# Asymptotic Normality of a Nonparametric Conditional Quantile

Authors: Sidi Ali Ould Abdi; Sophie Dabo-Niang; Aliou Diop; Ahmedoune Ould Abdi
Journal: Advances in Decision Sciences
 (2011)

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## Abstract

Given a stationary multidimensional spatial process(Zi=(Xi,Yi)∈ℝd×ℝ,i∈ℤN), we investigate a kernel estimate of the spatial conditional quantile function of the response variable Yi given the explicative variable Xi. Asymptotic normality of the kernel estimate is obtained when the sample considered is an α-mixing sequence.


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## Body

## 1. Introduction

In this paper, we are interested in nonparametric conditional quantile estimation for spatial data. Spatial data are modeled as finite realizations of random fields that is stochastic processes indexed inℤN, the integer lattice points in the N-dimensional Euclidean space (N≥1). Such data are collected from different locations on the earth and arise in a variety of fields, including soil science, geology, oceanography, econometrics, epidemiology, environmental science, forestry, and many others; see Chilès and Delfiner [1], Guyon [2], Anselin and Florax [3], Cressie [4], or Ripley [5].In the context of spatial data, the analysis of the influence of some covariates on a response variable is particularly difficult, due to the possibly highly complex spatial dependence among the various sites. This dependence cannot be typically modeled in any adequate way.Conditional quantile analysis is of wide interest in modeling of spatial dependence and in the construction of confidence (predictive) intervals. There exist an extensive literature and various nonparametric approaches in conditional quantile estimation in the nonspatial case(N=1) for independent samples and time-dependent observations; see, for example, Stute [6], Samanta [7], Portnoy [8], Koul and Mukherjee [9], Honda [10], Cai [11], Gannoun et al. [12], and Yu et al. [13]. Extending classical nonparametric conditional quantile estimation for dependent random variables to spatial quantile regression is far from being trivial. This is due to the absence of any canonical ordering in the space and of obvious definition of tail sigma-fields.Although potential applications of conditional spatial quantile regressions are without number, only the papers of Koenker and Mizera [14], Hallin et al. [15], Abdi et al. [16], and Dabo-Niang and Thiam [17] have paid attention to study these regression methods. Hallin et al. [15] gave a Bahadur representation and an asymptotic normality results of a local linear conditional quantile estimator. The method of Koenker and Mizera [14] is a spatial smoothing technique rather than a spatial (auto) regression one and they do not take into account the spatial dependency structure of the data. The work of Abdi et al. [16] deals with 2r—mean (r∈ℕ*) and almost complete consistencies of a kernel estimate of conditional quantiles. The paper of Dabo-Niang and Thiam [17] gives the L1 norm consistency and asymptotic normality of a kernel estimate of the spatial conditional quantile, but this estimate is less general than the one considered here.However, conditional mean regression estimation for spatial data has been considered in several papers; some key references are Carbon et al. ([18, 19]), Biau and Cadre [20], Lu and Chen ([21, 22]), Hallin et al. ([23, 24]), Lahiri and Zhu [25], Carbon et al. [26], and Dabo-Niang and Yao [27].The rest of the paper is organized as follows. In Section2, we provided the notations and the kernel quantile estimate. Section 3 is devoted to assumptions. The asymptotic normality of the kernel estimate is stated in Section 4. Section 5 contains a prediction application based on quantile regression and applied to simulated data. Proofs and preliminary lemmas are given in the last section.


## 2. Kernel Conditional Quantile Estimator

Let(Zi=(Xi,Yi),i∈ℤN)(N≥1) be an ℝd×ℝ-valued measurable strictly stationary spatial process (d≥1), with same distribution as the vector of variables (X,Y) and defined on a probability space (Ω,𝒜,ℙ). A point i=(i1,…,iN) in ℤN will be referred to as a site and may also include a time component.We assume that the process under study(Zi) is observed over a rectangular domain ℐn={i=(i1,…,iN)∈ℤN,1≤ik≤nk,k=1,…,N}, n=(n1,…,nN)∈ℤN. We will write n→∞ if min{nk}→∞ and |nj/nk|<C for a constant C such that 0<C<∞ for all j,k such that 1≤j,k≤N. In the sequel, all the limits are considered when n→∞. For n=(n1,…,nN)∈ℤN, we set n̂=n1,…,nN.LetS be a set of sites. ℬ(S) will denote in what follows, the Borel σ-field generated by Zi,i∈S. We assume that the regular version of the conditional probability of Y given X exists and has a bounded density with respect to Lebesgue’s measure over ℝ. For all x∈ℝd, we denote by Fx (resp., fx) the conditional distribution function (resp., the conditional density) of Y given X=x. In the following, x is a fixed point in ℝd and we denote by Vx a neighborhood of this point. For x∈ℝd, we denote by ∥x∥ the Euclidian norm of x. We suppose that the marginal and joint densities fX and fX,Y of, respectively, X and (X,Y) exist with respect to Lebesgue measures on ℝd and ℝd+1.Forα∈]0,1[, the conditional quantile of order α of Fx, denoted by qα(x), can be written as a solution of the equation Fx(qα(x))=α.To insure thatqα(x) exists and is unique, we assume that Fx is strictly increasing. Let Ki=K1((x-Xi)/h1) and Hi(y)=K2((y-Yi)/h2). The conditional distribution Fx and the corresponding density fx can be estimated by the following respective estimators:(2.1)F̂x(y)={∑i∈InKiHi(y)∑i∈InKiif∑i∈InKi≠0,0else,f̂x(y)={(1/h2)∑i∈InKiK((y-Yi)/h2)∑i∈InKiif∑i∈InKi≠0,0else,
where K1 is a kernel density, K2 is a distribution function, K is the first derivative of K2, and h1=h1,n (resp., h2=h2,n) are sequences of positive real numbers tending to 0 when n→∞. Remark that we can write F̂x(y)=gn(x,y)/f̂(x), where gn(x,y)=(1/n̂h1d)∑i∈ℐnKiHi(y) and f̂(x)=(1/n̂h1d)∑i∈ℐnKi are, respectively, the estimates of g(x,y)=Fx(y)fX(x) and fX(x). The kernel estimate of the conditional quantile is related to the conditional distribution estimator. A natural estimator q̂α(x) of qα(x) is defined such that(2.2)F̂x(q̂α(x))=α.Remark 2.1.
Another alternative characterization of theαth conditional quantile (see, e.g., Gannoun et al. [12]) is
(2.3)qα(x)=argminθ∈R⁡E[(2α-1)(Y-θ)+|Y-θ|∣X=x].
Then, one can consider the alternative local constant estimator (see Hallin et al. [15]) defined by
(2.4)q̃α(x)=argminθ∈R∑i∈In(|Yi-θ|+(2α-1)(Yi-θ))K1(x-Xih1).
Let us mention that it can be shown that (2.4) is equivalent to
(2.5)q̃α(x)=inf⁡{t∈R,F̃x(t)≥α},
where F̃x(·) is the same estimator of Fx(·) as F̂x(·) except that Hi(y)=1{Yi≤y}.
In this paper, we will focus on the study of the asymptotic behavior ofq̂α(x), since in practice some simulations permit to remark that the differences between this estimator and the local linear one are too small to affect any interpretations; see also Dabo-Niang and Thiam [17] and Gannoun et al. [12].


