STAT 613 Exam 1February 21, 2018

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INSTRUCTIONS:

• This is an closed-book, closed-notes exam. However, a formula page is provided at the back.

• Total time is 75 minutes (3:00 P.M to 4:15 P.M.)

• Show all work, clearly and in order, if you want to receive full credit. When you use yourcalculator, explain all relevant mathematics. I reserve the right to take o↵ points if I cannotsee how you arrived at your answer (even if your final answer is correct).

• Circle or otherwise indicate your final answers.

• Answer all the questions in the space provided. You may attach additional sheets

if necessary.

• This test has 4 problems and is worth 80 points. It is your responsibility to make sure thatyou have all of the problems.

• Good luck!

Prob. No. Max Points Earned Pts.

1 20

2 20

3 20

4 20

TOTAL:

Question 1. (20 pts.) Let X1, X2, . . . , Xn be i.i.d discrete uniform distribution on {1, 2, . . . , N}where N is an unknown parameter taking values in the set of all positive integers, i.e. {1, 2, . . .}.Recall that P (X1 = j | N) = 1/N for all j = 1, . . . , N .

a) (8 points) Find a minimal su�cient statistic for N . No need to actually show that it is minimalsu�cient.

b) (12 points) Is the statistic obtained in a) complete? Explain your answer.

a) to ( a ,. - . ixnln ) = ntn 1[×a, en ]

.

Hence Xcn ,is an Mss

b)

Tosho=×cn)isC.

for N 71 , Re 1,2 ,.

. . in

PN ( Xa , =k) = Pn ( Xcn ,E K ) - PN ( Xcn )

Eh ' )

= an . #"

1 ifN =L

,k=1

Henn. Pnlxurr ) = { ⇐,n . rent it III.In

Start with g : { 1,2 ,.

. - . } → R such that

En { g(×a , ) } =° tn=i , 3 . --

SetN= ⇒ g ( i ) = 0

Sane ⇒ gG÷ +gc 2) ( I - In )=o ⇒ g (2) =o

Setty ⇒ gg , p(×a , =D + 9k )P( xcn )- 2) +9C 3)

PCXLN)=3 )

⇒ g (3) = 0 since PCXAD = 3) 40 . =o

Continuing like this gck ) = O t K =L , 4 . --

⇒ PN ( g Cxcn , ) =o ) = it N 31

⇒ xcn ) is CSS

Question 2. (20 pts.) Suppose we take one observation, X, from the discrete distribution, withprobability mass function given in the following Table where ✓ 2 [0, 1].

Table 1: Probability mass function of X

x �2 �1 0 1 2P (X = x | ✓) (1� ✓)/4 ✓/12 1/2 (3� ✓)/12 ✓/4

a) (4 points) Find a real-valued statistic T (X) with E✓[T (X)] = ✓ (T is an unbiased estimator of✓, but might take values outside [0, 1]).

b) (8 points) Obtain the maximum likelihood estimator (MLE) ✓(X) of ✓ and show that it is notnecessarily unique.

c) (8 points) Is any choice of MLE unbiased?

a) Let TGD = 12 if ×= -1

= 0 otherwise

to [ Tcx ) ] = 12 P(×= - , ) = 12 . 01,2=0

b) MLE depends onwhich X you

observe

of C- 2) =0

,€ ( t ) =i

,£ ( o ) is any

number in

[oD

EG ) =o,

I (a) =L

C)@ For MLE to be unbiased ,we must have

to [ Ecx ) ] = a t off ,i ]

⇒ o÷ + €62 + Eat °

@ ( → =a÷⇒ 8¥)

=

2¥⇒

→ contradiction Sinn

estimator Can't defend on

parameter .

Question 3. (20 points) Let X1, X2, . . . , Xn be iid N(µ,�2) with unknown mean µ 2 R and knownvariance �2 > 0. Recall that X = (1/n)

Pni=1 Xi is a complete and su�cient statistic for µ.

a) (8 points) For fixed and known t 6= 0, find the uniformly minimum variance unbiased estimatorof etµ.

b) (12 points) Show that the variance of the uniformly minimum variance unbiased estimatorobtained in a) does not achieve the Cramer-Rao lower bound, but it is asymptotically e�cient.

a) In Mm , oyn ) e[ etx ] = etnt t÷I

⇒ E [ et'T - t÷T ] =etr

⇒ et× ' t 'Tis fmuiowof c Ss & unbiased for

etm,

So UMVUE for etm .

b)CRLB = Seenetm }

2

teenage 7¥,

= need

Fnwgf =" In Var ( etx - t÷ )then

= e. ¥ { ecentx ) -ETETD }

= e- t¥ { ezntt' ' IF

.

ezrtt tiny= eat { et¥ . , }

£252

µm var ( et 'T - II )=

et -1=

z- it -

n → A CRLBn , A t~÷

So UMWE is asymptotically efficient .

