Applied Statistical Methods

It’s homework for Applied Statistical Methods. Please do all except the last question(PhD problem)..Please read all questions carefully. The accuracy of the assignment must be 98%+. Only bid if you are very confident that you can answer.

Homework 7
Due Thursday, 3/25/21 on Canvas.
1. Let p1; : : : ; pm be p-values for null hypotheses H0;1; : : : ; H0;m, where pj  U[0; 1] if H0; j is
true. Define 0 2 [0; 1] to be the fraction of the m hypotheses that are true.
(a) Show that the Bonferroni procedure controls the FWER regardless of the dependence
between p1; : : : ; pm. That is, show that for any  2 (0; 1),
P

9 j = 1; : : : ;m such that H0; j is true and pj  =m

 0:
(b) If p1; : : : ; pm are independent, show that the Bonferroni procedure provides exact control
of the FWER for small . That is, show that for any m  1 and significance level
2 (0; 1),
P

9 j = 1; : : : ;m such that H0; j is true and pj  =m

= 0 f1 + o(1)g
as  ! 0.
(c) Is (b) necessarily true if p1; : : : ; pm are dependent? If yes, prove it. If not, find a
counterexample.
2. (Preliminaries for estimation in random eects models) Let Y 2 Rn be a random vector with
E(Y ) = 0 and Var(Y ) = () 2 Rnn, where  2 Rp is an unknown parameter. Assume
that () is continuously dierentiable as a function of , i.e. there exist continuous matrix
functionsMj() 2 Rnn for j = 1; : : : ; p such that
( + ) ? () =
Xp
j=1
jMj() + o (kk2) :
Let
` () = ?
1
2
log [det f()g] ?
1
2
Y T f()g?1 Y
be the log-likelihood for the normal distribution (up to additive constants that do not depend
on ). Note that we are NOT assuming Y is normally distributed.
(a) Let V 2 Rnn be a symmetric, positive definite matrix. For any symmetric matrix
A 2 Rnn, prove the following:
log fdet (V + A)g ? log fdet(V )g =  Tr

AV ?1

+ o()
uT (V + A)?1 u ? uTV ?1u = ?uTV ?1AV ?1u + o()
for  > 0 and u 2 Rn.
1
(b) Use part (a) to show that
[r` ()]j = ?
1
2
Tr
h
Mj () f()g?1
i
+
1
2
Y T f()g?1Mj () f()g?1 Y :
Conclude that a root of r` () is a suitable estimator for . That is, show that
E fr` ()g = 0
regardless of whether or not Y is normally distributed.
3. A study was done to compare the yields of 56 varieties of wheat in a randomized complete
block design (RCBD) with four blocks of size 56. The data for the experiment are in the file
“wheat56.txt”. The four blocks are observations 1-56, 57-112, 113-168 and 169-224. The
varieties, yields, latitudes and longitudes of each plot (latitudes and longitudes in unstated
units) are given. Although the units are unstated, keep in mind that agricultural field trials
like this are carried out at a single farm so that the weather is essentially the same at all
plots. The labeling of the varieties as 1-56 “in order” in Block 1 is for convenience; you may
assume that in fact the variety assignments were properly randomized in all four blocks.
(a) Estimate the variety eects using the standard model for an RCBD treating blocks
and varieties as fixed eects. Using appropriate tables and/or plots, summarize your
findings about the dierences between varieties. As part of your analysis, include an
F-test for the hypothesis of no variety eects.
(b) Find a 95% confidence interval for the mean yield of varieties 1-20 minus the mean
yield of varieties 21-56.
(c) Plot the residuals as a function of the geographic coordinates of the plots. Discuss any
patterns you see and comment on the reasonableness of the assumptions underlying the
analyses in (a). Can you identify any varieties whose yields (relative to other varieties)
might be over or underestimated because of the plots to which they were assigned?
Comment.
(d) Reanalyze the data including a linear function of the coordinates in your mean function.
What eect does this change have on your inferences about variety eects? In
particular, which estimated variety eects change the most from the analysis in (c)?
Plot the residuals as a function of the geographic coordinates of the plots. To what
extent are any problems you noted with the residual plot in (c) fixed?
(e) Answer the same questions as in (d), but this time including a quadratic function of the
coordinates (i.e., a second order polynomial in latitude and longitude) in your mean
function.
(f) Do you think the design used for this study was well-chosen? Discuss any problems
you see and describe how the study might have been designed dierently to avoid or
reduce these problems.
2
4. PhD problem: Proof of She´e. Let X 2 Rnp be a full rank design matrix and Y 
N

