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MAST20005/MAST90058 Assignment

创建日期：2024-06-03 15:49:24

所属分类：数学mathematics

标签： MAST20005

Assignment

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MAST20005/MAST90058 Assignment

1. Robert inherits an ancient coin from his granddad which is said to be the fairest coin ever minted in
history; he is determined to test whether the coin is indeed fair, by using the number of heads Y turning
up based on 36 independent tosses. Formally, if p is the chance of the coin turning up a head, he tests
the hypothesis
H0 : p = 0.5 vs H1; p 6= 0.5.
His test is such that he would reject H0 whenever |Y − 18| ≥ 4.
(a) (R) Find the size of the test. You should first explicitly identify the particular distribution involved
for its computation, and then show how you can find it using appropriate function(s) in R. [2]
(b) (R) Plot a power function of the test in the range p ∈ [0.5, 1]. Label your y-axis as “power in p” and
x-axis as “p”. [1]
2. Let θˆ be a statistic that estimate a parameter θ. Assume that se(θˆ) is a standard error of θˆ and θˆ−θ
se(θˆ)
is
approximately distributed as N(0, 1) for any θ ∈ R. Let α ∈ (0, 1).
(a) Give an approximate two-sided (1− α)-confidence interval for θ. [1]
(b) Give a test with approximate size α for the null hypothesis H0 : θ = θ0, for a given value θ0 ∈ R.
You have to specify the rejection region and the test statistic used. [1]
(c) Show that H0 is rejected by your test in (b) if and only if θ0 falls outside of your confidence interval
in (a). [1]
(Moral: Any value in a confidence interval can be considered an “acceptable” value for the parameter
θ, and hence there are generally many acceptable values for θ. This gives another reason why modern
hypothesis testing abstains from “accepting a null hypothesis” )
3. Let X1, . . . , Xn be a random sample of X with a population density fθ(x) parametrized by θ ∈ R. Recall
that the Fisher information is defined as
In(θ) = −
n∑
i=1
Eθ
[
∂2
∂θ2
ln fθ(Xi)
]
= −nEθ
[
∂2
∂θ2
ln fθ(X)
]
,
where the expectation is taken with respect to the distribution under the parameter value θ.
(a) Suppose g : R → R is a smooth invertible function, and by defining φ = g(θ), let fφ(x) be the
population density of X1, . . . , Xn re-parametrized in φ. Show that the Fisher information with
respect to φ has the form
In(φ) = In(g(θ)) =
In(θ)(
g′(θ)
)2 ,
where g′(·) is the derivative of the function g(·).
Hint: You can use the facts (i) Eθ
[
d
dθ ln fθ(Xi)
]
= 0 and (ii) if h(φ) = g−1(φ) and h′(·) is the
derivative of h(·), then h′(φ) = (g′(g−1(φ)))−1 = (g′(θ))−1. [3]
(b) Let Y1, . . . , Yn denote a random sample of size n from a Poisson distribution with mean λ. For large
n, find an approximate one-sided 100(1 − α)% confidence interval for g(λ) = e−λ = P (Y = 0) that
is an upper bound, based on normal quantiles.
Hint: Use the fact that the MLE for λ is Y¯ = n−1
∑n
i=1 Yi, In(λ) =
n
λ and part (a). [3]
MAST20005/90058 Assignment 2 Semester 2, 2023
4. Let Y1, . . . , Ym be m random variables, whose means and covariances are denoted by
µi = E[Yi] for 1 ≤ i ≤ m and σij = E[YiYj ]− E[Yi]E[Yj ] for 1 ≤ i, j ≤ m;
in particular, σii = var(Yi). Moreover, define the mean vector and covariance matrix,
µ = (µ1, . . . , µm)
T and Σ = (σij)1≤i≤m
1≤j≤m
,
for the random vector Y = (Y1, . . . , Ym)
T . The variables Y1, . . . , Ym are said to have a joint normal
distribution, if the random vector Y has a probability density function of the form
f(y) =
1
(2pi)m/2
√
det(Σ)
exp
(
−(y − µ)TΣ−1(y − µ)
2
)
for y ∈ Rm.
This is denoted as Y ∼ Nm(µ,Σ) and generalizes the bivariate normal distribution learnt in MAST20004;
in particular, any component Yi of Y is distributed as N(µi, σii). Moreover, if Y ∼ Nm(µ,Σ), it is
well-known properties (so you can take for granted) that
• For any k-by-m constant matrix B ∈ Rk×m, BY ∼ Nk(Bµ,BΣBT ).
• Y1, . . . , Ym are mutually independent if and only if all the off-diagonal entries of Σ are zero.
The following parts assume X1, . . . , Xn is a random sample from a (univariate) normal distribution with
mean µ and variance σ2. In other words, X ∼ Nn(µ1n, σ2In), where X = (X1, . . . , Xn)T , 1n is a column
vector of length n filled with 1’s and In is an n-by-n identity matrix. Let A be an n-by-n matrix defined
by
A =

