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MAST20005/MAST90058 Assignment

创建日期：2024-06-04 17:43:39

所属分类：数学mathematics

标签： MAST20005

Assignment

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MAST20005/MAST90058 Assignment

1. This problem involves analysis-of-variance decompositions for regression. Consider the regression model
Yi = α+ βxi + i for i = 1, . . . , n,
where x1, . . . , xn are treated as fixed, and 1, . . . , n are independent errors with
E[i] = 0 and var(i) = σ2 for each i.
We will use the notation x¯ = n−1
∑n
i=1 xi.
(a) For K =
∑n
i=1(xi − x¯)2, recall the analysis-of-variance formula
n∑
i=1
(
Yi − µ(xi)
)2
=
n∑
i=1
(Yi − Yˆi)2 + n(αˆ0 − α0)2 +K(βˆ − β)2.
Show that
cov(αˆ0, Yi − Yˆi) = 0.
(Moral: This completes Week 8’s tutorial problem 2(b).) [1]
(b) Consider the alternative analysis-of-variance formula
n∑
i=1
(Yi − Y¯ )2︸︷︷︸
SS(TO)
=
n∑
i=1
(Yˆi − Y¯ )2︸︷︷︸
SS(R)
+
n∑
i=1
(Yi − Yˆi)2︸︷︷︸
SS(E)
. (1)
(i) Prove that SS(R) is independent of SS(E). Hint: You can take for granted the fact that βˆ is
independent of the SS(E), as shown by Week 8’s tutorial problem 2(b). [1]
(ii) Show that under β = 0, SS(R)/σ2 has a χ21 disbribution. Hint: Use (i), the fact that SS(E)/σ
2 ∼
χ2n−2, and an moment-generating-function trick. [2]
(Moral of (i) and (ii): SS(R)/1SS(E)/(n−2) has a F1,n−2 distribution under β = 0.)
(iii) Recall we can use the t-statisitc
Tβ=0 =
βˆ
σˆ/
√∑n
i=1(xi − x¯)2
to test the hypothesis H0 : β = 0, where σˆ
2 = SS(E)/(n− 2). Show the algebraic identity
T 2β=0 =
MS(R)
MS(E)
,
where MS(R) = SS(R)/1 and MSE(E) = SS(E)/(n− 2). [1]
(Moral: a square of a tn−2-distributed random variable is F1,n−2-distributed. )
(c) Let y1, . . . , yn be the observed values for Y1, . . . , Yn respectively. By defining the sample variances and
covariances sxx = (n− 1)−1
∑n
i=1(xi− x¯)2, syy = (n− 1)−1
∑n
i=1(yi− y¯)2, sxy = (n− 1)−1
∑n
i=1(xi−
x¯)(yi − y¯), one can alternatively write βˆ as
βˆ =
∑n
i=1(xi − x¯)(yi − y¯)∑n
i=1(xi − x¯)2
=
sxy
sxx
MAST20005/MAST90058 Assignment 3 Semester 2, 2023
(i) Show that
n∑
i=1
(yˆi − y¯)2 = [
∑n
i=1(xi − x¯)(yi − y¯)]2∑n
i=1(xi − x¯)2
Hint: Recall αˆ0 = y¯ and yˆi = αˆ0 + βˆ(xi − x¯). [1]
(ii) Recall that the sample correlation r =
sxy√
sxxsyy
. Show that r2 =
βˆsxy
syy
. [1]
(Moral: Together with Week 8 Tutorial problem 5(b), from (i) one can also conclude the analysis of
variance formula (1) holds. Further, with (i) and (ii), one can readily see that r2 =
∑n
i=1(yˆi−y¯)2∑n
i=1(yi−y¯)2 , so
r2 represents the proportion of variation in yi’s explained by the xi’s via regression. )
2. Consider the one-way ANOVA model, where for each group i = 1, . . . , k, we observe ni normally distributed
Xij ∼ N (µi, σ2), j = 1, . . . , ni; observations across the groups are also independent. Let n =
∑k
i=1 ni,
X¯i· = 1ni
∑ni
j=1Xij and X¯·· =
1
n
∑k
i=1
∑ni
j=1Xij =
1
n
∑k
i=1 niX¯i·, and define the “sums of squares”
SS(T ) =
k∑
i=1
ni∑
j=1
(X¯i· − X¯··)2 =
k∑
i=1
ni(X¯i· − X¯··)2,
SS(E) =
k∑
i=1
ni∑
j=1
(Xij − X¯i·)2 =
k∑
i=1
(ni − 1)S2i
and SS(TO) =
k∑
i=1
ni∑
j=1
(Xij − X¯··)2,
as well as the decomposition SS(TO) = SS(T ) + SS(E). Show that, if µ1 = · · · = µk, then SS(T )/σ2 is
χ2k−1 distributed.
Hint: You can take for granted that SS(TO)/σ2 ∼ χ2n−1, SS(E)/σ2 ∼ χ2n−k, and the independence
between SS(E) and SS(T ). Use moment generating functions. [1]
