Adaptive Covariance Estimation with model selection

Transcript

1 Adaptive Covariace Estimatio with model selectio Rolado Biscay, Hélèe Lescorel ad Jea-Michel Loubes arxiv:03007v [mathst Mar 0 Abstract We provide i this paper a fully adaptive pealized procedure to select a covariace amog a collectio of models observig iid replicatios of the process at fixed observatio poits For this we geeralize the results of [3 ad propose to use a data drive pealty to obtai a oracle iequality for the estimator We prove that this method is a extesio to the matricial regressio model of the work by Baraud i [ Keywords: covariace estimatio, model selectio, adaptive procedure Itroductio Estimatig the covariace fuctio of stochastic processes is a fudametal issue i statistics with may applicatios, ragig from geostatistics, fiacial series or epidemiology for istace we refer to [0, [8 or [5 for geeral refereces While parametric methods have bee extesively studied i the statistical literature see [5 for a review, oparametric procedures have oly recetly received attetio, see for istace [6, 3, 4, ad refereces therei I [3, a model selectio procedure is proposed to costruct a o parametric estimator of the covariace fuctio of a stochastic process uder mild assumptios However their method heavily relies o a prior kowledge of the variace I this paper, we exted this procedure ad propose a fully data drive pealty which leads to select the best covariace amog a collectio of models This result costitutes a geeralizatio to the matricial regressio model of the selectio methodology provided i [ Cosider a stochastic process X t t T takig its values i R ad idexed by T R d, d N We assume that E [X t = 0 t T ad we aim at estimatig its covariace fuctio σ s, t = E [X s X t < for all t, s T We assume we observe X i t j where i { } ad j { p} Note that the observatio poits t j are fixed ad that the X i s are idepedet copies of the process X Set x i = X i t,, X i t p i { } ad deote by Σ the covariace matrix of X at the observatios poits Σ =E x i x i = σ t j, t k j p, k p Followig the methodology preseted i [3, we approximate the process X by its projectio oto some fiite dimesioal model For this, cosider a coutable set of fuctios g λ λ Λ which may be for istace a basis of L T ad choose a collectio of models M P Λ For m M, a fiite umber of idices, the process ca be

2 approximated by X t λ m a λ g λ t Such a approximatio leads to a estimator which depeds o the collectio of fuctios m, deoted by ˆΣ m Our objective is to select i a data drive way, the best model, ie the oe close to a oracle m 0 defied as the miimizer of the quadratic risk, amely [ Σ m 0 arg mir m = arg mie ˆΣm This result is achieved usig a model selectio procedure The paper falls ito the followig parts The descriptio of the statistical framework of the matrix regressio is give i Sectio Sectio 3 is devoted to the mai statistical results Namely we recall the results of the estimate give i [3 ad prove a oracle iequality with a fully data drive pealty Sectio 4 states techical results which are used i all the paper, while the proofs are postpoed to the Appedix Statistical model ad otatios We cosider a R-valued process X t idexed by T a subset of R d with expectatio equal to 0 We are iterested i its covariace fuctio deoted by σ s, t = E [X s X t We have at had the observatios x i = X i t,, X i t p for i where X i are idepedet copies of the process ad t j are determiistic poits We ote Σ R p p the covariace matrix of the vector x i Hece we observe x i x i = Σ + U i, i where U i are iid error matrices with expectatio 0 We deote by S the empirical covariace of the sample : S = i= x ix i We use the Frobeius orm defied by A = Tr AA for all matrix A Recall that for a give matrix A R p q, veca is the vector i R pq obtaied by stackig the colums of A o top of oe aother We deote by A the reflexive geeralized iverse of the matrix A, see for istace i [9 or [7 The idea is to cosider that we have a quite good approximatio of the process i the followig form X t a λ g λ t, λ m where m is a fiite subset of a coutable set Λ, a λ λ Λ are radom coefficiets i R ad g λ λ Λ are real valued fuctios We will cosider models m amog a fiite collectio deoted by M We ote G m R p m where G m jλ = g λ t j ad a m the radom vector of R m with coefficiets a λ λ m Hece, we obtai the followig approximatios : x = X t,, X t p G m a m