X;2In

for some  2 Rp. Let S =
n
c 2 Rp :
Pp
j=1 cj = 0
o
.
(a) If ˆ is the the ordinary least squares estimate for , show that for any  2 (0; 1),
P
2666666664
8>>><>>>:
cT ˆ ? cT
se

cT ˆ

9>>>=>>>;
2
 q1?8c 2 S
3777777775

where se

cT ˆ

=
q
ˆ 2cT (XTX)?1 c and q1? is the 1? quantile of the (p?1)Fp?1;n?p
distribution.
(b) Use (a) to derive simultaneous 1? confidence intervals for every point in
n
cT : c 2 S
o
.
3

Variety Yield Latitude Longitude 1 585 4.3 19.2 2 631 4.3 20.4 3 701 4.3 21.6 4
602 4.3 22.8 5 661 4.3 24.0 6 605 4.3 25.2 7 704 4.3 26.4 8 388 8.6 1.2 9 487 8.6
2.4 10 511 8.6 3.6 11 502 8.6 4.8 12 492 8.6 6.0 13 509 8.6 7.2 14 268 8.6 8.4 15
633 8.6 9.6 16 513 8.6 10.8 17 632 8.6 12.0 18 446 8.6 13.2 19 684 8.6 14.4 20 422
8.6 15.6 21 560 8.6 16.8 22 566 8.6 18.0 23 514 8.6 19.2 24 635 8.6 20.4 25 840
8.6 21.6 26 618 8.6 22.8 27 658 8.6 24.0 28 481 8.6 25.2 29 564 8.6 26.4 30 597
12.9 1.2 31 580 12.9 2.4 32 418 12.9 3.6 33 526 12.9 4.8 34 517 12.9 6.0 35 479
12.9 7.2 36 506 12.9 8.4 37 542 12.9 9.6 38 513 12.9 10.8 39 504 12.9 12.0 40 368
12.9 13.2 41 437 12.9 14.4 42 540 12.9 15.6 43 631 12.9 16.8 44 610 12.9 18.0 45
639 12.9 19.2 46 611 12.9 20.4 47 545 12.9 21.6 48 598 12.9 22.8 49 656 12.9 24.0
50 557 12.9 25.2 51 486 12.9 26.4 52 563 17.2 1.2 53 539 17.2 2.4 54 502 17.2 3.6
55 605 17.2 4.8 56 403 17.2 6.0 9 556 17.2 8.4 16 569 17.2 9.6 4 455 17.2 10.8 32
534 17.2 12.0 42 513 17.2 13.2 45 549 17.2 14.4 14 620 17.2 15.6 17 498 17.2 16.8
39 513 17.2 18.0 43 648 17.2 19.2 50 624 17.2 20.4 56 552 17.2 21.6 5 693 17.2
22.8 1 570 17.2 24.0 28 589 17.2 25.2 3 611 17.2 26.4 36 536 21.5 1.2 47 477 21.5
2.4 26 548 21.5 3.6 30 602 21.5 4.8 35 495 21.5 6.0 31 507 21.5 7.2 10 520 21.5
8.4 27 500 21.5 9.6 33 587 21.5 10.8 13 572 21.5 12.0 19 534 21.5 13.2 38 505
21.5 14.4 55 675 21.5 15.6 48 446 21.5 16.8 29 561 21.5 18.0 34 691 21.5 19.2 12
748 21.5 20.4 6 580 21.5 21.6 44 624 21.5 22.8 37 742 21.5 24.0 22 590 21.5 25.2
24 627 21.5 26.4 49 404 25.8 1.2 8 528 25.8 2.4 41 513 25.8 3.6 46 638 25.8 4.8 7
621 25.8 6.0 11 615 25.8 7.2 51 543 25.8 8.4 18 606 25.8 9.6 54 634 25.8 10.8 15