1√
n
1√
n
1√
n
1√
n
· · · 1√
n
1√
n
1√
2
−1√
2
0 0 · · · 0 0
1√
2·3
1√
2·3
−2√
2·3 0 · · · 0 0
...
...
...
...
...
...
...
1√
(n−1)n
1√
(n−1)n · · · · · · · · ·
1√
(n−1)n
−(n−1)√
(n−1)n

;
in particular, if we use a1, . . . ,an to respectively denote column vectors that are the 1-st to n-th rows of
A so that A = [a1| · · · |an]T , we have a1 = ( 1√n , . . . , 1√n)T , and for 2 ≤ k ≤ n
ak =
(
1√
(k − 1)k , · · · ,
1√
(k − 1)k ,
−(k − 1)√
(k − 1)k︸︷︷︸
k entries
, 0, · · · , 0︸︷︷︸
n−k entries
)T
.
It is easy to see that
AX = (X¯
√
n,U1, U2, · · · , Un−1)T (1)
where U1, U2, . . . , Un−1 are combinations in X1, . . . , Xn with coefficients in A and X¯ = n−1
∑n
i=1Xi.
(a) Check that AAT = In. (Show your steps) [3]
(b) Using (a), conclude that the components in the vector AX are mutually independent normal random
variables with variance σ2. [1]
2
MAST20005/90058 Assignment 2 Semester 2, 2023
(c) Using (a), conclude that
∑n−1
i=1 U
2
i =
∑n
i=1(Xi − X¯)2.
Hint: Note
∑n
i=1(Xi − X¯)2 =
∑n
i=1X
2
i − nX¯2, and that AAT = In also implies ATA = In. [1]
(d) Using (b) and (c), conclude the independence between X¯ and S2 = (n− 1)−1∑ni=1(Xi − X¯)2. [1]
(e) Using (b) and (c), conclude that ∑n
i=1(Xi − X¯)2
σ2
=
(n− 1)S2
σ2
is chi-square distributed with (n− 1) degrees. (This is an alternative way of finding the distribution
of (n−1)S
2
σ2
, where you have done it using mgfs in Tutorial Week 3, question 11.) [1]
5. A random number generator (RNG) can generate numbers according to a normal distribution N(µ, 1),
and it purports that the mean µ for its generated numbers is exactly zero. Robert suspects the latter
claim, and consider the hypotheses
H0 : µ = 0 vs H1 : µ 6= 0.
Robert decides to reject H0 if |X| ≥ Φ−1(0.975), where Φ(·) denotes the standard normal cumulative
distribution function and Φ−1(·) denotes its inverse, by using a single number X requested from the RNG;
he expects this test to have a size of 0.05 given what he has learnt in MAST20005.
Turns out, the RNG is glitchy, but not in the way Robert has suspected: While it can indeed generate
numbers according to the N(0, 1) distribution, a number won’t be reported to Robert until the RNG produces
the first number with absolute value larger than 0.4. Precisely, the RNG will respond to Robert’s request
and generate a first number, called Y1, which is distributed according to N(0, 1). If |Y1| > 0.4, the
RNG will report Y1 to Robert; if |Y1| ≤ 0.4, the RNG will automatically generate another independent
number Y2 ∼ N(0, 1) and only report Y2 to Robert if |Y2| > 0.4. If not (i.e. |Y2| ≤ 0.4), this process will
repeat itself until the RNG produces its first number with absolute value larger than 0.4. Since Robert
is oblivious to this internal mechanism of the RNG, he will take as X the first such Yi with |Yi| > 0.4.
Mathematically, by letting (Yi)
∞
i=1 be a hypothetical sequence of independent N(0, 1) random numbers
being generated, then X = Yt, where t is defined as t ≡ min{i : |Yi| > 0.4}.
(a) (R) Use a simulation experiment with 1 million generated instances of X to numerically approximate
the actual size of Robert’s test. Remember to show your R code and result. [2]
(b) Give an exact mathematical expression for the actual size of Robert’s test. (The evaluated value of
your mathematical expression should be very close to your numerically computed approximate value
in (a).) [2]
(Moral: Selection bias can produce invalid test size)
Total marks = 23

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ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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