Goodness–of–fit tests for regression models: the functional data case

Goodness–of–fit tests for regression models: thefunctional data case

Wenceslao González Manteiga

Joint work with Juan Cuesta–Albertos, Eduardo García–Portuguésand Manuel Febrero–Bande

Department of Statistics and O.R., University of Santiago de Compostela, Spain

ICMAT, Madrid, 29/01/2015

Goodness–of–fit tests for regression models

Our Research group: “Optimization Models, Decision,Statistics and Applications (MODESTYA)"

The group is formed by 25 persons dis-tributed as follows: 15 professors, 5PhD students, 5 contracted persons un-der transfer projects and

also 12 collaborators from other univer-sities.

ICMAT 2015 – W. González-Manteiga et al. Goodness–of–fit tests for regression models 2/46

http://eio.usc.es/pub/gi1914/descargas/modestya.pdf


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1 Introduction2 GOF for regression models in Functional Data3 The test4 Simulation study5 Real data application



Introduction

Goodness of fit

The term Goodness-of-fit (GOF) was introduced by Pearson at the begin-ning of the 20th century and it refers to tests that check how a distributionfit to a data set in an omnibus way.

The basic idea consists in comparing a nonparametric pilot estimator forthe unknown distribution F or the density f , with a consistent parametricestimator under the null hypothesis.



Introduction

Two basic references

Durbin (1973) and Bickel and Rosenblatt (1973) settled the beginnings ofthe mathematical developments for GOF tests, based on the estimation ofthe cdf and the density function, respectively.



Introduction

Extension

These ideas have been extended in the nineties to the general case of aregression model (c.f. Härdle and Mammen (1993), GM and Cao (1993)).

Given a regression model (fixed or random design):

Yi = m(Xi) + εi, i = 1, . . . , n

where E(Yi|Xi) = m(Xi). The goal is to test

H0 : m ∈M = {mθ}θ∈Θ⊂Rq , vs. Ha : m /∈M

with m(x) = E(Y |X = x) the regression function of Y over X,σ2 = Var(Y |X = x) and f the density of the explanatory variable (ifexists).



Introduction

The integrated regression function

In a similar way to the tests for the distribution function F (x) =∫ x

−∞f(t)dt, we may consider the integrated regression function:

I(x) =

∫ x

−∞m(t)dF (t) = E(Y · I(X ≤ x))



Introduction

The integrated regression function can be nonparametrically estimatedas follows:

In(x) =1

n

n∑i=1

Yi · I(Xi ≤ x)

with associated empirical process:

Rn(x) =√n(In(x)− Eθ(In(x))) =

1√n

n∑i=1

I(Xi ≤ x)εi

This empirical process was the basis for a broad class of test statistics(see Stute, 1997 and references in the last years).



Introduction

More generally in the last fifteen years (semiparametric ornonparametric) “null hypothesis” have been considered:

- Partial linear models:

H0 : Yi = Xti θ +m(Zi) + εi, i = 1, . . . , n

- Generalized partial linear models:

H0 : E(Yi|Xi, Zi) = G(Xti θ +m(Zi))

with G a known link function.- Significance test:

H0 : E(Yi|Xi, Zi) = E(Yi|Xi)

- Testing additivity:

E(Yi|Xi1, . . . , Xip) =

p∑j=1

mj(Xij)



Introduction

Even more generally, “complex models” have been also consideredrecently

ä Goodness–of–Fit Tests for Interest Rate Models

ä Goodness–of–Fit Tests for Directional Data

ä Goodness–of–Fit Tests for Functional Data

(see GM and Crujeiras (2013) for a big review in the topic)



Functional data

Functional data

Each observation is a curve! Real examples in:

Meteorology: temperature, precipitation, wind speed,. . .

Finance: evolution of asset prices/returns.

Spectrometry

. . .



Functional data

850 900 950 1000 1050

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Wavelength (mm)

Abs

orba

nces

Figure : Tecator dataset. Absorbance coloured by fat content.



Functional data

Day

Tem

pera

ture

(ºC

)

j F m A M J J A S O N D

510

1520

2530

Figure : Yearly evolution of temperatures in 76 AEMET monitoring stations.



