Post on 21-Dec-2015
1
STATISTICAL INFERENCEPART II
SOME PROPERTIES OF ESTIMATORS
SOME PROPERTIES OF ESTIMATORS
• θ: a parameter of interest; unknown
• Previously, we found good(?) estimator(s) for θ or its function g(θ).
• Goal:
• Check how good are these estimator(s). Or are they good at all?
• If more than one good estimator is available, which one is better?
2
3
SOME PROPERTIES OF ESTIMATORS
• UNBIASED ESTIMATOR (UE): An estimator is an UE of the unknown parameter , if
ˆE for all
Otherwise, it is a Biased Estimator of .
ˆ ˆBias E
Bias of for estimating
If is UE of , ˆ 0.Bias
4
SOME PROPERTIES OF ESTIMATORS
• ASYMPTOTICALLY UNBIASED ESTIMATOR (AUE): An estimator is an AUE of the unknown parameter , if
ˆ ˆ0 lim 0n
Bias but Bias
5
SOME PROPERTIES OF ESTIMATORS
• CONSISTENT ESTIMATOR (CE): An estimator which converges in probability to an unknown parameter for all is called a CE of .
ˆ .p
• MLEs are generally CEs.
For large n, a CE tends to be closer to the unknown population parameter.
6
EXAMPLE
For a r.s. of size n,
E X X is an UE of . By WLLN,
pX
X is a CE of .
7
MEAN SQUARED ERROR (MSE)• The Mean Square Error (MSE) of an
estimator for estimating is
22ˆ ˆ ˆ ˆMSE E Var Bias
If is smaller, is a better estimator
of .
ˆMSE
1 2ˆ ˆ ,For two estimators, and of if
1 2ˆ ˆ ,MSE MSE
1 is better estimator of .
8
MEAN SQUARED ERROR CONSISTENCY
is called mean squared error consistent (or consistent in quadratic mean) if E{ }2 0 as n.
Theorem: is consistent in MSE iff
i) Var( )0 as n.
If E{ }20 as n, is also a CE of .
)ˆ(lim) Eii
n
9
EXAMPLES
X~Exp(), >0. For a r.s of size n, consider the following estimators of , and discuss their bias and consistency.
1ˆ,ˆ 12
11
n
X
n
Xn
ii
n
ii
10
SUFFICIENT STATISTICS
• X, f(x;),
• X1, X2,…,Xn
• Y=U(X1, X2,…,Xn ) is a statistic.
• A sufficient statistic, Y, is a statistic which contains all the information for the estimation of .
11
SUFFICIENT STATISTICS
• Given the value of Y, the sample contains no further information for the estimation of .
• Y is a sufficient statistic (ss) for if the conditional distribution h(x1,x2,…,xn|y) does not depend on for every given Y=y.
• A ss for is not unique:
• If Y is a ss for , then any 1-1 transformation of Y, say Y1=fn(Y) is also a ss for .
12
SUFFICIENT STATISTICS• The conditional distribution of sample rvs
given the value of y of Y, is defined as
1 21 2
; , , ,, , ,
;n
n
L x x xh x x x y
g y
1 21 2
, , , , ;, , ,
;n
n
f x x x yh x x x y
g y
• If Y is a ss for , then
1 21 2 1 2
; , , ,, , , , , ,
;n
n n
L x x xh x x x y H x x x
g y
ss for may include y or constant.
Not depend on for every given y.
• Also, the conditional range of Xi given y not depend on .
13
SUFFICIENT STATISTICS
EXAMPLE: X~Ber(p). For a r.s. of size n, show that is a ss for p.
n
1iiX
14
SUFFICIENT STATISTICS
• Neyman’s Factorization Theorem: Y is a ss for iff
1 2 1 2; , , , nL k y k x x x
where k1 and k2 are non-negative functions.
The likelihood function Does not depend on xi
except through y
Not depend on (also in the range of xi.)
15
EXAMPLES
1. X~Ber(p). For a r.s. of size n, find a ss for p if exists.
16
EXAMPLES
2. X~Beta(θ,2). For a r.s. of size n, find a ss for θ.
17
SUFFICIENT STATISTICS
• A ss, that reduces the dimension, may not exist.