3 xx G m a m a mg m Σ G m E [ a m a m G m Thus, this poit of view leads us to approximate Σ by a matrix i the subset S G m = { G m ΨG m/ψ symmetric i R m m } R p p 3 Hece, for a model m, a atural estimator for Σ is give by the projectio of S oto S G m We ca prove usig stadard algebra see i [3 for a geeral proof that it has the followig form : Σ m = Π m SΠ m m M R p p, 4 where are orthogoal projectio matrices Set Π m = G m G m G m G m R p p 5 = T r Π m Π m which is the dimesio of S G m assumed to be positive, ad Σ m = Π m ΣΠ m the projectio of Σ oto this subspace Hece we obtai the model selectio procedure defied i [3 The estimatio error for a model m M is give by Σ E Σm = Σ Π m ΣΠ m + δ m, 6 where δm = Tr Π m Π m Φ, Φ=V vec x x by where Give θ > 0, it is thus atural to defie the pealized covariace estimator Σ = Σ m { m = arg mi } x i x i Σ m + pe m, i= pe m = + θ δ m 7 The followig result proved i [3 states a oracle iequality for the estimator Σ Theorem Let q > 0 be give such that there exists > + q satisfyig E x x < The, for some costats K θ > ad C θ,, q > 0 we have that E Σ Σ q /q q + [K θ if Σ Π m ΣΠ m + δ m + δ sup, 3

4 where ad q = C θ,, q E x x δm Dm / q δ sup = max { δ m : m M } However the pealty defied here depeds o the quatity δ m which is ukow i practice sice it relies o the matrix Φ = V vec xx Our objective is to study a covariace estimator built with a ew pealty ivolvig a estimator of Φ More precisely, we will replace pem by a empirical versio pem, where ad pe m = + θ δ m, 8 δ m = Tr Π m Π m Φ, with Φ a estimator of Φ The objective is to geeralize Theorem ad to costruct a fully adaptive pealized procedure to estimate the covariace fuctio 3 Mai result : adaptive pealized covariace estimatio Here we state the oracle iequality obtaied for the ew covariace estimator itroduced previously Set y i = vec x i x i, i, which are vectors i R p ad deote by S vec = i= y i their empirical mea Cosider the followig costat C if = if Tr Π m Π m Φ, ad assume that the collectio of models is chose such that C if > 0 Set Φ = δ m = Tr i= yi yi S vec Svec, Π m Π m Φ 4

5 Give θ > 0, we cosider the covariace estimator Σ = Σ m with { } m = arg mi x i x i Σ m + pe m, where i= pe m = + θ δ m 9 Theorem 3 Let q > 0 be give such that there exists β > max + q, 3 + q satisfyig E xx β < The, for a costat C depedig o θ, β ad q, we have for β, θ, C if, Σ, ad + q ; mi β, β 4[ : [ Σ E Σ q /q C if Σ Σ m + δ m + C [ [ [ xx β E β β + Σ + δ sup 0 where [ xx q β E = c θ, β, q β q = C θ,, q E xx δm β Dm β/ q δm Dm / q ad δ sup = max { δ m : m M } We have obtaied i Theorem 3 a oracle iequality sice the estimator Σ has the same quadratic risk as the oracle estimator except for a additive term of order O ad a costat factor Hece, the selectio procedure is optimal i the sese that it behaves as if the true model were at had The proof of this theorem is divided ito two parts First, as i the of Theorem proved i [3, we will cosider a vectorized versio of the model I this techical part we will obtai a oracle iequality uder some particular assumptios for a geeral pealty I a secod part, we will prove that our particular pealty verifies these assumptios by usig properties of the estimator Φ 4 Techical results 4 Vectorized model Here we cosider the vectorized versio of model I this case, we observe the followig vectors i R p : y i = f i + ε i i 5

6 Here y i correspods to vec x i x i i the model, fi to vect Σ ad ε i to vec U i We set f = f,, f, y = y,, y ad ε = ε,,, ε which are vectors i R p We estimate f by a estimator of the form f m = P m y m M, where P m is the orthogoal projectio oto a subspace S m of dimesio We ote f m = P m f ad we cosider the empirical orm f = i= f i f i with the correspodig scalar product, First we state the vectorized form of Theorem Write δ m = Tr P m I Φ, δ sup = max { δ m : m M } Give θ > 0, defie the pealized estimator f = f m, where { y } m = arg mi fm + pe m, with pe m = + θ δ m The, the proof of Theorem relies o the followig propositio proved i [3: Propositio 4 : Let q > 0 be give such that there exists > + q satisfyig E ε < The, for some costats K θ > ad C θ,, q > 0 we have that E f f q /q [ q + K θ if f P m f + δ m + δ sup, 3 where q = C θ,, q E ε δm Dm / q The ew estimator Σ defied previously correspods here to the estimator f = f m, where { y } m = arg mi fm + pe m, with pe m = + θ δ m, ad δ m is some estimator of δ m Next Propositio gives a oracle iequality for this estimator uder ew assumptios o the model As Propositio 4, it is ispired by the paper [ 6