Functional data

Sessions

SP

500

inde

x

1260

1270

1280

1290

1300

1310

11/08/2008 12/08/2008 13/08/2008 14/08/2008 15/08/2008 18/08/2008 19/08/2008 20/08/2008 21/08/2008 22/08/2008

Figure : SP500 index. Each day is a functional datum.



Functional data

Objective

Propose a Goodness–of–Fit test for the null hypothesis of the functionallinear model,

Y = 〈X , β〉+ ε =

∫X (t)β(t)dt+ ε,

with ε a centred r.v. independent from X . This is equivalent to test H0 :m(·) ∈ {〈·, β〉 : β ∈ H}, being H = L2[0, T ] the Hilbert space of squareintegrable functions.



Functional data

ä Let X (t) be a functional r.v. taking values in H = L2[0, T ].

ä Let {Ψj}∞j=1 be a basis of H. Then for each observation Xi,i = 1, . . . , n of X , we can express

Xi =

∞∑j=1

xijΨj .

ä Let {Ψj}pj=1 the p–truncate basis with the first p elements of{Ψj}∞j=1. The representation of X ∈ H in this truncated basis isdenoted by

X (p)i =

p∑j=1

xijΨj .

ä The basis choice and the number of elements of the truncated basisare crucial in order to capture correctly the information given by thefunctional process.



Functional data

Functional linear model

ä The Functional Linear Model states that

Y = 〈X , β〉+ ε =

∫X (t)β(t)dt+ ε,

with ε a centred r.v. independent from X .

ä The estimation of the functional parameter β is done by minimisingthe RSS:

β = arg minβ∈H

n∑i=1

(Yi − 〈Xi, β〉)2.

ä Different methods have been proposed to search for the β thatminimizes the RSS (see Ferraty and Romain (2011)):

1 Using a basis representation of B–splines or Fourier functions.2 Using Principal Components (PC).3 Using Partial Least Squares (PLS).



The test

ä Let (X , Y ) be r.v.’s in H× R and the sample {(Xi, Yi)}ni=1.

ä We want to test this null composite hypothesis:

H0 : m ∈ {〈·, β〉 : β ∈ H} ,

versus the general alternative

H1 : P {m /∈ {〈·, β〉 : β ∈ H}} > 0.

ä The simple hypothesis, i.e. checking for a specific functional linearmodel, is also of interest

H0 : m (X ) = 〈X , β0〉 , for a fixed β0 ∈ H

and it includes the important case of no interaction between thefunctional covariate and the scalar response (β0 = 0).



The test

Lemma

Let β ∈ H. The following statements are equivalent:

I m (X ) = 〈X , β〉 , ∀X ∈ H.

II E [Y − 〈X , β〉 |X = x] = 0, for a.e. x ∈ H.

III E [Y − 〈X , β〉 | 〈X , γ〉 = u] = 0, for a.e. u ∈ R and ∀γ ∈ SH.

III’ E [Y − 〈X , β〉 | 〈X , γ〉 = u] = 0, for a.e. u ∈ R and ∀γ ∈ SpH, ∀p ≥ 1.

IV E[(Y − 〈X , β〉)1{〈X ,γ〉≤u}

]= 0, for a.e. u ∈ R and ∀γ ∈ SH.

IV’ E[(Y − 〈X , β〉)1{〈X ,γ〉≤u}

]= 0, for a.e. u ∈ R and ∀γ ∈ SpH,

∀p ≥ 1.

Lemma based on the results of Patilea et al. (2012).



The test

ä A possible way to measure the deviation of the data from H0

(following the ideas of Stute (1997) in scalar case) is by theprojected empirical process:

Rn(u, γ) = n−12

n∑i=1

(Yi −

⟨Xi, β

⟩)1{〈Xi,γ〉≤u}.

ä To measure the distance of the empirical process from zero:Cramér–von Mises and Kolmogorov–Smirnov norms, adapted to theprojected space Π = R× SH:

PCvMn =

∫Π

Rn(u, γ)2 Fn,γ(du)ω(dγ),

PKSn = sup(u,γ)∈Π

|Rn(u, γ)| ,

where Fn,γ is the ecdf of {〈Xi, γ〉}ni=1 and ω represents a functionalmeasure on SH.



