Presentation of Bassoum Abou on Stein's 1981 AoS paper

Introduction
Basic Formulas
Basic Formal Bayes Estimates
Choice of a scalar factor
Application
Another estimates
The case of unknown variance
Conclusion

Reading Seminar on Classics

presented by
Bassoum Abou
Article

Estimation of the Mean of a Multivariate Normal Distribut

Suggested by C. Robert

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Plan

1 Introduction
2 Basic Formulas
3 Basic Formal Bayes Estimates
4 Choice of a scalar factor
5 Application
6 Another estimates
7 The case of unknown variance
8 Conclusion

Introduction
Basic Formulas
Application
Another estimates
Conclusion

1 Introduction
2 Basic Formulas
3 Basic Formal Bayes Estimates
4 Choice of a scalar factor
5 Application
6 Another estimates
7 The case of unknown variance
8 Conclusion

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Presentation of the article

Estimation of the Mean of a Multivariate Normal Distribution
Authors: Charles M Stains
Source: The Annals of Statistics, Vol.9, No. 6(Nov., 1981),
1135-1151
Implemented in FORTRAN

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Presentation of the article

Estimation of the means of independant normal random variable is
considered, using sum of squared errors as loss.
The central problem studied in this paper is tha tof estimating the
mean of multivariate normal distribution with the squared length of
the error as as loss when the covariance matrix is khown to be the
identity matrix.

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Lemme 1

Let fuction Y be a fuction N(0, 1) real random variable and let
g : R → R be an indeﬁnite integral of a Lebesgue
measurement fuction g , essentially the derivate of g . Suppose
also that E|g(Y)| < ∞ . Then

E(g (Y)) = E(Yg(Y))

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Deﬁnition 1

A function h : Rp → R will be called almost differentiable if
there exist a function h : Rp → Rp such that for all z ∈ Rp

h(x + z) − h(x) = z. h(x + tz)dt

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Lemme 2

If h : Rp → R is a almost differentiable function with
Eξ X < ∞ then Eξ h(X) = Eξ (X − ξ)h(X)

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Theorem 1

Consider the estimate X+g(X) for ξ such that g : p → p

is an almost differentiable function for wich

Eξ Σ| i gi (X)| <∞

Then for each i ∈ (1, ........, p)

Eξ (Xi + gi (X) − ξi )2 = 1 + Eξ (g2 (X) + 2 gi (X))
i

and consequently

Eξ X + g(X) − ξi 2 = p + Eξ g(X) 2 + 2 g(X))

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Theorem 2
Let f : p → + ∩ (0) be an almost differentiable function

for wich f : p → p can be taken to be almost

differentiable, and suppose also that
1 2 f (X)|)
Eξ ( f (X) Σ| i <∞

and

Eξ logf (X) 2 <∞

then
2
√
(f (X)
Eξ X + logf (X) − ξ 2 = p + 4Eξ ( √
(f (X)

Introduction
Basic Formulas
Application
Another estimates
Conclusion

General

Let X be a random vector in p , conditionally narmally distributed
given ξ with conditionnal mean ξ , with the identity as
conditionnal covariance matrix. Then the unconditionnal density of X
with respect to Lebesgue measure in p is given by
−1
1
f (x) = p e 2 |x − ξ|dΠ(ξ)
(2Π) 2

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Formal bayes estimates

The Bayes estimate φn (X) of ξ which is deﬁned by the condition
that φ = φn minimizes
−|x−ξ|dΠ(ξ)
ξ−φ(X) 2 e 2
E ξ − φ(X) 2 = EEx ξ − φ(X) 2 = E( )
−|x−ξ|dΠ(ξ)
e 2

is given by

φn (X) = Ex ξ = X + logf (X)

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Nox let us compare the unbiased estimate of risk of the normal Bayes
estimate φn (X) of ξ given by Theorem 2 with the the formal
posterior risk E ξ − φ(X) 2 . From Theorem 2 , the unbiased
estimate of the risk is given by
2 f (X) f (X) 2
ρ(X) = p + 2 f (X) − f 2 (X)

For the formal posterior risk we have
2 f (X)
Ex ξ − φ(X) 2 =p+ − logf (X) 2
f (X)

and we have at the end
2 f (X)
Ex ξ − φ(X) 2 = ρ(X) − f (X)

Introduction
Basic Formulas
Application
Another estimates
Conclusion

If f is superharmonic then the formal posterior risk Ex ξ − φ(X) 2 is
an overestimated of the estimate φn (X) given by the last formula in
the sense that

Ex ξ − φ(X) 2 ≥ ρ(X)

Now if the prior measure Π has a superharmonic density π , the f is
also superharmonic and thus φn (X) is a minimax estimate of ξ