7 Propositio 4 Let q > 0 be give such that there exists > + q satisfyig E ε < } For α 0; [, set Ω = { δ m α δm Assume that A E [ δ m δm A P Ω c C α γ for some γ q q/ The, for a costat C depedig o, θ ad q, ad we have where ad [ f E f q = q /q C if f P m f + δ m [ [ + C E [ ε + f + δ sup C α q with α = α θ is fixed i 0; [ q = C θ,, q E ε δ m D / q m 4 5 Theorem 3 is thus a direct applicatio of Propositio 4 Hece oly remai to be checked the two assumptios A ad A 4 Auxiliary cocetratio type lemmas Here we state some propositios required i the proofs of the previous results To our kowledge, the first is due to vo Bahr ad Essee i [ Lemma 43 Let U,, U idepedet cetred variables with values i R we have : [ E U i 8 E [ U i i= The ext propositio is proved i [3 i= For ay Propositio 44 Give N, k N, let Ã R Nk Nk {0} be a o-egative defiite ad symmetric matrix ad ε,, ε N iid radom vectors i R k with E ε = 0 ad V ε = Φ Write ε =, ε,, ε N ζ ε = ε Ãε, ad δ = TrÃI N Φ TrÃ For all β such that E ε β < it holds that, for all x > 0, E ε P ζ ε δ Tr Ã + δ Tr Ã ρ Ã x + δ ρ Ã β Tr Ã x C β, δ β ρ Ã x β/ where the costat C β depeds oly o β 7 6

8 5 Appedix 5 Proof of Propositio 4 This proof follows the guidelies of the proof of Theorem 6 i [ The followig lemma will be helpful for the proof of this propositio Lemma 5 Choose η { = η θ > 0 ad α = α θ 0; [ such that + θ α f + η Set H m f = f [ θ f f m + Dm δ } m where θ = The, for m 0 miimizig m f f m + Dm δ m i m M where was defied i Propositio 4 E [H m0 f q Ω q δ q Proof Lemma 5 First, remark that o the set Ω, for all m M sup η 7 q + θ pe m α + θ δ m + η δ m Set pem = + η δ m Dm, which correspods to the pealty of Propositio 4 The proof of this lemma is based o the proof of Propositio 4 i [3 I fact, it is sufficiet to prove that for each x > 0 ad P H f Ω where we have set H f = Ideed, for each m M, + η x δ m c, η E ε [ f f + 4 { f fm0 η + pe m 0 } + δ m, 8 / η + x f f m0 + pe m 0 = f f m0 + + θ δ m 0 0 the we get that for all q > 0, Usig the equality E [H q f Ω = + θ f f m0 + δ m 0 0 H q f Ω H q m 0 f Ω 9 0 qu q P H q f Ω > u du 8

9 ad followig the proof of Proposito 4 i [3 we obtai the upper boud 7 of Lemma 5 Now we tur to the proof of 8 For ay g R p we defie the empirical quadratic loss fuctio by γ g = y g Usig the defiitio of γ we have that for all g R p, f g = γ g + g y, ε + ε ad therefore f f f P m0 f = γ f Usig the defiitio of f, we kow that γ f + pe m γ g + pe m 0 γ P m0 f + f Pm0 f, ε 0 for all g S m0 The γ f γ P m0 f pe m 0 pe m So we get from 0 ad that f f f P m0 f + pe m 0 pe m + f P m0 f, ε + P m f f, ε + f P m f, ε I the followig we set for each m M, B m = {g S m : g }, G m = sup t B m u m = Sice f = P m f+ P m ε, gives g, ε = P m ε, { Pm f f P m f f if P m f f 0 0 otherwise f f f P m0 f + pe m 0 pe m + f P m0 f u m0, ε + f P m f u m, ε + G m 3 Usig repeatedly the followig elemetary iequality that holds for all positive umbers ν, x, z xz νx + ν z 4 9