610 25.8 12.0 20 487 25.8 13.2 23 522 25.8 14.4 25 599 25.8 15.6 21 656 25.8 16.8
2 563 25.8 18.0 52 654 25.8 19.2 53 738 25.8 20.4 40 368 25.8 21.6 34 623 25.8
24.0 23 539 25.8 25.2 50 616 25.8 26.4 42 438 30.1 1.2 8 592 30.1 2.4 18 485 30.1
3.6 3 542 30.1 4.8 19 421 30.1 6.0 11 479 30.1 7.2 31 546 30.1 8.4 36 600 30.1 9.6
28 690 30.1 10.8 17 662 30.1 12.0 56 564 30.1 13.2 40 516 30.1 14.4 54 679 30.1
15.6 24 607 30.1 16.8 51 378 30.1 18.0 29 678 30.1 19.2 37 675 30.1 20.4 45 679
30.1 21.6 38 500 30.1 22.8 1 562 30.1 24.0 44 500 30.1 25.2 21 606 30.1 26.4 4
337 34.4 1.2 52 342 34.4 2.4 46 191 34.4 3.6 13 30 34.4 4.8 39 255 34.4 6.0 26 443
34.4 7.2 20 384 34.4 8.4 53 471 34.4 9.6 5 501 34.4 10.8 10 665 34.4 12.0 41 480
34.4 13.2 16 635 34.4 14.4 27 481 34.4 15.6 15 769 34.4 16.8 22 517 34.4 18.0 6
656 34.4 19.2 25 702 34.4 20.4 32 621 34.4 21.6 55 663 34.4 22.8 47 580 34.4 24.0
33 643 34.4 25.2 2 818 34.4 26.4 30 360 38.7 1.2 7 43 38.7 2.4 43 75 38.7 3.6 48
59 38.7 4.8 9 174 38.7 6.0 14 221 38.7 7.2 49 247 38.7 8.4 35 449 38.7 9.6 12 538
38.7 10.8 40 471 38.7 13.2 41 580 38.7 14.4 49 553 38.7 15.6 44 480 38.7 16.8 9
515 38.7 18.0 25 471 38.7 19.2 42 613 38.7 20.4 11 564 38.7 21.6 1 568 38.7 22.8
7 574 38.7 24.0 34 515 38.7 25.2 37 450 38.7 26.4 52 185 43.0 1.2 30 486 43.0 2.4
15 99 43.0 3.6 2 74 43.0 4.8 26 294 43.0 6.0 35 272 43.0 7.2 36 246 43.0 8.4 6 350
43.0 9.6 4 303 43.0 10.8 47 471 43.0 12.0 32 390 43.0 13.2 48 530 43.0 14.4 18
416 43.0 15.6 10 506 43.0 16.8 50 348 43.0 18.0 39 453 43.0 19.2 17 632 43.0 20.4
56 339 43.0 21.6 33 625 43.0 22.8 46 473 43.0 24.0 27 509 43.0 25.2 23 549 43.0
26.4 19 291 47.3 1.2 21 121 47.3 2.4 16 21 47.3 3.6 31 128 47.3 4.8 45 102 47.3
6.0 43 356 47.3 7.2 38 443 47.3 8.4 8 307 47.3 9.6 54 240 47.3 10.8 5 500 47.3
12.0 29 442 47.3 13.2 3 586 47.3 14.4 14 469 47.3 15.6 55 558 47.3 16.8 12 632
47.3 18.0 53 604 47.3 19.2 24 606 47.3 20.4 20 406 47.3 21.6 51 593 47.3 22.8 13
531 47.3 24.0 28 512 47.3 25.2 22 538 47.3 26.4
1

 
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