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 on the test plots can be viewed as iid observations from some log-normal population, so that the statistical model for the experiment is given by fðy; m; s2Þ ¼ 1

ð Þ 2p n=2 sn Qn i¼1 yi exp  1 2s2 S n

i¼1

ðlnðyiÞ  m Þ

2 P

n i¼1 Ið0;1Þ ðyiÞ

for m 2 ð1;1Þ and s>0:

a. Define minimal sufficient statistics for f(y;m,s2

).

b. Are the sufficient statistics you defined in

(a) complete sufficient statistics?

c. Is t1(Y) ¼ n1 Pn i¼1 lnð Þ Yi the MVUE of the parameter m? Is it consistent? Why?

d. Is t2(Y) ¼ (n1)1Pn i¼1 lnð Þ Yi n1Pn i¼1 lnð Þ Yi

 2 the MVUE of the parameter s2

? Is it consistent?

e. Define a consistent estimator of qð Þ¼ m; s emþs2=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 calculate. See D.

J. Finney (1941), “On the distribution of a variate whose logarithm is normally distributed.” Roy. Statistical Society, Series B, 7, pp. 155–161.)

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 X

100 i¼1 lnð Þ Yi ¼ 375:00;

X 100 i¼1

ðln ðYiÞÞ2 ¼ 1455:75:

Generate an estimate of q m; s2  

ð Þ 21 ¼ m s2 using the MVUE of q(m,s2

). Generate an estimate of the mean of the lognormal distribution using a consistent estimator.