The test

ä A p–truncated version of this statistic is:

Rn,p

(u, γ(p)

)= n−

12

n∑i=1

(Yi − xTi,p Ψ bp

)1{xTi,p Ψ gp≤u}

= Rn,p (u,gp) ,

where bp are the coefficients of β in the p–truncated basis {Ψj}pj=1,Ψ = (〈Ψi,Ψj〉)ij (Ip if the basis is orthonormal) and gp are thecoefficients of γ in the truncated basis.

ä A simplified version of the statistics PCvMn considering the uniformdistribution of the sphere is:

PCvMn,p =

∫Sp×R

|R|−1Rn,p(u,R

−1gp)2 Fn,R−1gp(du) dgp,

where R is the p× p matrix such that Ψ = RTR



The test

By calculus analogous to those of Escanciano (2006),

PCvMn,p = n−2n∑i=1

n∑j=1

n∑r=1

εiεjAijr,

where:

Aijr = A(0)ijr

πp/2−1

Γ(p2 + 1

) |R|−1,

A(0)ijr =

2π, x′i,p = x′j,p = x′r,p,π, x′i,p = x′j,p,x

′i,p = x′r,p or x′j,p = x′r,p,∣∣∣∣π − arccos

((x′i,p−x′r,p)T (x′j,p−x′r,p)

||x′i,p−x′r,p||·||x′j,p−x′r,p||

)∣∣∣∣ , else.

E. García-Portugués, W. González-Manteiga and M. Febrero-Bande (2014).A Goodness–of–Fit test for the functional linear model with scalar response.Journal of Computational and Graphical Statistics, Vol. 23 Issue 3, 761–778.



The test

Simple hypothesis

Calibration of the test procedure for simple hypothesis

Let be {(Xi, Yi)}ni=1 an iid sample:1 Express Xi(·) (and optionally β0(·)) in a p-truncated basis.2 Construct εi = Yi − 〈Xi, β0〉.3 Compute PCvMn,p = n−2

∑ni=1

∑nj=1

∑nr=1 εiεjAijr

4 Bootstrap procedure (Wild bootstrap):ä Construct ε∗i = Viεi, with E [Vi] = 0, E

[V 2i

]= 1.

ä Compute PCvM∗n,p = n−2∑n

i=1

∑nj=1

∑nr=1 ε

∗i ε

∗jAijr

5 p–value≈ #{

PCvMn,p ≤ PCvM∗n,p}/B



The test

Simple hypothesis

Other competing methods for the simple hypothesis

ä Delsol et al. (2011) propose a test statistic for H0 : m(X ) = m0(X )using ideas of Härdle and Mammen (1993):

Tn =1

n

n∑j=1

(n∑i=1

εiK

(d(Xj ,Xi)

h

))2

ω(Xj),

where K is a kernel function, d is a semimetric (L2), h is thebandwidth (0.25, 0.50, 0.75 and 1.00) and ω a weight function(uniform).

ä González-Manteiga et al. (2012) extends the ideas of the classicalF–test to the functional framework, resulting a statistic to test thenull hypothesis of no interaction inside the functional linear model:

Dn =

∣∣∣∣∣∣∣∣∣∣ 1n

n∑i=1

(Xi − X

) (Yi − Y

)∣∣∣∣∣∣∣∣∣∣H

.



The test

Composite hypothesis

Calibration of the test procedure for composite hypothesis

Let be {(Xi, Yi)}ni=1 an iid sample:1 Set Xi(·) := Xi(·)− Xi(·) and Yi := Yi − Y .2 Express Xi(·) and β(·) in a p-truncated basis (bspline, PC or PLS).

3 Construct εi = Yi −⟨Xi, β

⟩.

4 Compute PCvMn,p = n−2∑ni=1

∑nj=1

∑nr=1 εiεjAijr

5 Bootstrap procedure (Wild bootstrap):

ä Construct Y ∗i =

⟨Xi, β

⟩+ ε∗i , where ε∗i = Viεi, with E [Vi] = 0,

E[V 2i

]= 1.

ä Estimate β∗(·) by basis representation, PC or PLS with kn elements.

ä Construct ε∗∗i = Y ∗i −

⟨Xi, β

∗⟩

.