3. Consider a two-way anova model with one observation per cell, where we assume Xij ∼ N (µij , σ2),
i = 1, . . . , a, j = 1, . . . , b are ab independent normal random variables with means defined by
µij = µ+ αi + βj ,
where
∑a
i=1 αi = 0 and
∑n
j=1 βj = 0. Let
X¯·· =
1
ab
a∑
i=1
b∑
j=1
Xij , X¯i· =
1
b
b∑
j=1
Xij , X¯·j =
1
a
a∑
i=1
Xij ,
and consider the null hypothesis
H0B : β1 = · · · = βb = 0
for the main effects of the factor B. Recall the standard decomposition:
a∑
i=1
b∑
j=1
(Xij − X¯··)2︸︷︷︸
SS(TO)
= b
a∑
i=1
(X¯i· − X¯··)2︸︷︷︸
SS(A)
+ a
b∑
j=1
(X¯·j − X¯··)2︸︷︷︸
SS(B)
+
a∑
i=1
b∑
j=1
(Xij − X¯i· − X¯·j + X¯··)2︸︷︷︸
SS(E)
(2)
2
MAST20005/MAST90058 Assignment 3 Semester 2, 2023
(a) Show that the “cross-terms”
C1 =
a∑
i=1
b∑
j=1
(X¯i· − X¯··)(X¯·j − X¯··), (3)
C2 =
a∑
i=1
b∑
j=1
(X¯i· − X¯··)(Xij − X¯i· − X¯·j ,+X¯··) (4)
and
C3 =
a∑
i=1
b∑
j=1
(X¯·j − X¯··)(Xij − X¯i· − X¯·j + X¯··) (5)
all equal zero. It is OK you just show (3) and (4), because one can readily conclude (5) from (4) by
a symmetric argument. [2]
(Moral: Hence, one can conclude the decomposition (2).)
(b) Show that the decomposition below is true:
a∑
i=1
b∑
j=1
(Xij − X¯··)2︸︷︷︸
SS(TO)
= b
a∑
i=1
(X¯i· − X¯··)2︸︷︷︸
SS(A)
+
a∑
i=1
b∑
j=1
(Xij − X¯i·)2.
Hint: Show that the “cross terms” in
∑a
i=1
∑b
j=1((Xij − X¯i·) + (X¯i· − X¯··))2 vanish. [1]
(c) Show that the term σ−2
∑a
i=1
∑b
j=1(Xij − X¯i·)2 is χ2a(b−1) distributed if H0B is true. [2]
(d) Show that SS(B)/σ2 is χ2(b−1) distributed if the hypothesis H0B is true. [2]
(e) Using (b)− (d), show that SS(B)/(b−1)SS(E)/((a−1)(b−1)) is Fb−1,(a−1)(b−1) distributed under H0B. [4]
Hint 1 : You can use the following theorem:
Theorem (Hogg and Craig, 1958). Let Q = Q1 +Q2, where Q, Q1 and Q2 are all quadratic forms
in N independent and identically distributed N (0, σ2) random variables, i.e.
Q =
N∑
i=1
N∑
j=1
cijYiYj and Qk =
N∑
i=1
N∑
j=1
cij,kYiYj for k = 1, 2,
where Y1, . . . , YN ∼i.i.d. N (0, σ2), and {cij , cij,1, cij,2}1≤i,j≤N are certain constant coefficients.
Suppose
• Q/σ2 is χ2r distributed,
• Q1/σ2 is χr1 distributed with r1 ≤ r, and
• Q2 is non-negative.
Then the following conclusions are true:
• Q1 and Q2 are independent, and
• Q2/σ2 is χ2r−r1 distributed.
Hint 2 : For constant coefficients c1, . . . , cN and i.i.d. N (0, σ2) variables Y1, . . . , YN , the expression
E = (c1Y1 + · · ·+ cNYN )2 is necessarily a quadratic form in the Yi’s because it is readily seen to be
so upon expansion; you can take this fact for granted. Another fact you can take for granted: If E1
and E2 are two quadratic forms in Y1, . . . , YN , then E1 + E2 is also a quadratic form in Y1, . . . , YN .
3
MAST20005/MAST90058 Assignment 3 Semester 2, 2023
4. The problem considers further aspects of the Wilcoxon tests.
(a) To test the null hypothesis H0 that a given continuous and symmetrically distributed random variable
X has median 0, we could use the Wilcoxon signed rank test statistic W =
∑n
i=1 sgn(Xi)rank(Xi),
where X1, . . . , Xn is a random sample of X. Under H0, we have learn that W has the same distri-
bution as the sum
∑n
i=1Wi, where W1, . . . ,Wn are independent variables with the properties that
Pr(Wi = i) = Pr(Wi = −i) = 1/2 for each i = 1, . . . , n.
(i) Show that, under H0, the moment generating function E[exp(sW )] of W has the form
2−n
n∏
i=1
(e−si + esi)