## 3. Assumptions

We denote byg(j) the derivative of order j of a function g. In what follows, C and C′ will denote any positive constant.


### 3.1. General Assumptions

We will use the following four regularity conditions (see, e.g., Gannoun et al. [12] or Ferraty et al. [28]).(H1)
The jointfX,Y and marginal fX densities are, respectively, continuous on ℝd+1 and ℝd. Moreover fX(x)>0 and fx(qα(x))>0.Iffx(qα(x))=0, then one can use a condition like Fx(·) is of class 𝒞j, Fx(k)(qα(x))=0 for 1≤k<j, and 0<C<|Fx(j)(y)|<C′<∞ as in Ferraty et al. [28].(H2)
Forall(x1,x2)∈Vx×Vx, ∀(y1,y2)∈ℝ2(3.1)|Fx1(y1)-Fx2(y2)|≤C(‖x1-x2‖b1+|y1-y2|b2),b1>0,b2>0.(H3)
We assume that the conditional densityf(Xi,Xj) of (Yi,Yj) given (Xi,Xj) exists and is uniformly bounded in i,j.For simplicity, we assume the following condition on the kernelK1 (see, e.g., Devroye [29]).(H4)
There existC1 and C2, 0<C1<C2<∞ such that(3.2)C1I[0,1](∥t∥)<K1(t)<C2I[0,1](∥t∥).(H5)
K2 is of class 𝒞1 with a symmetric, Lipschitz, bounded, and compact support density K. In addition, we assume that the restriction of K2on{t∈ℝ,K2(t)∈(0,1)} is a strictly increasing function.AssumptionH5 is classical in nonparametric estimation and is satisfied by usual kernels such as Epanechnikov and Biweight, whereas the Gaussian density K is also possible; it suffices to replace the compact support assumption by ∫ℝd|t|b2K(t)dt<∞. Assumption H5 ensures the existence and the uniqueness of the quantile estimate q̂α(x).(H6)
n̂h1θ1h2θ2logn̂θ3→+∞ with θ>2N.(H7)
n̂h1θ1′h2θ2′logn̂θ3′→+∞ with θ>N(1+2β̃),whereθ and β̃ are defined in Section 3.2.1, and(3.3)θ1=dθθ-2N,θ2=θθ-2N,θ3=-1,θ1′=d(N+θ)θ-N(1+2β̃),θ2′=N+θθ-N(1+2β̃),θ3′=N-θθ-N(1+2β̃).
Hypotheses H6 or H7 on the bandwidths are similar to that of Carbon et al. [19] and imply the classical condition in nonparametric estimation, that is, n̂h1dh2/logn̂→+∞.(H8)
n̂h1d+2b1→0 and n̂h1dh22b2→0.(H9)
Letr>2 be a given nonnegative real number. One assume that there exits a sequence of integers q=qn tending to infinity and such that(i)
[n̂(h1d)1+(1-2/r)2N]1/(2N)→∞,(ii)
q=o([n̂(h1d)1+(1-2/r)2N]1/(2N)),(iii)
n̂q-θ→0,(iv)
(h1d)-(1-2/r)q-θ(1-2/r)+N→0,withθ>N/(1-2/r).


### 3.2. Dependency Conditions

In spatial dependent data analysis, the dependence of the observations has to be measured. Here we will consider the following two dependence measures.