10 we get for ay m M f P m f u m, ε ν f P m f + ν u m, ε 5 By Pythagora s Theorem we have f f = f P m f P + m f f = f P m f + G m 6 We derive from 3 ad 5 that for ay ν > 0 f f f P m0 f + ν f P m 0 f + ν u m 0, ε +ν f P m f + ν u m, ε + G m + pe m 0 pe m Now takig ito accout that by equatio 6 f P m f f = f G m the above iequality is equivalet to ν f f + ν f P m0 f + ν u m 0, ε + ν u m, ε + ν G m + pe m 0 pe m 7 We choose ν = 0, [, but for sake of simplicity we keep usig the otatio ν Let +η p ad p be two fuctios depedig o ν mappig M ito R + They will be specified as i [3 to satisfy pe m ν p m + ν p m m M 8 Remember that o Ω, pem pem m M Sice p ν m pe m ad + ν, we get from 7 ad 8 that o the set Ω ν f f + ν f P m0 f + pe m 0 + ν p m 0 + ν G m p m + u m, ε ν p m + um0, ε ν p m 0 f P m0 f + pe m 0 + ν G m p m + u m, ε ν p m + um0, ε ν p m 0 9 As = + 4 we obtai that ν η { ν H f Ω = ν f f ν + 4 f Pm0f η + pe m 0 } Ω + { = ν f f f P m0 f + pe m 0 } Ω + { ν G m p m + u m, ε ν p m + um, ε ν p m 0 } + 0

11 For ay x > 0, P ν H f Ω xδ m P m M : ν G m p m xδ m 3 + P m M : um, ε ν p m xδ m 3 P ν P m ε p m xδ m 3 m M + P um, ε ν p m xδ m 3 m M := P,m x + P,m x 30 m M m M From ow o, the proof of Lemma 5 is exactly the same as the ed of the proof of Propositio 4 i [3 with L m = ν Proof Propositio 4 [ f We first provide a upper boud for E f q Ω, where the set Ω depeds o α chose as i Lemma 5 As q, we have a + b q a q + b q Together with Lemma 5 we deduce that [ f E f q Ω q δ q Usig the covexity of x x q [ f E f q sup + E q [ [ θ q f f m0 + D q m 0 δ m 0 together with the Jese iequality, we obtai [ [ q Ω /q δsup + /q E θ f f m0 + D m 0 δ m 0, ad by usig the assumptio A we have that [ f E f q [ q Ω /q δsup + /q θ f f m0 + D m 0 δ m 0 3 [ f Now we eed to fid a upper boud for the quatity E f First, remark that f f = f P m y = f P mf + P m f y q Ω c f P m f + ε = f P mf + ε Ad thus f f f + ε

12 So we have [ f E f q Ω c f q P Ωc + E [ ε q Ω c Usig Hölder s iequality with q > we obtai E [ ε q Ω c E [ ε q P Ω c q But E [ ε = E i= ε i, ad as, we ca use Mikowsky s iequality to obtai E [ ε E [ ε i i= = E [ ε, that is So we have [ f E f q Ω c E [ ε E [ ε [E [ ε q + f q P Ω c q, ad with assumptio A As γ [ f E f q Ω c q, we deduce that q/ [E [ ε q q + f q Cα γ [ f E f q Ω c q q [ E [ ε + f Cα q q 3 To coclude, we use agai the covexity of x x q ad the iequality 3 to get [ f E f q q 4 q [ E [ ε + f +4 /q δsup +4 /q [ θ f f m0 + D m 0 δ m 0 Cα q q

13 5 Proof of Theorem 3 Recall that β > max + q, 3 + q ad + q ; mi β, β 4[ I order to use Propositio 4, we eed to prove the followig iequalities : A E [ δ m δm A P Ω c C α γ for γ q q/ First we prove A Remember that δ m = TrΠm Πm Φ By usig the liearity of the trace ad the equality E [ Φ = [ δ Φ, we obtai that E m = δ m which proves the result } For the secod, write Ω c = { δ m α δm We boud up the quatity P δ m α δm i the followig Propositio Propositio 5 For all m M, α 0; [ ad, β, α, C if, Σ we have for some costats C β, C β : P δ m α δm C γ β β+ + C β [ xx E β δ α β m β D β m, for γ q q/ This Propositio cocludes the proof of A with C α = C β β+ + C β [ xx E β α β δ β m D β m Proof Propositio 5 We start by dividig P m = P δ m α δm ito two parts with oe of them ivolvig a sum of idepedet variables with expectatio equal to 0 P m = P Tr Π m Π m Φ α Tr Π m Π m Φ P m = P Tr Π m Π m Φ Φ + µµ + µµ αtr Π m Π m Φ P m P Tr P m P Tr Π m Π m Π m Π m yi yi T Φ µµ + µµ S vec Svec αtr Π m Π m Φ i= yi yi T Φ µµ α Tr Π m Π m Φ i= 3