Question 4. (20 points) Suppose X1, . . . , Xn are independent, with Xi ⇠ N(✓i, 1), i = 1, . . . , n.In vector notation, we may write X ⇠ Nn(✓, In), where X = (X1, . . . , Xn)T is the observable,✓ = (✓1, . . . , ✓n)T is the unknown mean vector and In is the n⇥ n identity matrix.

a) (12 points) Find the MLE of = k✓k2, the squared norm of ✓ (prove that it is the MLE !).

b) (2 points) Show that the MLE of is biased.

c) (3 points) What happens to the bias as n ! 1? Explain the phenomenon.

d) (3 points) Find the Cramer-Rao lower bound for an unbiased estimator of .

a) l ( AID = . nzhog th - ÷ ( x - a)'

( x - o )

% KMD = +( x . a) =o ⇒ d '

- X

- E[off ,log A Cola ] = In .n .

d- . xisthe MLE . By invariant ,

⇒ zx ,

.2 is he MLE for

4

b) E ( Exp ) = [ oi2 in

= 4 + n

Bias = Ettzxiy - Y =n

c) |Bias| → a as n→& .

This is because #

of parameters dependon n .

- )

d) 04 = 20 .

CRLB =JYTICO ) 04

= 4110112

Table of Distributions

Distribution PMF/PDF and Support Expected Value Variance MGF

Bernoulli

Bern(p)

P (X = 1) = p

P (X = 0) = q = 1� p p pq q + pet

Binomial

Bin(n, p)

P (X = k) =�nk

�pkqn�k

k 2 {0, 1, 2, . . . n} np npq (q + pet)n

Geometric

Geom(p)

P (X = k) = qkp

k 2 {0, 1, 2, . . . } q/p q/p2 p1�qet

, qet < 1

Negative Binomial

NBin(r, p)

P (X = n) =�r+n�1

r�1

�prqn

n 2 {0, 1, 2, . . . } rq/p rq/p2 (p

1�qet)r, qet < 1

Hypergeometric

HGeom(w, b, n)

P (X = k) =⇣wk

⌘⇣b

n�k

⌘/⇣w+bn

⌘

k 2 {0, 1, 2, . . . , n} µ =nwb+w

⇣w+b�nw+b�1

⌘nµ

n (1� µn ) messy

Poisson

Pois(�)

P (X = k) = e���k

k!

k 2 {0, 1, 2, . . . } � � e�(et�1)

Uniform

Unif(a, b)

f(x) = 1b�a

x 2 (a, b) a+b2

(b�a)2

12etb�eta

t(b�a)

Normal

N (µ,�2)

f(x) = 1�p2⇡

e�(x � µ)2/(2�2)

x 2 (�1,1) µ �2 etµ+�2t2

2

Exponential

Expo(�)

f(x) = �e��x

x 2 (0,1)1�

1�2

���t , t < �

Gamma

Gamma(a,�)

f(x) = 1�(a) (�x)

ae��x 1x

x 2 (0,1)a�

a�2

⇣�

��t

⌘a, t < �

Beta

Beta(a, b)

f(x) = �(a+b)�(a)�(b)x

a�1(1� x)b�1

x 2 (0, 1) µ =a

a+bµ(1�µ)(a+b+1) messy

Log-Normal

LN (µ,�2)

1x�

p2⇡

e�(log x�µ)2/(2�2)

x 2 (0,1) ✓ = eµ+�2/2 ✓2(e�2 � 1) doesn’t exist

Chi-Square

�2n

12n/2�(n/2)

xn/2�1e�x/2

x 2 (0,1) n 2n (1� 2t)�n/2, t < 1/2

Student-ttn

�((n+1)/2)pn⇡�(n/2)

(1 + x2/n)�(n+1)/2

x 2 (�1,1) 0 if n > 1n

n�2 if n > 2 doesn’t exist