#### 3.2.1. Mixing Condition

The spatial dependence of the process will be measured by means ofα-mixing. Then, we consider the α-mixing coefficients of the field (Zi,i∈ℤN), defined by the following: there exists a function φ(t)↓0 as t→∞, such that EandE′ subsets of ℤN with finite cardinals are(3.4)α(B(E),B(E′))=sup⁡B∈B(E),C∈B(E′)|P(B∩C)-P(B)P(C)|≤ψ(Card(E),Card(E′))φ(dist⁡(E,E′)),
where Card(E) (resp., Card(E′)) is the cardinality of E (resp., E′), dist(E,E′) the Euclidean distance between E and E′, and ψ:ℕ2→ℝ+ is a symmetric positive function nondecreasing in each variable. Throughout the paper, it will be assumed that ψ satisfies either(3.5)ψ(n,m)≤Cmin⁡(n,m),∀n,m∈N
or(3.6)ψ(n,m)≤C(n+m+1)β̃,∀n,m∈N
for some β̃≥1 and some C>0. We assume also that the process satisfies a polynomial mixing condition:(3.7)φ(t)≤Ct-θ,θ>0,t∈R+*.Ifψ≡1, then Zi is called strongly mixing. Many stochastic processes, among them various useful time series models, satisfy strong mixing properties, which are relatively easy to check. Conditions (3.5)-(3.6) are used in Tran [30] and Carbon et al. [18, 19]. See Doukhan [31] for discussion on mixing and examples.


#### 3.2.2. Local Dependence Condition

Since we aim to get the same rate of convergence as in thei.i.d. case, we need some local dependency assumptions. Then, we assume the following local dependency condition used in Tran [30].The joint probability densityf(Xi,Xj) of (Xi,Xj) exists and satisfies(3.8)|f(Xi,Xj)(u,v)-fXi(u)fXj(v)|≤C
for some constant C and for all u,v,i,j.In addition, let the densityf(Zi,Zj) of (Zi,Zj) exist and(3.9)|f(Zi,Zj)(z,t)-fZi(z)fZj(t)|≤C;∀z,t∈Rd+1,i,j,
where C is a positive constant.In the following, the notations→𝒟 and →P mean, respectively, convergences in distribution and in probability.


## 3.1. General Assumptions

We will use the following four regularity conditions (see, e.g., Gannoun et al. [12] or Ferraty et al. [28]).(H1)
The jointfX,Y and marginal fX densities are, respectively, continuous on ℝd+1 and ℝd. Moreover fX(x)>0 and fx(qα(x))>0.Iffx(qα(x))=0, then one can use a condition like Fx(·) is of class 𝒞j, Fx(k)(qα(x))=0 for 1≤k<j, and 0<C<|Fx(j)(y)|<C′<∞ as in Ferraty et al. [28].(H2)
Forall(x1,x2)∈Vx×Vx, ∀(y1,y2)∈ℝ2(3.1)|Fx1(y1)-Fx2(y2)|≤C(‖x1-x2‖b1+|y1-y2|b2),b1>0,b2>0.(H3)
We assume that the conditional densityf(Xi,Xj) of (Yi,Yj) given (Xi,Xj) exists and is uniformly bounded in i,j.For simplicity, we assume the following condition on the kernelK1 (see, e.g., Devroye [29]).(H4)
There existC1 and C2, 0<C1<C2<∞ such that(3.2)C1I[0,1](∥t∥)<K1(t)<C2I[0,1](∥t∥).(H5)
K2 is of class 𝒞1 with a symmetric, Lipschitz, bounded, and compact support density K. In addition, we assume that the restriction of K2on{t∈ℝ,K2(t)∈(0,1)} is a strictly increasing function.AssumptionH5 is classical in nonparametric estimation and is satisfied by usual kernels such as Epanechnikov and Biweight, whereas the Gaussian density K is also possible; it suffices to replace the compact support assumption by ∫ℝd|t|b2K(t)dt<∞. Assumption H5 ensures the existence and the uniqueness of the quantile estimate q̂α(x).(H6)
n̂h1θ1h2θ2logn̂θ3→+∞ with θ>2N.(H7)
n̂h1θ1′h2θ2′logn̂θ3′→+∞ with θ>N(1+2β̃),whereθ and β̃ are defined in Section 3.2.1, and(3.3)θ1=dθθ-2N,θ2=θθ-2N,θ3=-1,θ1′=d(N+θ)θ-N(1+2β̃),θ2′=N+θθ-N(1+2β̃),θ3′=N-θθ-N(1+2β̃).
Hypotheses H6 or H7 on the bandwidths are similar to that of Carbon et al. [19] and imply the classical condition in nonparametric estimation, that is, n̂h1dh2/logn̂→+∞.(H8)
n̂h1d+2b1→0 and n̂h1dh22b2→0.(H9)
Letr>2 be a given nonnegative real number. One assume that there exits a sequence of integers q=qn tending to infinity and such that(i)
[n̂(h1d)1+(1-2/r)2N]1/(2N)→∞,(ii)
q=o([n̂(h1d)1+(1-2/r)2N]1/(2N)),(iii)
n̂q-θ→0,(iv)
(h1d)-(1-2/r)q-θ(1-2/r)+N→0,withθ>N/(1-2/r).


## 3.2. Dependency Conditions

In spatial dependent data analysis, the dependence of the observations has to be measured. Here we will consider the following two dependence measures.


### 3.2.1. Mixing Condition

The spatial dependence of the process will be measured by means ofα-mixing. Then, we consider the α-mixing coefficients of the field (Zi,i∈ℤN), defined by the following: there exists a function φ(t)↓0 as t→∞, such that EandE′ subsets of ℤN with finite cardinals are(3.4)α(B(E),B(E′))=sup⁡B∈B(E),C∈B(E′)|P(B∩C)-P(B)P(C)|≤ψ(Card(E),Card(E′))φ(dist⁡(E,E′)),
where Card(E) (resp., Card(E′)) is the cardinality of E (resp., E′), dist(E,E′) the Euclidean distance between E and E′, and ψ:ℕ2→ℝ+ is a symmetric positive function nondecreasing in each variable. Throughout the paper, it will be assumed that ψ satisfies either(3.5)ψ(n,m)≤Cmin⁡(n,m),∀n,m∈N
or(3.6)ψ(n,m)≤C(n+m+1)β̃,∀n,m∈N
for some β̃≥1 and some C>0. We assume also that the process satisfies a polynomial mixing condition:(3.7)φ(t)≤Ct-θ,θ>0,t∈R+*.Ifψ≡1, then Zi is called strongly mixing. Many stochastic processes, among them various useful time series models, satisfy strong mixing properties, which are relatively easy to check. Conditions (3.5)-(3.6) are used in Tran [30] and Carbon et al. [18, 19]. See Doukhan [31] for discussion on mixing and examples.