14 ad Set Q = P Tr +P Πm Π m µµ S vec Svec α Tr Π m Π m Φ Tr Π m Π m yi yi T Φ µµ α Tr Π m Π m Φ i= Tr Q = P Πm Π m µµ S vec Svec α Tr Π m Π m Φ Study of Q First we use Markov s iequality to obtai A [ Tr β E Πm Π m i= yi yi T Φ µµ β αtr Π m Π m Φ β We must cosider the two followig cases : If β, Rosethal s iequality gives +C E Tr Π m Π m y i yi T i= β Φ µµ [ Tr Πm Π m y y Φ µµ β β C E β β [ Tr E Πm Π m y y Φ µµ β 4 As β, obtai β β 4 C E β ad we ca use Jese s iequality o the secod term to Tr Π m Π m y i yi T i= E β Φ µµ [ Tr Πm Π m y y Φ µµ β β 4 If β, we use Lemma 43 of subsectio 4 to get E Tr Π m Π m y i yi T i= 8 β E β Φ µµ [ Tr Πm Π m y y Φ µµ β 4

15 I both cases, we ca use the fact that x x β is a covex ad icreasig fuctio to obtai [ Tr E Πm Π m y y Φ µµ β β [E [ Tr Πm Π m y y β + Tr Πm Π m Φ + µµ β Ad by usig the Jese s iequality o the secod term we have that [ E Tr Π m Π m y y Φ µµ β [ Tr β E Πm Π m y y β Now cosider the followig lemma Lemma 53 If Ψ is symmetric o-egative defiite, the From this fact we get that Tr Π m Π m Ψ [0; Tr Ψ 33 Tr Πm Π m y y β Tr y y β = y β = xx β I coclusio, we have Q C β [ xx E β α β δmd β β, 34 γ m with γ = mi β 4, β ad C β = C β if β 4 where C β is the costat i Rosethal s iequality ad C β = 8 if β 4 Remark that β 4 4 ad β 4, so γ 4 Set Study of Q Recall that Q = P Tr Πm Π m µµ S vec Svec α Tr Π m Π m Φ B = Tr Π m Π m µµ S vec S vec Usig the properties of the trace, we ca write B = Tr Π m Π m µµ Tr Π m Π m S vec S vec = Tr µ Π m Π m µ Tr S vec Π m Π m S vec But Π m Π m is a orthogoal projectio matrix, the 5

16 B = Tr µ Π m Π m Π m Π m µ Tr S vec Π m Π m Π m Π m S vec B = Π m Π m µ Π m Π m S vec B = Π m Π m µ Π m Π m S vec Π m Π m µ + Π m Π m S vec Hece B Π m Π m µ S vec Π m Π m µ + Π m Π m S vec B Π m Π m µ S vec + Π m Π m µ S vec Π m Π m µ B Π m Π m µ S vec + Π m Π m µ S vec µ Fially Q P +P Π m Π m µ S vec α 4 Tr Π m Π m Φ Π m Π m µ S vec α 8 µ Tr Π m Π m Φ 35 Now we eed to provide a upper boud for the quatities P Π m Π m µ S vec t For this we will use the deviatio boud provided by Propositio 44 stated i subsectio 4 Set Id p Id p G = Id p Id p Rp p Id p Id p The G y f = S vec µ Now, if Π m Π m Π m Π m 0 H m = Id Π m Π m = p, Rp 0 0 Π m Π m we have H m S vec µ = Π m Π m S vec µ 6

