .. "XnJ is generated from a geometric population distribution. The estimator t(X) = (52 + X) will be used to estimate EX; = p-2.
b. The random sample (XI, ... , Xn) is generated from an exponential population distribution. The estimator t(XJ = (1/2) L~=I X'fIn will be used to estimate var(Xi) = 02 •
a. The random sample (XI," "Xn) is generated from a Gamma population distribution. The estimator t(X) = L~=I Xdn will be used to estimate EXi = a{3.
12. In each case below, determine whether the estimator under consideration is unbiased, asymptotically unbiased, and/or consistent.
Generate an estimate of q(J..L,a2l!zxll = [~ ] using the MVUE of qlJ..L, a 2 ). Generate an estimate of the mean of the lognormal distribution using a consistent estimator.
f. An overworked, underpaid, gaunt-looking research assistant hands you an envelope that contains only summary information on the results of the experiment.In particular, the information is that 100
e. Define a consistent estimator of qlJ..L,a) = e!1+u2/2, which is the mean of the log-normal population.Justify your answer. (The MVUE of the mean exists, but it is quite complicated to define and
d. Is t2(Y) = (n - 1)-1 L~=dln Yi - n-I L~=lln Yd2 the MVUE of the parameter a2 ? Is it consistent?
b. Are the sufficient statistics you defined in (a) complete sufficient statistics?c. Is tl(Y) = n- I L~=I In Yi the MVUE of the parameter J..L? Is it consistent? Why?
a. Define minimal sufficient statistics for fly; J..L, a 2 ).
11. One hundred one-acre test plots are being used to assess the yield potential of a new variety of wheat genetically engineered by Washington State University.The actual yield-per-acre observations
f. If LJ~ Xi = 257, generate a MVUE estimate of the variance of waiting times.
e. Is the sample variance, S~, the MVUE for the population variance 02?
d. Define the MVUE of 02 by finding an appropriate function of the complete sufficient statistic-if you can.
c. Does there exist an unbiased estimation of 02 whose variance is equal to the CRLB?
b. Define the CRLB for unbiasedly estimating 02•
a. Define a complete sufficient statistic for fIx; 0).
10. Suppose random sampling is from an exponential population distribution representing the waiting time between customer arrivals at a bank, so that the statistical model is given by nI(x; 0) =
n. What, if anything, can you say to the Detroit manufacturer to convince the company to buy your trigger mechanisms?
1. Define an asymptotic distribution for the estimator(Xn)-I of p. Is the estimator asymptotically efficient?m. Use the estimator (Xn)-I to estimate p, and use this estimated value in the asymptotic
What is your estimate of the expected number of impacts needed to obtain the first failure?k. Is (Xn)-l a consistent estimator of p, the probability that a trigger successfully signals deployment of
j. The 10,000 observations resulted in L:~'IOOO Xi =1.5 X 107
i. Define an appropriate asymptotic distribution for the sample mean in this case. Is the sample mean asymptotically efficient?
h. Use Theorem 7.17 on the attainment of the Cramer-Rao lower bound to derive the MVUE of the expected number of impacts needed to obtain the first failure.
g. Is the sample mean a MVUE? Why?
f. Derive the Cramer-Rao lower bound for the variance of unbiased estimators of the expected number of impacts to obtain the first failure. (Hint: You may use the alternative (second-derivative) form
e. Is the sample mean a BLUE (or equivalently, a MVLUE)? Why?
d. Is the sample mean a consistent estimator? Why?
c. Is the sample mean an asymptotically unbiased estimator?Why?
b. Is the sample mean an unbiased estimator in this case? Why?
a. Define an appropriate statistical model for the sampling experiment, and justify your choice.
One large Detroit automobile manufacturer said that it would be willing to purchase trigger mechanisms from your company if you could provide convincing support for the statement that, in repeated
9. Your company sells trigger mechanisms for air bags that are used in many modern domestic and foreignbuilt passenger cars. The reliability of such trigger mechanisms is obviously critical in the
e. A random sample of size n = 1000 from I(z; 8) results in L:~ Xi = 4,100. Use your estimatorto estimate the value of 8. Usingyourestimateof8, what is the estimated probability that z E (4.05, 4.15)?
d. Define an asymptotic distribution for your estimator.
c. Is your estimator a consistent estimator for 8? Why or why not?
b. Is the estimator you defined in (a) a BLUE for 8? If not, find a BLUE for 8, if it exists.
a. Based on the random sample, define an unbiased estimator of the parameter 8.