### 3.2.2. Local Dependence Condition

Since we aim to get the same rate of convergence as in thei.i.d. case, we need some local dependency assumptions. Then, we assume the following local dependency condition used in Tran [30].The joint probability densityf(Xi,Xj) of (Xi,Xj) exists and satisfies(3.8)|f(Xi,Xj)(u,v)-fXi(u)fXj(v)|≤C
for some constant C and for all u,v,i,j.In addition, let the densityf(Zi,Zj) of (Zi,Zj) exist and(3.9)|f(Zi,Zj)(z,t)-fZi(z)fZj(t)|≤C;∀z,t∈Rd+1,i,j,
where C is a positive constant.In the following, the notations→𝒟 and →P mean, respectively, convergences in distribution and in probability.


## 3.2.1. Mixing Condition

The spatial dependence of the process will be measured by means ofα-mixing. Then, we consider the α-mixing coefficients of the field (Zi,i∈ℤN), defined by the following: there exists a function φ(t)↓0 as t→∞, such that EandE′ subsets of ℤN with finite cardinals are(3.4)α(B(E),B(E′))=sup⁡B∈B(E),C∈B(E′)|P(B∩C)-P(B)P(C)|≤ψ(Card(E),Card(E′))φ(dist⁡(E,E′)),
where Card(E) (resp., Card(E′)) is the cardinality of E (resp., E′), dist(E,E′) the Euclidean distance between E and E′, and ψ:ℕ2→ℝ+ is a symmetric positive function nondecreasing in each variable. Throughout the paper, it will be assumed that ψ satisfies either(3.5)ψ(n,m)≤Cmin⁡(n,m),∀n,m∈N
or(3.6)ψ(n,m)≤C(n+m+1)β̃,∀n,m∈N
for some β̃≥1 and some C>0. We assume also that the process satisfies a polynomial mixing condition:(3.7)φ(t)≤Ct-θ,θ>0,t∈R+*.Ifψ≡1, then Zi is called strongly mixing. Many stochastic processes, among them various useful time series models, satisfy strong mixing properties, which are relatively easy to check. Conditions (3.5)-(3.6) are used in Tran [30] and Carbon et al. [18, 19]. See Doukhan [31] for discussion on mixing and examples.


## 3.2.2. Local Dependence Condition

Since we aim to get the same rate of convergence as in thei.i.d. case, we need some local dependency assumptions. Then, we assume the following local dependency condition used in Tran [30].The joint probability densityf(Xi,Xj) of (Xi,Xj) exists and satisfies(3.8)|f(Xi,Xj)(u,v)-fXi(u)fXj(v)|≤C
for some constant C and for all u,v,i,j.In addition, let the densityf(Zi,Zj) of (Zi,Zj) exist and(3.9)|f(Zi,Zj)(z,t)-fZi(z)fZj(t)|≤C;∀z,t∈Rd+1,i,j,
where C is a positive constant.In the following, the notations→𝒟 and →P mean, respectively, convergences in distribution and in probability.