17 I coclusio, with A m = H m G = Π m Π m Π m Π m Π m Π m Π m Π m Π m Π m Π m Π m p Rp, we have that A m y f = Π m Π m S vec µ Moreover, A m is a orthogoal projectio matrix ad we have the followig equalities A m y f = Π m Π m S vec µ = y f A m y f, Tr A m = Tr Π m Π m =, Tr A m Id Φ = Tr Π m Π m Φ = Tr Π m Π m Φ Now we ca use Propositio 44 with Ã = A m, ε i = y i µ, Tr A m =, ρ A m =, δ = δm ad β This gives for all x > 0 P y f A m y f Tr Π m Π m Φ ad that is [ + x [ P Π m Π m S vec µ Tr Π m Π m Φ + x I order to use this deviatio boud to obtai the iequalities with γ m M C β E[ y µ β β D + m, TrΠ β m Π mφ x β P Π m Π m µ S vec α 4 Tr Π m Π m Φ C γ P Π m Π m µ S vec α 8 µ Tr Π m Π m Φ C γ q, we eed to fid x > 0 satisfyig the three followig facts q/ m M α 4 C β E[ y µ β β D + m TrΠ β m Π mφ x β x + 36 α α Tr Π m Π m Φ C if x µ 8 µ D β + m Tr Π m Π m Φ β x β 7 = δ β mx β C γ 38

18 36 ad 37 hold for the choice x = r with r < ad if is large eough to have + α 4 39 ad + α C if 8 µ 40 I order to obtai 38 with x = r, we use the iequality which gives δ β mx β δ β md β m rβ/ Moreover [ E y µ β [ E y + µ β, ad by usig properties of covexity we obtai E [ y µ β [ β E y β + µ β With the Jese s iequality we get: [ E y µ β [ β E y β I coclusio, with r = + β < we obtai for, β, α, C if, Σ E Q β+ C β [ xx β δ β md β m /4 4 where C β is the costat which appears i Propositio 44 I coclusio, combiig 34 ad 4 P δ m α δm C /4 β β+ + C β [ xx E β δ α β m β D β m for, β, α, C if, Σ To coclude, remark that q q/ = > +q q as q Proof Lemma 53 Recall that Π m Π m is a orthogoal projectio matrix Hece there exists a orthogoal matrix P m such that P m Π m Π m P m = D, with D a diagoal matrix with D ii = if i, ad D ii = 0 otherwise The if Ψ is symmetric o-egative defiite we have : Tr Π m Π m Ψ = Tr DP mψp m 8

19 p p = D kl P m ΨP m l= k= = l= P m ΨP m kl = p ll D ll P m ΨP m ll l= [0; Tr Ψ Ideed, P mψp m is o-egative defiite so all its diagoal etries are o-egative Refereces [ Y Baraud Model selectio for regressio o a fixed desig Probability theory related fields, 74: , 000 [ J Bigot, R Biscay, J-M Loubes, ad L M Alvarez Group lasso estimatio of high-dimesioal covariace matrices Joural of Machie Learig Resarch, 0 [3 J Bigot, R Biscay, J-M Loubes, ad L Muñiz-Alvarez Noparametric estimatio of covariace fuctios by model selectio Electro J Stat, 4:8 855, 00 [4 J Bigot, R Biscay Lirio, J-M Loubes, ad L Muiz Alvarez Adaptive estimatio of spectral desities via wavelet thresholdig ad iformatio projectio preprit hal , May 00 [5 N A C Cressie Statistics for spatial data Wiley Series i Probability ad Mathematical Statistics: Applied Probability ad Statistics Joh Wiley & Sos Ic, New York, 993 Revised reprit of the 99 editio, A Wiley-Itersciece Publicatio [6 S N Eloge, O Perri, ad C Thomas-Aga No parametric estimatio of smooth statioary covariace fuctios by iterpolatio methods Stat Iferece Stoch Process, :77 05, 008 [7 H Egl, M Hake, ad A Neubauer Regularizatio of iverse problems, volume 375 of Mathematics ad its Applicatios Kluwer Academic Publishers Group, Dordrecht, 996 [8 A G Jourel Krigig i terms of projectios J Iterat Assoc Mathematical Geol, 96: , 977 [9 G A F Seber A matrix hadbook for statisticias Wiley Series i Probability ad Statistics Wiley-Itersciece [Joh Wiley & Sos, Hoboke, NJ, 008 [0 M L Stei Iterpolatio of spatial data Spriger Series i Statistics Spriger- Verlag, New York, 999 Some theory for Krigig [ B vo Bahr ad C-G Essee Iequalities for the rth absolute momet of a sum of radom variables, r A Math Statist, 36:99 303, 965 9