Transcript Document

Sufficient
Statistics
Dayu
11.11
Some Abbreviations
• i.i.d. : independent, identically
distributed
Content
• Estimator, Biased, Mean Square Error
(MSE) and Minimum-Variance Unbiased
Estimator (MVUE)
When MVUE is unique?
• Lehmann–Scheffé Theorem
– Biased
– Complete
– Sufficient
• the Neyman-Fisher factorization criterion
How to construct MVUE is unique?
• Rao-Blackwell theorem
Estimator
• The probability mass function (or
density) of X is partially unknown,
i.e. of the form f(x;θ) where θ is a
parameter, varying in the parameter
space Θ.
Unbiased
• An estimator ˆ  t(x)is said to be
unbiased for a function ˆ if it equals in
expectation i.e. E{ˆ}  
• E.g using mean of a sample to estimate
mean of the population
x is unbiased
n
1 n
1
1 n
1
E ( x )  E (  xi )  E ( xi )   E ( xi )   n    
n i 1
n i 1
n i 1
n
Mean Squared Error (MSE)
• MSE of an estimator T of an
unobservable parameter θ is
MSE(T)=E[(T- θ)2]
• Since E(Y2)=V(Y)+[E(Y)]2
MSE(T)=var(T)+[bias(T)]2
where bias(T)=E(T- θ)=E(T)- θ
• For the unbiased one, MSE=V(T)
since biasd(T)=0
Examples
Two estimators for σ2 :
Results from MLE, biased, but
smaller variance
Unbiased, but bigger variance
Minimum-Variance Unbiased
Estimator (MVUE)
• An unbiased estimator of minimum
MSE also has minimum variance.
• MVUE is an unbiased estimator of
parameters, whose variance is minimized
for all values of the parameters.
• Two theorems
– Lehmann-Scheffé theorem can show that
MVUE is unique.
– Constructing a MVUE: Rao-Blackwell theorem
Lehmann–Scheffé Theorem
• any estimator that is complete,
sufficient, and unbiased is the unique
best unbiased estimator of its
expectation.
• The Lehmann-Scheffé Theorem
states that if a complete and
sufficient statistic T exists, then the
UMVU estimator of g(θ) (if it exists)
must be a function of T.
Completeness
• Suppose a random variable X has a probability
distribution belonging to a known family of
probability distributions, parameterized by θ,
• A function g(X) is an unbiased estimator of zero if
the expectation E(g(X)) remains zero regardless
of the value of the parameter θ. (by the definition
of unbiased)
• Then X is a complete statistic precisely if it
admits (up to a set of measure zero) no such
unbiased estimator of zero except 0 itself.
Example of Completeness
• suppose X1, X2 are i.i.d. random variables,
normally distributed with expectation θ and
variance 1.
• Not complete: Then X1 — X2 is an unbiased
estimator of zero. Therefore the pair (X1, X2)
is not a complete statistic.
• Complete: On the other hand, the sum X1 +
X2 can be shown to be a complete statistic.
That means that there is no non-zero function
g such that E(g(X1 + X2 )) remains zero
regardless of changes in the value of θ.
Detailed Explanations
• X1 + X2~(2θ,2)
Sufficiency
• Consider an i.i.d. sample X1, X2,.. Xn
• Two people A and B:
– A observe the entire sample X1, X2,.. Xn
– B observes only one number T,
T=T(X1, X2,.. Xn)
• Intuitionly, Who has more
information?
• Under what condition, B will have as
much information about θ as A has?
Sufficiency
• Definition:
– A statistic T(X) is sufficient for θ precisely if the
conditional probability distribution of the data X
given the statistic T(X) does not depend on θ.
• How to find?: the Neyman-Fisher
factorization criterion: If the probability
density function of X is f(x;θ), then T
satisfies the factorization criterion if and
only if functions g and h can be found such
that
• h(x): a function that does not depend on θ
• g(T(x),θ): a function that depends on data
only throught T(x)
• E.g.
• T=x1+x2+.. +xn is a sufficient statistic for p
for Bernoulli Distribution B(p)
g(T(x),p)∙1 h(x)=1
Example 2
Test
T=x1+x2+.. +xn
for Poisson Distribution Π(λ):
h(x): independent of λ
Hence, T=x1+x2+.. +xn is sufficient!
g(T(x), λ)
Notes on Sufficient
Statistics
• Note that the sufficient statistic is
not unique. If T(x) is sufficient, so
are T(x)/n and log(T(x))
Rao-Blackwell theorem
• named after
– C.R. Rao (1920- ) is a famous Indian
statistician and currently professor emeritus at
Penn State University
– David Blackwell (1919-) is Professor Emeritus
of Statistics at the UC Berkeley
• describes a technique that can transform
an absurdly crude estimator into an
estimator that is optimal by the meansquared-error criterion or any of a variety
of similar criteria.
Rao-Blackwell theorem
• Definition: A Rao–Blackwell estimator δ1(X) of an
unobservable quantity θ is the conditional
expected value E(δ(X) | T(X)) of some estimator
δ(X) given a sufficient statistic T(X).
– δ(X) : the "original estimator"
– δ1(X): the "improved estimator".
• The mean squared error of the Rao–
Blackwell estimator does not exceed that
of the original estimator.
Conditional Expectation
B  {x  X | f ( x)  b}
E ( f ( x) | B)   P( x | x  B)  f ( x)
xB
 0, x  B
 P( x)
P( x | x  B)  
xB
 P( x  B)
Example I
• Phone calls arrive at a switchboard
according to a Poisson process at an
average rate of λ per minute.
• λ is not observable
• Observe: the numbers of phone calls that
arrived during n successive one-minute
periods are observed.
• It is desired to estimate the probability e−λ
that the next one-minute period passes
with no phone calls.
Original estimator:
t=x1+x2+.. +xn is sufficient
Example II
• To estimate λ for X1 … Xn ~ P(λ)
• Original estimator: X1
We know t= X1 +…+ Xn is sufficient
• Improved estimator by R-B theorem:
E[X1| X1 +…+ Xn =t]  cannot compute directly
We know Σ[E(Xi| X1 +…+ Xn =t)]
=E(ΣXi| X1 +…+ Xn =t)=t
• Since X1 … Xn are i.i.d. so every term is t/n
In fact, it’s x
Thank you!