## 4. Consistency Results

This section contains results on asymptotic normality of the conditional quantile estimate. The main result of this paper is given by the following theorem.Theorem 4.1.
Under assumptionsH1–H5andH8-H9, (3.4), (3.7), (3.8), (3.9), and (3.5) combined with H6 or (3.6) combined with H7, one has
(4.1)((fX,Y(x,qα(x)))2n̂h1dV(x))1/2(q̂α(x)-qα(x)-Cn(x))⟶DN(0,1),
where V(x)=α(1-α)fX(x)∫ℝd(K1(t))2dt and Cn(x)=(1/fX,Y(x,qα(x)))E[αf̂(x)-gn(x,qα(x))], with gn and f̂ being the estimates defined in Section 2.The proof of this theorem is based on the following three lemmas.Lemma 4.2.
Under assumptionsH2–H5andH9, (3.4), (3.7), (3.8), (3.9), and (3.5) or (3.6), one has
(4.2)[n̂h1dV(x)]1/2([αf̂(x)-gn(x,qα(x))]-E[αf̂(x)-gn(x,qα(x))])⟶DN(0,1).Lemma 4.3.
Under assumptionsH2,H4, and H5, one has:
(4.3)E[αf̂(x)-gn(x,qα(x))]=O(h1b1+h2b2).Lemma 4.4.
Under assumptions of Theorem4.1, one has:
(4.4)gn(1)(x,qα*(x))⟶fX,Y(x,qα(x))inprobability,
where qα*(x) is an element of the interval of extremities qα(x) and q̂α(x).Proof of Theorem4.1.
By assumptionH5,gn(x,·) is of class 𝒞1. Then a Taylor expansion on a neighborhood of qα(x) gives
(4.5)gn(x,q̂α(x))=gn(x,qα(x))+gn(1)(x,qα*(x))(q̂α(x)-qα(x)),
where qα*(x) is an element of the interval of extremities qα(x) and q̂α(x).
It follows that(4.6)q̂α(x)-qα(x)=1gn(1)(x,qα*(x))(gn(x,q̂α(x))-gn(x,qα(x)))=1gn(1)(x,qα*(x))(αf̂(x)-gn(x,qα(x))).
Then, we have
(4.7)((fX,Y(x,qα(x)))2n̂h1dV(x))1/2(q̂α(x)-qα(x)-Cn(x))=((fX,Y(x,qα(x)))2n̂h1dV(x))1/2×(1gn(1)(x,qα*(x))(αf̂(x)-gn(x,qα(x)))-1fX,Y(x,qα(x))E[αf̂(x)-gn(x,qα(x))])=[n̂h1dV(x)]1/2×fX,Y(x,qα(x))gn(1)(x,qα*(x))×((αf̂(x)-gn(x,qα(x)))-gn(1)(x,qα*(x))fX,Y(x,qα(x))E(αf̂(x)-gn(x,qα(x))))=fX,Y(x,qα(x))gn(1)(x,qα*(x))[n̂h1dV(x)]1/2×((αf̂(x)-gn(x,qα(x)))-E(αf̂(x)-gn(x,qα(x))))+fX,Y(x,qα(x))gn(1)(x,qα*(x))[n̂h1dV(x)]1/2×(1-gn(1)(x,qα*(x))fX,Y(x,qα(x)))E(αf̂(x)-gn(x,qα(x))).
Lemmas4.2 and 4.4 and Slutsky’s theorem imply that the first term of the right-hand side of the last equality above tends in distribution to N(0,1). In addition, Lemma 4.3 permits to write
(4.8)[n̂h1dV(x)]1/2E(αf̂(x)-gn(x,qα(x)))≤Cn̂h1d(h1b1+h2b2).
This last tends to 0 by H8. Thus, the second term goes to zero in probability. This yields the proof.Before going further, it should be interesting to give examples where all our conditions on the bandwidths are satisfied. That is done through the two following remarks.Remark 4.5.
Let us, for example, chooseb2=b1=1 and h2=h1=h=n̂-ν, where ν>0 is a real number such that H8 is verified. That is, n̂hd+2=n̂-ν(d+2)+1→0. Thus, it suffices to choose ν=(1/(d+2))+η, where η>0 is a real number arbitrarily small.
The hypothesisH6 holds if n̂hθ1+θ2lognθ3=n̂hd+((θ+2Nd)/(θ-2N))lognθ3→∞. Then, it suffices to have (θ+2Nd)/(θ-2N)<2. That is,  θ>2N(d+2). Similarly H7 holds if n̂hθ1′+θ2′lognθ3′=n̂hd+(2Nd(1+β̃)+N+θ)/(θ-N(1+2β̃))lognθ3′→∞. That is,  θ>2Nd(1+β̃)+N(3+4β̃).Remark 4.6.
Letγr=d(1-2/r)2N and h1=h2=h=n̂-(1/(d+2))-η be defined in the previous remark. Condition (i) of H9 is equivalent to n̂((2-γr)/(2N(d+2)))-η(d+γr)/2N→∞. Then, it suffices to have ((2-γr)/(2N(d+2)))-η(d+γr)/2N>0. As η is arbitrarily small, it is enough to have γr<2. That is, 2<r<2Nd/(Nd-1). In this case, for any positive real number τ such that ((2-γr)/2N(d+2))-η((d+γr+τ)/2N)>0, the choice q=⌊n̂((2-γr)/(2N(d+2)))-η((d+γr+τ)/2N)⌋+1(where⌊·⌋denotestheintegerpart) gives an example where H9(ii) is satisfied.
Letq*=n̂((2-γr)/(2N(d+2)))-η(d+γr+τ)/2N. We have n̂q-θ≤n̂q*-θ. Thus, to ensure that condition H9(iii) holds, it suffices to have  n̂q*-θ→0. That is,  θ>2N(d+2)/((2-γr)-η(d+2)(d+γr+τ)).
Lastly, the conditionH9(iv) is h-d(1-2/r)q-θ(1-2/r)+N→0. It can be written equivalently as h-γr/2Nq-θ(γr/2Nd)+N→0 or  h-γrdq-θγr+2N2d→0. Under the condition  θ>2N2d/γr, we have h-γrdq-θγr+2N2d≤h-γrdq*-θγr+2N2d. Then, to satisfy the condition  H9(iv), it is enough to have h-γrdq*-θγr+2N2d→0. That is, θ>(2N2d/γr)+((2N(d+ηd(d+2))(d+2))/((2-γr)-η(d+2)(d+γr+τ))).
Thus,H6,H8, and H9(resp.,H7–H9) are satisfied when 2<r<2Nd/(Nd-1) and
(4.9)θ>max⁡(N1-2/r,2N(d+2),2N(d+2)(2-γr)-η(d+2)(d+γr+τ),2N2dγr+2N(d+ηd(d+2))(2-γr)-η(d+2)(d+γr+τ)),(resp.θ>max⁡(N1-2/r,2Nd(1+β̃)+N(3+4β̃),2N(d+2)(2-γr)-η(d+2)(d+γr+τ),2N2dγr+2N(d+ηd(d+2))(2-γr)-η(d+2)(d+γr+τ))).


## 5. Application

In this section, we present a quantile prediction procedure and then apply it to simulated data.