• Jointly ss (Y1,Y2,…,Yk ) may be needed. Example: Example 10.2.5 in Bain and Engelhardt (page 342 in 2nd edition), X(1) and X(n)
are jointly ss for
• If the MLE of exists and unique and if a ss for exists, then MLE is a function of a ss for .
18
EXAMPLE
X~N(,2). For a r.s. of size n, find jss for and 2.
MINIMAL SUFFICIENT STATISTICS
• If is a ss for θ, then,
is also a ss
for θ. But, the first one does a better job in data reduction. A minimal ss achieves the greatest possible reduction.
19
))x(s),...,x(s()x(S~
k~
1~
))x(s),...,x(s),x(s()x(S~
k~
1~
0~
*
20
MINIMAL SUFFICIENT STATISTICS• A ss T(X) is called minimal ss if, for any
other ss T’(X), T(x) is a function of T’(x).• THEOREM: Let f(x;) be the pmf or pdf of
a sample X1, X2,…,Xn. Suppose there exist a function T(x) such that, for two sample points x1,x2,…,xn and y1,y2,…,yn, the ratio
is constant with respect to iff T(x)=T(y). Then, T(X) is a minimal sufficient statistic for .
1 2
1 2
, , , ;
, , , ;n
n
f x x x
f y y y
21
EXAMPLE
• X~N(,2) where 2 is known. For a r.s. of size n, find minimal ss for .
Note: A minimal ss is also not unique. Any 1-to-1 function is also a minimal ss.
22
ANCILLARY STATISTIC
• A statistic S(X) whose distribution does not depend on the parameter is called an ancillary statistic.
• Unlike a ss, an ancillary statistic contains no information about .
Example
• Example 6.1.8 in Casella & Berger, page 257:
Let Xi~Unif(θ,θ+1) for i=1,2,…,n
Then, range R=X(n)-X(1) is an ancillary statistic because its pdf does not depend on θ.
23
24
COMPLETENESS• Let {f(x; ), } be a family of pdfs (or pmfs)
and U(x) be an arbitrary function of x not depending on . If
requires that the function itself equal to 0 for all possible values of x; then we say that this family is a complete family of pdfs (or pmfs).
0 for all E U X
0 for all 0 for all .E U X U x x
i.e., the only unbiased estimator of 0 is 0 itself.
25
EXAMPLES
1. Show that the family {Bin(n=2,); 0<<1} is complete.
26
EXAMPLES
2. X~Uniform(,). Show that the family {f(x;), >0} is not complete.
27
COMPLETE AND SUFFICIENT STATISTICS (css)
• Y is a complete and sufficient statistic (css) for if Y is a ss for and the family
; ;g y is complete. The pdf of Y.
1) Y is a ss for .
2) u(Y) is an arbitrary function of Y. E(u(Y))=0 for all implies that u(y)=0 for all possible Y=y.
28
BASU THEOREM• If T(X) is a complete and minimal sufficient
statistic, then T(X) is independent of every ancillary statistic.
• Example: X~N(,2).
: the mss for X
(n-1)S2/ 2 ~2
1n Ancillary statistic for
By Basu theorem, and S2 are independent.X
S2
statisticcompleteaisX
.familycompleteis)n/,(Noffamilyand)n/,(N~X 22
BASU THEOREM
• Example:
• Let T=X1+ X2 and U=X1 - X2
• We know that T is a complete minimal ss.
• U~N(0, 22) distribution free of T and U are independent by Basu’s Theorem
29
X1, X2~N(,2), independent, 2 known.
Problems
• Let be a random sample from a Bernoulli distribution with parameter p.
a)Find the maximum likelihood estimator (MLE) of p.
b)Is this an unbiased estimator of p?
30
1 2, ,..., nX X X
Problems
• If Xi are normally distributed random variables with mean μ and variance σ2, what is an unbiased estimator of σ2?
31
Problems
• Suppose that are i.i.d. random variables on the interval [0; 1] with the density function,
where α> 0 is a parameter to be estimated from the sample. Find a sufficient statistic for α.
32
1 2, ,..., nX X X