Statistical Decision Theory
Abraham Wald
(1902 - 1950)
• Wald’s test
• Rigorous proof of the consistency of MLE
“Note on the consistency of the maximum
likelihood estimate”, Ann. Math. Statist., 20,
595-601.
Statistical Decision Theory
Unlike classical statistics which is only directed towards the use of
sampling information in making inferences about unknown numerical
quantities, an attempt in decision theory is made to combine the sampling
information with a knowledge of the consequences of our decisions.
A major use of statistical inference is its application to decision making
under uncertainty, say Parameter
and Hypothesis Testing.
estimation
Three elements in SDT
• State of Nature: θ, some unknown quantities, say parameters.
• Decision Space D : A space of all possible values of
decisions/actions/rules/estimators
• Loss function L(θ, d(X)):
– a non-negative function on Θ x D.
– a measure of how much we lose by choosing action d when θ is used.
– In estimation, a measure of the accuracy of estimators d of θ.
For example,
  0,1
θ = 0 means “nuclear warhead is NOT headed to UBC”
θ = 1 means “nuclear warhead is headed to UBC”
D ={0,1}={Stay in Vancouver, Leave
L(θ, d):
}
L(0,0) = 0
L(0,1) = cost of moving
L(1,1) = cost of moving + cost of belongings we cannot move
L(1,0) = loss of belongings +
Common loss functions
• Univariate
– L1 = |θ – d(x)|
– L2 = (θ – d(x))2
(absolute error loss)
(squared error loss)
• Multivariate
– (Generalized) Euclidean norm:
[θ – d(x)]T Q [θ – d(x)], where Q is positive definite
• More generally,
– Non-decreasing functions of L1 or Euclidean norm
Loss function, L(d(X), θ), is random
Frequentist
Bayesian
Risk E
or L
Posterior risk
(in Frequentist)
(in Bayesian)
Estimator Comparison
The Risk principle: the estimator d1(X) is better than another
estimator d2(X) in the sense of risk if R(θ,d1) ≤ R(θ,d2) for all θ, with
strict inequality for some θ.
Best estimator (uniformly minimum risk estimator)
d*(X) = arg min R(θ, d(X)) for all θ
d
However, in general, it does not exist
Class of all
estimators
Too Large
Shrink the class of estimators.
Then find the best estimator in
this class.
Smaller class of estimators
Class of estimators
For instance, only consider
mean-unbiased estimators.
In particular, UMVUE is the best
unbiased estimator when L2 is
used.
I am the best!!
Notice that the risk depends on θ. So, the risks of two estimators
often cross each other. This is another possibility that the best
estimator does not exist.
Weaken the optimality criterion by considering the maximum
value of the risk over all θ. Then choose the estimator with the
smallest maximum risk. The best estimator according to this
minimax principle is called minimax estimator.
R(θ,d)
d1
Winner
d2
Loser
θ
Alternatively, we can find the best estimator by minimizing
the average risk with respect to a prior π of θ in the Bayesian
framework.
Given a prior π of θ, the average risk of the estimator d(X) defined by
 R( , d ) ( )d
is called Bayes risk,
rπ(d)
The estimator having the smallest Bayes risk with a specific prior π
is called the Bayes estimator (with π).
 R( , d ) ( )
Under Bayes risk principle
i
i
i
Winner
Loser
Winner
Loser
R(θ, d1)
R(θ, d2)
d1
d2
θ
π(θ)
In general, it is not easy to find the Bayes estimator by
minimizing the Bayes risk.
However, if the Bayes risk of the Bayes estimator is
finite, then the estimator minimizing the posterior risk
and the Bayes estimator are the same.
Some examples for finding the Bayes estimator
(1) Squared error loss:
Minimize the posterior risk
min E[(Θ-d)2|x]
d
E[(Θ-d)2|x] = E[Θ2|x] -2d E[Θ|x] + d2
 f(d)
The minimizer of f(d) is E[Θ|x], i.e. the posterior mean.
Some examples for finding the Bayes estimator
(2) Absolute error loss:
min E[|Θ-d| |x]
d
The minimizer is med[Θ|x], i.e. the posterior median.
(3) Linear error loss:
L(θ,d) = K0(θ-d) if θ-d>=0 and
= K1(d-θ) if θ-d<0
The K0/(K0+K1) th quantile of the posterior is the Bayes
estimator of θ.
Relationship between minimax and
Bayes estimators
Denote by dπ the Bayes estimator with respect to π.
If the Bayes risk of dπ is equal to the maximum risk of dπ, i.e.
R( , d  )
 R( , d ) ( )d  sup

Then the Bayes estimator dπ is minimax .
In particular, if the Bayes estimator has a constant risk, then it is minimax.
Problems for the risk measure:
The risk measure is too sensitive to the choice of
loss function.
All estimators are assumed to have finite risks.
So, in general, the risk measure fails to use in
problems with heavy tails or outliers.
Other measures:
(1) Pitman measure of closeness, PMC:
θ
d1
d2
Pθ( || d1   ||Q || d 2   ||Q )≥1/2
d1 is Pitman-closer to θ than d2 if the above condition holds for all θ.
Other measures:
(2) Universal domination, u.d.:
d1(X) is said to universally dominate d2(X) if,
for all nondecreasing functions h and all θ,
Eθ[h(|| d1(X) - θ ||Q)] ≤ Eθ[h(|| d2(X) - θ ||Q)].
(3) Stochastic domination, s.d.:
d1(X) is said to stochastically dominate d2(X) if,
for every c > 0 and all θ,
Pθ[|| d1(X) - θ ||Q≤ c] ≥ Pθ[|| d2(X) - θ ||Q≤ c].
Problems:
(1) Pitman measure of closeness, PMC:
d3
θ
d1
d2
Problems:
(2) Universal domination, u.d.:
d1(X) is said to universally dominate d2(X) if,
for all nondecreasing functions h and all θ,
Eθ[h(|| d1(X) - θ ||Q)] ≤ Eθ[h(|| d2(X) - θ ||Q)].
Expectation is a linear operator
h(t) = at + b, a>0
For all nondecreasing functions h,
E
Tθ[h(|| d1(X) - θ ||Q)]
where T has a property that
T[h(y)] =h[T(y)]
Thank You!!