### 5.1. Prediction Procedure

An application where a multidimensional stationary spatial process may be observed is the case of prediction of a strictly stationaryℝ-valued random field (ξi,i∈ℤN) at a given fixed point i0 when observations are taken from a subset of ℐn, not containing i0, see Biau and Cadre [20] or Dabo-Niang and Yao [27].Assume that the value of the field at a given location depends on the values taken by the field in a vicinity𝒰i0 of i0 (i0∉𝒰i0); then the random variables whose components are the {ξi,i∈𝒰i0} are an ℝd-valued spatial process, where d is the cardinal of 𝒰i0. In other words, we expect that the process (ξi) satisfies a Markov property; see, for example, Biau and Cadre [20] or Dabo-Niang and Yao [27]. Moreover, we assume that 𝒰i0=𝒰+i0, where 𝒰 is a fixed bounded set of sites that does not contain 0.Suppose that(ξi) is bounded and observed over a subset 𝒪n of ℐn. The aim of this section is to predict ξi0, at a given fixed point i0 not in 𝒪n⊂ℕN. It is well known that the best predictor of ξi0 given the data in 𝒰i0 in the sense of mean-square error is(5.1)E(ξ̂i0∣ξi,i∈Ui0).
Let 𝒰i=𝒰+i={u+i,u∈𝒰} for each i∈ℕN. To define a predictor of ξi0, let us consider the ℝd-valued random variables ξ̃i={ξu,u∈𝒰i⊂𝒪n}. The notation of the previous sections is used by setting Xi=ξ̃i,Yi=ξi,i∈ℕN.As a predictor ofξi0, one can take the conditional median estimate ξ̂i0=q̂0.5(ξ̃i0).We deduce from the previous consistency results the following corollary that gives the convergence of the predictorξ̂i0.Corollary 5.1.
Under the conditions of Theorem4.1(5.2)((f̂ξ̃i0(q0.5(ξ̃i0)))2f̂(ξ̃i0)n̂h1dV̂(ξ̃i0))1/2(ξ̂i0-q0.5(ξ̃i0))⟶DN(0,1),
where
(5.3)V̂(x)=α(1-α)f̂(x)∫Rd(K1(t))2dt(α=0.5).This consistency result permits to have an approximation of a confidence interval and predictive intervals that consists of the(α2-α1)100% confidence intervals with bounds q̂α1(ξ̃i0) and q̂α2(ξ̃i0), (α1<α2).