ä Compute PCvM∗n,p = n−2∑n

i=1

∑nj=1

∑nr=1 ε

∗∗i ε

∗∗j Aijr

6 p–value≈ #{

PCvMn,p ≤ PCvM∗n,p}/B



The test


The weak convergence of the process Rn(u, γ) indexed inΠ = R× SH is a very difficult task!

A related process is given by

Rn(u) = n−12

n∑i=1

(Yi −

⟨Xi, β

⟩)1{〈Xi,γ〉},

with a fixed but randomly chosen γ ∼ ω.

If ω is a non–degenerated Gaussian measure, then:

H0 ⇔ E [Y − 〈X , β〉 | 〈X , γ〉] = 0, for some β ∈ H.

In this way it is possible to obtain the weak convergence of Rnindexed in u ∈ R and obtain statistical tests for testing H0.



The test


Testing significance: H0 : m(X ) = c

Under regularity conditions:

Rn(x) = n−1/2n∑i=1

(Yi − Y

)1{〈Xi,γ〉≤u}

d−→ G

where G is a centred gaussian process with covariance function

K (s, t) = Γ(s ∧ t) + Var (Y )F γ(s)F γ(t)

−F γ(t)E[1{〈X ,γ〉≤s} (Y − c)2

]−F γ(s)E

[1{〈X ,γ〉≤t} (Y − c)2

]with Γ(x) =

∫ x−∞Var (Y/ 〈X , γ〉 = u) dF γ(u) and F γ the distribution

function of 〈X , γ〉



The test


Testing Functional Linear Model: H0 : m(X ) = 〈X , β〉Under regularity conditions:

Rn(x) = n−1/2n∑i=1

(Yi −

⟨Xi, β

⟩)1{〈Xi,γ〉≤u}

has the same distribution as:

n−1/2n∑i=1

εi1{〈Xi,γ〉≤u} + n−1/2⟨E[1{〈Xi,γ〉≤u}X

], β − β

⟩ä One important case: the estimation with Principal ComponentAnalysis (PCA) (Working in progress!!)



The test


ä Using the process Rn(x) we can define different tests:∣∣∣∣∣∣Rn(x)∣∣∣∣∣∣KS

= supu |Rn(u)| (Kolmogorov–Smirnov)∣∣∣∣∣∣Rn(x)∣∣∣∣∣∣CvM

=∫R Rn(u)2dF γn (u) (Cramer–von Mises) being F γn

the empirical estimation of F γ

. . .

ä The big advantage of using random projections is that the empiricalprocess is indexed in real valuesä The disadvantage is that possibly we are suffering of some loss ofpowerä This possibility can be alleviated by chosing several projectionsγ1, . . . , γk and selecting an appropiated way to mix the obtainedp-values. For example, using the FDR method proposed in Benjaminiand Yekutieli (2001).



The test


Calibration of the test procedure for the functional linear model with PCA

Let be {(Xi, Yi)}ni=1 an iid sample:

ä Estimate β by PCA for a chosen p = pn elements of the basis andobtain the fitted residuals: εi = Yi −

⟨Xi, β

⟩, i = 1, . . . , n

ä Compute∣∣∣∣∣∣Rn∣∣∣∣∣∣ with || || = || ||KS or || || = || ||CvM

ä Resample using Wild Bootstrap and obtain estimations of β with thebootstrap samples for b = 1, . . . , B.

ä Compute∣∣∣∣∣∣R∗n,b(x)

∣∣∣∣∣∣ , b = 1, . . . , B

ä Approximate the p-value by 1B

∑Bb=1 1{||R∗n,b||≤||Rn||}

Repeat the procedure k times and use FDR for the k p-values.



Simulation study


Simulation scenarios and deviations from the null hypothesis.Scenario Coefficient β(t) Process X Deviation

S1 (2ψ2(t) + 4ψ2(t) + 5ψ3(t))/√

2 BM δk∆1, δ =(0, 1

4, 34

)S2 (2ψ3(t) + 4ψ5(t) + 5ψ7(t))/

√2 BM δk∆1, δ =

(0, 1

20, 15

)S3 (2ψ2(t) + 4ψ3(t) + 5ψ7(t))/

√2 BM −δk∆1, δ =

(0, 1

5, 12

)S4 log

(15t2 + 10

)+ cos(4πt) BM δk∆1, δ =

(0, 1

5, 1)