,Xn) from the population distribution I(z; 8) to answer the questions below.
8. The diameters of blank compact disks manufactured by the Dandy Disk Co. can be represented as outcomes of a random variable 1Z ~ I(z; 8) = 8114,4+Eldz), for some 8 > 0, where z is measured in
d. Can you foresee any practical problems in using t(X)to generate estimates of the population mean?424 Chapter 7 Elements of Point Estimation Theory
c. Define the MSEs of the estimators. Is there any validity to the statement that "for an appropriate choice of k" t(X) will be superior to X in terms of MSE? Explain.
b. Define asymptotic distributions for both estimators.On the basis of their asymptotic distributions, do you favor one estimator over the other?
a. We know that X is unbiased, asymptotically unbiased, BLUE, and consistent for estimating the mean of the population distribution. Which of these properties apply to the alternative estimator?
7. Two economics professors are arguing about the appropriate estimator to use in estimating the mean of the population distribution of incoming freshmen's LQ.'s.The estimators will be based on a
e. Polly summarized the outcome of the random sample as 'L:~~ Xi = 670, where Xj = 1 indicates that the ith sample voter was in favor of the initiative, and Xi = 0 otherwise. Estimate the proportion
d. Assume there are two million voters in the state.What is the probability that the estimator you defined in (b) generates an estimate that is within ±.03 of the true proportion of voters favoring
c. Is the estimator that you defined in (b) a consistent estimator of the proportion of voters in favor of the initiative? Is it a CAN estimator? Is it asymptotically efficient?
b. Define the MVLUE for the proportion of voters in favor of the initiative. Justify that your estimator really is a MVLUE.
a. Define a statistical model for the problem of estimating the proportion of voters in favor of the ini·tiative.423
6. Polly Pollster wants to estimate the proportion of voters in the state of Washington that are in favor of an antitax initiative. She will be using a random sample(with replacement) of 1,000
e. Define the CRLB for estimating P(z = 0) = e-)..Does there exist an unbiased estimator of e-). that achieves the CRLB? Why or why not?
d. Is X a member of the CAN class of estimators? Is X asymptotically efficient?
c. Use the CRLB attainment theorem to derive the MVUE for estimating A. Suppose n = 100 and'L:~ Xj = 283. Estimate A using the MVUE.
b. Derive the CRLB for unbiased estimation of the parameter A. Is X the MVUE for estimating A? Why or why not?
a. Show that the Cramer-Rao lower bound regularity conditions hold for the joint density of the random sample.
5. The number of customers that enter the corner grocery store during the noon hour has a Poisson distribution, i.e., e-). AZ I(z; A) = -,-I{O,I,l,3, ... )(z). z.Assume that (X1,Xl,oo.,Xn ) is a
(b) attain the Cramer-Rao lower bound? (The CRLB regularity conditions hold for the joint density of the random sample. Furthermore, the alternative form of the CRLB, expressed in terms of
d. Does the variance of the estimator you defined in
c. Is the MVUE estimator a consistent estimator of EZ = 48? Why or why not?
b. Define the MVUE for estimating EZ = 48. Justify the MVUE property of your estimator. Generate an estimate of EZ using the MVUE.
a. Define a minimal, complete sufficient statistic for estimating the expected operating life of the electric motors produced by the AJAX CO.
4. The operating life of a small electric motor manufactured by the AJAX Electric Co. can be represented as a random variable having a probability density given as Z ~ f(z;e) =
1 fIx;e) = elIO,BI(X).g. You are sampling from a Beta distribution 1f(Xi 11., P) = B( IX, p) x·-1 (1 - x)P-l ~(O, 1) (x).
e. You are random sampling from an N(Il, a2 ) population distribution.£. You are random sampling from a continuous uniform density
d. You are random sampling from a negative binomial density f( )- (x-I)! 10(1 )X-101 ( )xiIO,P - (IO -1)!(x- Io)! P - P (10.10+1 .... ) X I where IO is a known positive integer.
c. You are random sampling from a Poisson population distribution e-A).x fIx; ).) = -,-x1. 10,1,2, ... I(X).
b. You are random sampling from a "power function"population distribution given by f(z; ).) = ).zA-1IIO,II(Z), where). > O.
a. You are random sampling from a log-normal population distribution given by f(z; Il, a 2) = (2]f)~/2az exp ( - 2~2 (lnz - 1l)2) 110,001(z), where Il E (-00,00) and a2 > O.

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