### 5.2. Numerical Properties

In this section, we study the performance of the conditional quantile predictor introduced in the previous section towards some simulations. Let us denote byGRF(m,σ2,s) a Gaussian random field with mean m and covariance function defined by(5.4)ϑ(h)=σ2exp⁡{-(‖h‖s)2},h∈R2.We consider a random field(ξi)i∈ℕ2 from the three following models.Model 1.
One has(5.5)ξi=Ui*(sin⁡(2Xi)+2exp⁡{-16Xi2})+Zi,i∈N2.Model 2.
One has(5.6)ξi=Ui*(2+2cos⁡(2Xi)+exp⁡{-4Xi2})+Zi,i∈N2.Model 3.
One has(5.7)ξi=Ui*(1+(23)exp⁡{Xi})+Zi,i∈N2,whereX=(Xi,i∈ℕ2) is a GRF(0,5,3), Z=(Zi,i∈ℕ2) is a GRF(0,0.1,5) independent of X, and Ui=(1/n̂)∑j∈Inexp(-∥i-j∥/2). The choice of Ui in the models is motivated by a reinforcement of the spatial local dependency. Set(5.8)In={i=(i,j)∈(N*)2,1≤i≤61,1≤j≤61}.
We supposed that the field (ξi,i∈ℕ2) is observable over the rectangular region In, observed over a subset 𝒪n and nonobserved in a subset Vn. A sample of size n̂=61*61=3721 obtained from each model is plotted in Figure 1.The observation sites and the random fields(ξi) of Models  1, 2, and 3 in, respectively, Panels (a), (b), (c), and (d).
(a)(b)(c)(d)For the prediction purpose, subsets𝒪n of size M=441 and Vn (different with respect to the model) and the quantiles of order 0.5,0.025,0.975 have been considered.We want to predict the valuesξi1,…,ξim at given fixed sites i1,…,im in Vn, with m=10. We provide the plots of the kernel densities estimators of the field of each model in Figure 2. The distributions of the models look asymmetric and highly heteroskedastic in Model  3. These graphics exhibit a strongly bimodal profile of Model  3. That means that a simple study of conditional mean or conditional median can miss some of the essential features of the dataset.The densities of the random fields(ξi) of Models  1–3 in, respectively, Panels (a), (b), and (c).
(a)(b)(c)As explained above, for anyk∈{1,…,m}, we take the conditional median estimate ξ̂ik, as a predictor of ξik. We compute these predictors with the vicinity 𝒰={-1,0,1}×{-1,0,1}∖{(0,0)} and select the standard normal density as kernel K1 and the Epanechnikov kernel as K. For the bandwidth selection, we use two bandwidth choices.(1)
We consider first the rule developed in Yu and Jones [32] and take h1=h2=hn:
(5.9)hn=hmean(α(1-α)ϕ(Φ-1(α))2)1/(4+d),
where hmean is the bandwidth for kernel smoothing estimation of the regression mean obtained by cross-validation, and ϕ and Φ are, respectively, the standard normal density and distribution function.(2)
The second rule is to choose firsth1=hn, and once h1 is computed, use it and compute h2 by cross-validation using the conditional distribution function estimate F̂x(·).Remark 5.2.
The selection of the appropriate bandwidths in double-kernel smoothing conditional quantile is an important task as in classical smoothing techniques.
As we see above, one can choose eitherh1=h2 or different bandwidths. The first choice can be motivated by the paper of Gannoun et al. [12] who dealt with a double kernel estimator of the conditional quantile function with a single bandwidth. They compute the common value of the bandwidth by the use of the rule developed in Yu and Jones [32] for local linear quantile regression. One can also consider the rule where (see Gannoun et al., 2003) h1=h2=hn where hmean, instead of being the bandwidth obtained by cross-validation of the regression mean estimate, is
(5.10)hmean=(CR1(K1)σ(x)n̂R2(K1)2(m′′(x))2g(x))1/(4+d),
where
(5.11)R1(K1)=∫-∞∞K12(u)du,R2(K1)=∫-∞∞u2K1(u)du,m(x) and σ(x) are, respectively, the conditional mean and conditional variance, and C is a constant. One can also choose the bandwidths using the following mean square error result of the conditional quantile estimate:
(5.12)MSE(q̂α(x))≈R1(K1)α(1-α)n̂h1dg(x)fx(qα(x))2+o(h12b1+h22b2+h1b1h2b2).
In the case of different bandwidthsh1andh2, we may still use the same rule as above for h1 (h1=hn). Therefore, only the choice of h2 seems to deserve more future theoretical investigation since the simulations (see Tables 4 and 5) suggest that different choices of these two bandwidths are appreciable. This theoretical choice of h2 (by, e.g., cross-validation) is beyond the scope of this paper and deserves future investigations.To evaluate the performance of the quantile predictorξ̂ik (with h1=h2) or ξ̂ik2 (with h1≠h2) and compare it to the mean regression predictor (see Biau and Cadre [20]):(5.13)ξ̆ik=∑i∈OnξiK((ξ̃ik-ξ̃i)/hK)∑i∈OnK((ξ̃ik-ξ̃i)/hK),
we compute the mean absolute errors:(5.14)MAE=110∑k=110|ξik-ξikn|,whereξikn=ξ̂ik,ξ̂ik2,orξ̆ik.Tables1–6 give the quantile estimates for ξik,k=1,…,m,p∈{0.025,0.5,0.975}, 𝒰={-1,0,1}×{-1,0,1}∖{(0,0)}, and the quantile and mean regression prediction errors of the predictors ξ̂ik and ξ̃ik. In each table, Inf (resp., Sup) is the lower (resp., upper) bound of the confidence interval (see above) of the median estimator. The bandwidth choice considered in these first three tables is the first choice h1=h2=hn. Clearly, the predictors give good results. The median prediction errors are rather smaller than those of mean regression for the first two models (less than 0.038 (resp., 0.012) for Model  1 (resp., Model  2)). Notice that the quantile and mean regression prediction results of Model 3 are similar and rather worse than those of Models  1 and 2. This can be explained by the fact that multimodality of the field of Model  3 cannot be captured by conditional median and the conditional mean.Table 1
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) and mean regression predictor of Model  1 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ikSupα=0.975ξ̆ik−0.341−0.267−0.204−0.170−0.0740.056−0.160−0.701−0.556−0.560−0.483−0.410−0.150−0.481−0.558−0.317−0.232−0.246−0.1750.017−0.278−0.369−0.118−0.107−0.072−0.0260.196−0.056−0.649−0.452−0.225−0.319−0.185−0.002−0.4810.0080.1920.2920.2710.3490.5130.270−0.402−0.273−0.235−0.185−0.0970.068−0.140−0.701−0.603−0.489−0.483−0.363−0.088−0.507−0.613−0.486−0.353−0.382−0.278−0.130−0.457−0.613−0.529−0.463−0.437−0.345−0.193−0.444MAE0.0380.073Table 2
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) and mean regression predictor of Model  1 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ikSupα=0.975ξ̆ik−0.486−0.311−0.228−0.248−0.1850.033−0.248−0.190−0.165−0.107−0.100−0.0360.080−0.095−0.138−0.314−0.077−0.0730.168−0.006−0.079−0.508−0.615−0.366−0.367−0.119−0.285−0.368−0.266−0.349−0.162−0.1710.008−0.011−0.1760.1540.1960.2510.2580.3200.3720.258−0.285−0.267−0.118−0.136−0.004−0.057−0.144−0.432−0.494−0.338−0.336−0.177−0.032−0.3160.046−0.3510.2120.2150.7810.3580.213−0.0150.0430.1620.1540.2660.3720.148MAE0.0080.012Table 3
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) and mean regression predictor of Model  2 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ikSupα=0.975ξ̆ik−0.508−0.331−0.269−0.276−0.222−0.140−0.303−0.0570.0690.1230.1200.1720.2910.140−0.487−0.428−0.357−0.367−0.306−0.175−0.399−0.550−0.484−0.397−0.402−0.321−0.267−0.373−0.0110.1360.1770.1770.2180.3730.1530.0110.0930.1320.1320.1710.2910.125−0.0690.1110.1500.1430.1750.3240.1340.1540.2660.3140.3160.3650.4100.3090.0680.1600.2270.2270.2940.3660.237−0.402−0.328−0.276−0.278−0.228−0.128−0.296MAE0.0040.02Table 4
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) for different bandwidth choices and the conditional median ξ̂ik obtained with the same bandwidths h1 and h2 of Model 1 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ik2Supα=0.975ξ̂ik−1.13−1.27−0.71−0.73−0.2−0.19−1.13−0.67−3.780.02−0.23.370.18−0.67−1.13−2.73−0.44−0.711.32−0.03−0.920.330.740.930.830.931.220.33−0.44−0.470.09−0.060.360.18−0.160.180.410.280.561.350.970.180.350.680.890.841.011.410.460.110.540.620.690.841.380.330.180.280.240.530.790.790.180.640.820.950.951.091.250.64MAE0.150.37Table 5
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) for different bandwidth choices and the conditional median ξ̂ik obtained with the same bandwidths h1 and h2 of Model 2 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ik2Supα=0.975ξ̂ik−2.18−1.97−1.4−1.69−1.4−1.17−1.97−1.02−0.48−0.96−0.39−0.310.38−1.02−0.98−0.88−0.81−0.76−0.63−0.39−1.12−0.88−0.84−0.44−0.59−0.35−0.04−0.88−0.160.30.280.410.510.91−0.25−0.88−0.55−0.52−0.43−0.30.23−0.880.641.211.521.291.371.940.57−0.76−0.41−0.09−0.140.140.54−0.590.030.420.730.540.671.29−0.04−0.76−0.55−0.45−0.39−0.22−0.04−0.76MAE0.180.48Table 6
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) for different bandwidth choices and the conditional median ξ̂ik obtained with the same bandwidths h1 and h2 of Model 3 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ik2Supα=0.975ξ̂ik0.190.70.50.790.881.330.79−0.180.240.070.350.471.190.35−0.48−1.7−0.48−0.141.420.29−0.14−2.35−3.02−2.54−2.17−1.31−1.68−2.17−2.35−2.29−2.1−2.17−2.05−1.41−2.170.280.650.770.750.851.650.77−2.35−6.14−1.44−1.686.11−0.47−1.68−0.66−0.060.230.210.491.120.21−0.040.320.620.420.531.190.42−0.66−0.63−0.3−0.10.430.69−0.1MAE0.20.2We derive from the results of the tables95% confidence and predictive intervals where the extremities are the 2.5% and 97.5% quantiles estimates, for each of the 10 prediction sites. Note that the confidence interval is generally more precise than the predictive one.Tables4–6 give the same estimates (for Models  1–3) as the first three tables but using both the same and the different choices of h1 and h2 described the aftermentioned rules. As proved by the numerical experiments (all the results are not presented here for seek of simplicity), the prediction results are improved when the sample size increases but notice that the improvement depends on the number of sites which are dependent on the prediction site in interest. Depending on the position of the sites, the conditional mean regression gives sometimes better prediction than the conditional median and vice versa. A next step would be to apply the predictor to a spatial real data that deserves future investigations.