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Let us look at estimates of the form

ξ = X − λ(X)AX

then the risk of the estimate ξ deﬁned by the previous formula with
1
λ(X) = xT Bx

is given by
1 T 2
Eξ X − X T BX
AX −ξ 2 = p − Eξ ( (X T A X2 )
X
BX)

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Application to symetric moving averages

Let X1 , ..., Xp be independently normally distribution with means
ξ1 , ......, ξp and variance 1, and suppose we plan to estimate the ξi by
ˆ 1
ξi = Xi − λ(X){Xi − 2 (Xi−1 + Xi+1 )}

where it is understood that X0 =Xp and Xp+1 =X1 and simalary for the ξ
’s. This is the special case of

Introduction
Basic Formulas
Application
Another estimates
Conclusion

−1

 2 si j − i ≡ 1 ( mod p )
Aij = 1 si j − i ≡ 0 ( mod p )
0 otherwise


The characteristics roots and vectors of A, the solution αj and yj of

Ayj = αj yj

where αj real and Rp are given with j varying over the intergers such
that

−p ≤ j <
2
p
2

by

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Application

j
αj = 1 − cos(2π p )

and for i ∈ {1.....}
1



√
p if j = 0
(−1)i −p



 √
p if j = 2
yij = 2 2πij −p

 p cos p if 2 <j<0


 2 2πij p

p sin p if 0 < j < 2

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Application

this being the ith coordinate of yi . The matrix A can be expressed as
A = yαyT where α is the diagonal matrix and the matrix B, given is by

B = {tr(A)I − 2A−1 }A2 = y(pI − 2α)−1 α2 yT

It is unreasonable to use a three-term moving average with weight
more extreme than ( 1 , 1 , 3 ) . Thus it seems appropriate to modify our
3 3
1

estimate to
ˆ
ξ = X − λ1 (X)AX

where

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Application

λ1 (X) = min( X T1 , 3 )
BX
2

The unbiased estimate of the improvement in the risk is changed from
XT 2
∆(X) = (X T A X2 given is
BX)

3
∆(X) if X T BX > 2
∆1 (X) = 4p 4
3 − 9 { 1 (Xi−1 + Xi+1 )}
2 if X T BX ≤ 3
2

Introduction
Basic Formulas
Application
Another estimates
Conclusion

ˆ
We consider the James Stein estimate ξ0 = (1 − p−22 )X
X
ˆ
Let ξ = X + g(X) where g : p → p is deﬁned by

a
− 2 X
(Xl2 ∧Zk ) l
if |X| ≤ Zk
gl (X) = a
− 2 Z sgnXl
(Xl2 ∧Zk ) k
if |X| > Zk

And the risk is
ˆ
Eξ ξ − ξ 2 = p − (k − 2)2 Eξ ( 1
2 )
(Xl2 ∧Zk )

Introduction
Basic Formulas
Application
Another estimates
Conclusion

We observed that the estimated improvement in the risk for the
ˆ
estimate ξ k is
(k−2)2
∆k (X) = (Xl2 ∧Zk )
2

and the estimated improvement in the risk for the James Stein
estimate is
(p−2)2
∆(X) = E (Xj2

Introduction
Basic Formulas
Application
Another estimates
Conclusion

ˆ
We would use the estimate ξ0 = X + g(X) where
g : p → p is homogeneous of degree -1 . We consider, for

ˆ
the present problem the modiﬁed estimate ξ = X + cSg(X) where c is
ξ
constant to be determined. Let Y = σ , η = σ , S∗ = σ2 .
X S

From theorem 1 we obtain

Eξ,σ X + S
−ξ 2 = σ2E p + n 2 ∗ .g(Y))
n+2 g(X) n+2 ( g(Y) +2

Introduction
Basic Formulas
Application
Another estimates
Conclusion

Conclusion
Different approaches to obtaining improved conﬁdence sets for ξ
are described by Morris(1977), Faith (1978) and .

Introduction
Basic Formulas
Application
Another estimates
Conclusion

References

Anderson, T.W ( 1971) The Statistical Analysis of Time Series.
Wiley, New York.
BERGER, J J.(1980) A robust generalized Bayes estimor and
conﬁdence region for a multivariate normal mean. Ann. Statist .
8.
EFRON B. and Morris, C. ( 1971) . Limiting the risk of Bayes
and empirical Bayes estimator, Part I: The Bayes case. J. Amer.
Statist. Assoc . 66 807-815.

Thank you for your attention !!!
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Presentation of Bassoum Abou on Stein's 1981 AoS paper

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Presentation of Bassoum Abou on Stein's 1981 AoS paper