presentation of bassoum abou on stein's 1981 aos paper

IntroductionBasic Formulas

Basic Formal Bayes EstimatesChoice of a scalar factor

ApplicationAnother estimates

The case of unknown varianceConclusion

Reading Seminar on Classics

presented byBassoum Abou

Article

Estimation of the Mean of a Multivariate Normal Distribution

Suggested by C. Robert





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





1 Introduction

2 Basic Formulas



5 Application

6 Another estimates


8 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





Presentation of the article

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





1 Introduction

2 Basic Formulas



5 Application

6 Another estimates


8 Conclusion





Lemme 1

Let fuction Y be a fuction N(0, 1) real random variable and letg : R → R be an indefinite integral of a Lebesgue

measurement fuction g′ , essentially the derivate of g . Supposealso that E|g(Y)| <∞ . Then

E(g′(Y)) = E(Yg(Y))





Definition 1

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

h(x + z)− h(x) =∫

z.∇h(x + tz)dt





Lemme 2

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





Theorem 1

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

is an almost differentiable function for wich

EξΣ|∇igi(X)| <∞

Then for each i ∈ (1, ........, p)

Eξ(Xi + gi(X)− ξi)2 = 1 + Eξ(g2

i (X) + 2∇gi(X))

and consequently

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





Theorem 2

Let f : <p → <+ ∩ (0) be an almost differentiable functionfor wich ∇f : <p → <p can be taken to be almostdifferentiable, and suppose also that

Eξ( 1f (X)Σ|∇2

i f (X)|) <∞

and

Eξ‖∇logf (X)‖2 <∞

then

Eξ‖X +∇logf (X)− ξ‖2 = p + 4Eξ(∇2√

(f (X)√(f (X)





1 Introduction

2 Basic Formulas



5 Application

6 Another estimates


8 Conclusion





General

Let X be a random vector in <p , conditionally narmally distributedgiven ξ with conditionnal mean ξ , with the identity asconditionnal covariance matrix. Then the unconditionnal density of Xwith respect to Lebesgue measure in <p is given by

f (x) = 1(2Π)

p2

∫e−12 |x− ξ|dΠ(ξ)





Formal bayes estimates

The Bayes estimate φn(X) of ξ which is defined by the conditionthat φ = φn minimizes

E‖ξ − φ(X)‖2 = EEx‖ξ − φ(X)‖2 = E(‖ξ−φ(X)‖2

∫e−|x−ξ|dΠ(ξ)

2∫e−|x−ξ|dΠ(ξ)

2)

is given by

φn(X) = Exξ = X +∇logf (X)





Nox let us compare the unbiased estimate of risk of the normal Bayesestimate φn(X) of ξ given by Theorem 2 with the the formalposterior risk E‖ξ − φ(X)‖2 . From Theorem 2 , the unbiasedestimate of the risk is given by

ρ(X) = p + 2∇2f (X)f (X) −

‖f (X)‖2

f 2(X)

For the formal posterior risk we have

Ex‖ξ − φ(X)‖2 = p + ∇2f (X)f (X) − ‖∇logf (X)‖2

and we have at the end

Ex‖ξ − φ(X)‖2 = ρ(X)− ∇2f (X)f (X)





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

Ex‖ξ − φ(X)‖2 ≥ ρ(X)

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





1 Introduction

2 Basic Formulas



5 Application

6 Another estimates


8 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

λ(X) = 1xT Bx

is given by

Eξ‖X − 1XT BX AX − ξ‖2 = p− Eξ( XT A2X

(XT BX)2 )





1 Introduction

2 Basic Formulas



5 Application

6 Another estimates


8 Conclusion





Application to symetric moving averages

Let X1, ...,Xp be independently normally distribution with meansξ1, ......, ξp and variance 1, and suppose we plan to estimate the ξi by

ξ̂i = Xi − λ(X){Xi − 12(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





Aij =

−12 si j− i ≡ 1 ( mod p )

1 si j− i ≡ 0 ( mod p )0 otherwise

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

Ayj = αjyj

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

− p2 ≤ j < p

2

by





Application

αj = 1− cos(2π jp)

and for i ∈ {1.....}

yij =

1√p if j = 0

(−1)i√

p if j = −p2√

2p cos 2πij

p if −p2 < j < 0√

2p sin 2πij

p if 0 < j < p2





Application

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

B = {tr(A)I − 2A−1}A2 = y(pI − 2α)−1α2yT

It is unreasonable to use a three-term moving average with weightmore extreme than (1

3 ,13 ,

13) . Thus it seems appropriate to modify our

estimate to

ξ̂ = X − λ1(X)AX

where





Application

λ1(X) = min( 1XT BX ,

23)

The unbiased estimate of the improvement in the risk is changed from∆(X) = XT A2X

(XT BX)2 given is

∆1(X) =

{∆(X) if XTBX > 3

24p3 −

49∑{1

2(Xi−1 + Xi+1)} if XTBX ≤ 32





1 Introduction

2 Basic Formulas



5 Application

6 Another estimates


8 Conclusion





We consider the James Stein estimate ξ̂0 = (1− p−2‖X‖2 )X

Let ξ̂ = X + g(X) where g : <p → <p is defined by

gl(X) =

{− a∑

(X2l ∧Z2

k )Xl if |X| ≤ Zk

− a∑(X2

l ∧Z2k )

ZksgnXl if |X| > Zk

And the risk is

Eξ‖ξ̂ − ξ‖2 = p− (k − 2)2Eξ( 1∑(X2

l ∧Z2k )

)





We observed that the estimated improvement in the risk for theestimate ξ̂k is

∆k(X) = (k−2)2∑(X2

l ∧Z2k )

and the estimated improvement in the risk for the James Steinestimate is

∆(X) = (p−2)2

E∑

(X2j





1 Introduction

2 Basic Formulas



5 Application

6 Another estimates


8 Conclusion





We would use the estimate ξ̂0 = X + g(X) whereg : <p → <p is homogeneous of degree -1 . We consider, for

the present problem the modified estimate ξ̂ = X + cSg(X) where c isconstant to be determined. Let Y = X

σ , η = ξσ , S∗ = S

σ2 .From theorem 1 we obtain

Eξ,σ‖X + Sn+2 g(X)− ξ‖2 = σ2E

[p + n

n+2(‖g(Y)‖2 + 2∇∗.g(Y))]





1 Introduction

2 Basic Formulas



5 Application

6 Another estimates


8 Conclusion





ConclusionDifferent approaches to obtaining improved confidence sets for ξare described by Morris(1977), Faith (1978) and .





References

Anderson, T.W ( 1971) The Statistical Analysis of Time Series.Wiley, New York.

BERGER, J J.(1980) A robust generalized Bayes estimor andconfidence region for a multivariate normal mean. Ann. Statist .8.

EFRON B. and Morris, C. ( 1971) . Limiting the risk of Bayesand empirical Bayes estimator, Part I: The Bayes case. J. Amer.Statist. Assoc . 66 807-815.

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