S5∑20

j=1 23/2(−1)jj−2φj(t) HHN (l = 1) −δk∆2, δ = (0, 1, 3)

S6∑20

j=1 23/2(−1)jj−2φj(t) HHN (l = 2) −δk∆2, δ = (0, 1, 3)

S7 (2ψ2(t) + 4ψ2(t) + 5ψ3(t))/√

2 BB −δk∆2, δ =(0, 2, 15

2

)S8 sin(2πt)− cos(2πt) OU −δk∆2, δ =

(0, 1

4, 1)

S9 t−(t− 3

4

)2 OU −δk∆3, δ =(0, 1

100, 110

)S10 t+ cos(2πt) OU δk∆3, δ =

(0, 1

100, 120

)S11 log

(15t2 + 10

)+ cos(4πt) GBM (s0 = 1) δk∆3, δ =

(0, 1

2, 2)

S12 π2(t2 − 1

3

)GBM (s0 = 2) δk∆3, δ =

(0, 1

2, 52

)Hk,d : Y = 〈X, βk〉+ δd∆d k

3e(X) + ε, with ∆1(X) = ||X||,

∆2(X) = 25∫ 10

∫ 10

sin(2πts)s(1− s)t(1− t)X(s)X(t) ds dt and ∆3(X) =⟨e−X,X2

⟩. The

noise ε is distributed as aN (0, σ2), where σ2 is chosen such that R2H0

is 0.95.



Simulation study


0.0 0.2 0.4 0.6 0.8 1.0

−2

02

46

810

−3

−2

−1

01

2

0.0 0.2 0.4 0.6 0.8 1.0

−5

05

10

−2

02

0.0 0.2 0.4 0.6 0.8 1.0

−5

05

−3

−2

−1

01

2

0.0 0.2 0.4 0.6 0.8 1.0

11.

52

2.5

33.

54

4.5

−2

02

4

0.0 0.2 0.4 0.6 0.8 1.0

−4

−2

02

46

−4

−2

02

4

0.0 0.2 0.4 0.6 0.8 1.0

−4

−2

02

46

−3

−2

−1

01

23

0.0 0.2 0.4 0.6 0.8 1.0

−4

−2

02

46

810

−2

−1

01

2

0.0 0.2 0.4 0.6 0.8 1.0

−1.

5−

1−

0.5

00.

51

1.5

−3

−2

−1

01

2

0.0 0.2 0.4 0.6 0.8 1.0

−1

−0.

50

0.5

1

−3

−2

−1

01

23

0.0 0.2 0.4 0.6 0.8 1.0

−1

−0.

50

0.5

11.

52

−3

−2

−1

01

2

0.0 0.2 0.4 0.6 0.8 1.0

11.

52

2.5

33.

54

4.5

02

46

8

0.0 0.2 0.4 0.6 0.8 1.0

−4

−2

02

46

05

1015

20

Figure : From right to left and up to down, functional coefficients β and underlyingprocesses for the twelve different scenarios, labelled S1 to S12.



Simulation study


Hk,δ KS 1 KS 5 KS 10 KS 25 KS 50 CvM 1 CvM 5 CvM 10 CvM 25 CvM 50 PCvM Mean pH1,0 0.055 0.046 0.035 0.028 0.033 0.055 0.049 0.044 0.041 0.039 0.046 3.417H2,0 0.066 0.061 0.063 0.050 0.039 0.066 0.049 0.046 0.048 0.042 0.071 8.783H3,0 0.056 0.057 0.052 0.044 0.043 0.056 0.048 0.052 0.054 0.045 0.062 8.243H1,1 0.554 0.500 0.495 0.496 0.477 0.554 0.534 0.502 0.490 0.484 0.696 3.396H2,1 0.339 0.299 0.299 0.286 0.279 0.339 0.323 0.303 0.300 0.285 0.447 8.748H3,1 0.939 0.952 0.955 0.957 0.950 0.939 0.948 0.948 0.955 0.954 0.990 8.109

Table : Part I. Empirical sizes and powers for the competing tests with α = 0.05, n = 100,

B = 5000, M = 1000 and estimation of β by data-driven PC’s. Hk,δ represents the k–th model with

δ deviation (δ = 0 for the null hypothesis). KS and CvM tests are taken with 1, 5, 10, 25 and 50

projections.