for any s ∈ (−∞,∞). Hint: The Wi’s are independent. [1]
(ii) For n = 2, expand the mgf in (i) to show that the null distribution of W is given by
k -3 -1 1 3
Pr(W = k) 1/4 1/4 1/4 1/4
Hint: Recall an mgf uniquely defines a random variable. [1]
(b) (R) Consider again the following dataset from the Week 10 tutorial, where yields from a particular
process using raw materials from two different sources, A and B, are tabulated as:
(A) 75.1 80.3 85.7 77.2 83.9 85.1 86.7 80.5 75.2 70.0
(B) 80.4 83.1 85.6 90.7 86.6 95.9 82.8 80.9 78.7 89.4
With the help of the wilcox.test function in R, perform an exact test at level 0.05 on the null
hypothesis that the source B does not render more yield than the source A. Provide your code and
conclusion. Hint: Read the documentation of wilcox.test if needed. [2]
4
MAST20005/MAST90058 Assignment 3 Semester 2, 2023
5. Let X be a multinomial random vector with number of trials n and success probabilities p1, . . . , pk, where∑k
i=1 pk = 1 for the k categories, i.e. if the integers x1, . . . , xk ≥ 0 are counts of the n trials falling into
the k different categories, then
Pr(X = x) =
n!
x1! . . . xk!
px11 · · · pxkk ,
where x = (x1, . . . , xk)
T . This naturally extends the binomial distribution to more than 2 categories.
Consider a generic null hypothesis H0 that specifies the success probabilities as
p1 = p
0
1, p2 = p
0
2 , . . . , pk = p
0
k,
for some non-negative values p01, . . . , p
0
k that sum to 1. Recall that the p-value is defined to be the
probability of the data being at least as “extreme” as what we have observed, under the null being true.
As such, if the observed value for X is x, it is natural to define the p-value for H0 as
p =
∑
x′:
PrH0 (X=x
′)≤PrH0 (X=x)
PrH0(X = x
′); (6)
the summation means we are summing over all the possible realizations x′ of X whose probability
PrH0(X = x
′) is less than or equal to the probability of observing x, PrH0(X = x), under the null.
The p-value defined in (6) is exact since no distributional approximation is involved.
This problem concerns “exact” versions of the goodness-of-fit tests we have introduced.
(a) (R) Let X1, . . . , Xn be n independent flips of a coin, i.e., Xi ∼iid Be(p), where p is the probability
of having the coin turning up head. We are to test the null hypothesis
H0 : p = 0.6.
Suppose n = 20, and we have observed X =
∑n
i=1Xi to be 13.
(i) Use the binom.test function in R to produce an exact p-value for H0. [1]
(ii) Use the function pbinom to compute the exact p-value, with respect to the formulation in (6);
show your code. You are precluded from using binom.test for this part, but your computed
p-value should match the one you have in (i). [2]
(b) (R) A group of rats, one by one, proceed down a ramp of one of three doors. We wish to test the
hypothesis that
H0 : p1 = p2 = p3 = 1/3,
where pi is the probability that a rat will choose door i, for i = 1, 2, 3. Suppose the rats were sent
down the ramp n = 90 times and the observed cell frequencies for the doors are n1 = 23, n2 = 36 and
n3 = 31 for door 1 to 3 respectively. Use the multinomial.test from the package EMT to perform
an exact test for H0 at level 0.05 and provide your conclusion. [1]
Total marks = 27

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CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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