Simulation study


Hk,δ KS 1 KS 5 KS 10 KS 25 KS 50 CvM 1 CvM 5 CvM 10 CvM 25 CvM 50 PCvM Mean pH1,2 0.991 0.998 0.999 1.000 1.000 0.991 0.999 1.000 1.000 1.000 1.000 3.275H2,2 0.995 0.997 0.998 0.997 0.996 0.995 0.998 1.000 0.998 0.998 1.000 8.496H3,2 0.993 1.000 1.000 1.000 1.000 0.993 0.999 1.000 1.000 1.000 1.000 7.673H1,3 0.993 0.999 1.000 1.000 1.000 0.993 0.999 1.000 1.000 1.000 1.000 3.181H2,3 0.995 0.999 1.000 1.000 1.000 0.995 0.999 1.000 1.000 1.000 1.000 7.871H3,3 0.997 1.000 1.000 1.000 1.000 0.997 0.999 1.000 1.000 1.000 1.000 6.131

Table : Part II. Empirical sizes and powers for the competing tests with α = 0.05, n = 100,

B = 5000, M = 1000 and estimation of β by data-driven PC’s. Hk,δ represents the k–th model with

δ deviation (δ = 2 and δ = 3). KS and CvM tests are taken with 1, 5, 10, 25 and 50 projections.



Real data application

Day

Tem

pera

ture

(ºC

)


510

1520

2530

Day

β(t)


−0.

002

−0.

001

0.00

00.

001

0.00

2

Figure : AEMET temperatures and β. Response: average wind speed.ä H0 : Y = 〈X , β〉+ ε.

NProj 10 25 50 100 PCvMp-value (CVM) 0.512 0.580 0.700 0.719 0.119p-value (KS) 0.443 0.440 0.382 0.442

ä Simple hypothesis H0 : β = 0

RPCvM(50) RPKS(50) PCvM F–test Delsol et al.p-value 0.00 0.00 0.062 0.002 0.000




850 900 950 1000 1050

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Wavelength (mm)

Abs

orba

nces

850 900 950 1000 1050

−0.

02−

0.01

0.00

0.01

0.02

0.03

0.04

0.05

Wavelength (mm)

d(A

bsor

banc

es,1

)850 900 950 1000 1050

−0.

004

−0.

002

0.00

00.

002

0.00

4

Wavelength (mm)

d(A

bsor

banc

es,2

)

Figure : Tecator dataset. Absorbance, and first derivative by fat content.

ä H0 : Y = 〈X , β〉+ ε. p–value is 0.0000 (nproj:10–100), evidence forrejecting FLM.

ä Significative dependence between Y and X , but not a linear one.

ä Same results for first and second derivative (X ′, X ′′).ä Same conclusions with previous alternative test.




Sessions

SP

500

inde

x

700

800

900

1000

1100

1200

1300

11/08/2008 03/03/2009 01/10/2009 12/05/2010 01/12/2010 04/08/2011

Time

β(t)

−5

05

1015

20

09:30 10:15 11:00 11:45 12:30 13:15 14:00 14:45 15:30

Figure : SP500 index and estimated functional coefficient β with 3 PLS.Response: mean index for the next session.

ä Idea: predict the mean of the tomorrow’s session (Yt+1 = Xt+1) withthe today’s curve, Xt.

NProj 10 25 50 100 PCvMp-value (CVM) 0.748 0.809 0.820 0.803 0.662p-value (KS) 0.530 0.0.627 0.644 0.604




Computation time

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●

●

0 200 400 600 800 1000

020

4060

80

Sample size

Tim

e (s

econ

ds)

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PCvMCvM and KS




Conclusions

ä We have proposed a goodness–of–fit test for the functional linearmodel based on empirical process using random projections as analternative of a previous test of García–Portugués et al (2014) withresults quite promising.

ä The test is easy to compute and simple to calibrate using wildbootstrap and it will be available through the R-package fda.usc ofFebrero–Bande and Oviedo–De la Fuente (2011).

ä Obvious extensions of this test can include other possible forms ofthe regression function (several covariates, non linear form,. . . )




Thanks for your attention!


Goodness–of–fit tests for regression models: the functional data case

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