## 5.1. Prediction Procedure

An application where a multidimensional stationary spatial process may be observed is the case of prediction of a strictly stationaryℝ-valued random field (ξi,i∈ℤN) at a given fixed point i0 when observations are taken from a subset of ℐn, not containing i0, see Biau and Cadre [20] or Dabo-Niang and Yao [27].Assume that the value of the field at a given location depends on the values taken by the field in a vicinity𝒰i0 of i0 (i0∉𝒰i0); then the random variables whose components are the {ξi,i∈𝒰i0} are an ℝd-valued spatial process, where d is the cardinal of 𝒰i0. In other words, we expect that the process (ξi) satisfies a Markov property; see, for example, Biau and Cadre [20] or Dabo-Niang and Yao [27]. Moreover, we assume that 𝒰i0=𝒰+i0, where 𝒰 is a fixed bounded set of sites that does not contain 0.Suppose that(ξi) is bounded and observed over a subset 𝒪n of ℐn. The aim of this section is to predict ξi0, at a given fixed point i0 not in 𝒪n⊂ℕN. It is well known that the best predictor of ξi0 given the data in 𝒰i0 in the sense of mean-square error is(5.1)E(ξ̂i0∣ξi,i∈Ui0).
Let 𝒰i=𝒰+i={u+i,u∈𝒰} for each i∈ℕN. To define a predictor of ξi0, let us consider the ℝd-valued random variables ξ̃i={ξu,u∈𝒰i⊂𝒪n}. The notation of the previous sections is used by setting Xi=ξ̃i,Yi=ξi,i∈ℕN.As a predictor ofξi0, one can take the conditional median estimate ξ̂i0=q̂0.5(ξ̃i0).We deduce from the previous consistency results the following corollary that gives the convergence of the predictorξ̂i0.Corollary 5.1.
Under the conditions of Theorem4.1(5.2)((f̂ξ̃i0(q0.5(ξ̃i0)))2f̂(ξ̃i0)n̂h1dV̂(ξ̃i0))1/2(ξ̂i0-q0.5(ξ̃i0))⟶DN(0,1),
where
(5.3)V̂(x)=α(1-α)f̂(x)∫Rd(K1(t))2dt(α=0.5).This consistency result permits to have an approximation of a confidence interval and predictive intervals that consists of the(α2-α1)100% confidence intervals with bounds q̂α1(ξ̃i0) and q̂α2(ξ̃i0), (α1<α2).


## 5.2. Numerical Properties

In this section, we study the performance of the conditional quantile predictor introduced in the previous section towards some simulations. Let us denote byGRF(m,σ2,s) a Gaussian random field with mean m and covariance function defined by(5.4)ϑ(h)=σ2exp⁡{-(‖h‖s)2},h∈R2.We consider a random field(ξi)i∈ℕ2 from the three following models.Model 1.
One has(5.5)ξi=Ui*(sin⁡(2Xi)+2exp⁡{-16Xi2})+Zi,i∈N2.Model 2.
One has(5.6)ξi=Ui*(2+2cos⁡(2Xi)+exp⁡{-4Xi2})+Zi,i∈N2.Model 3.
One has(5.7)ξi=Ui*(1+(23)exp⁡{Xi})+Zi,i∈N2,whereX=(Xi,i∈ℕ2) is a GRF(0,5,3), Z=(Zi,i∈ℕ2) is a GRF(0,0.1,5) independent of X, and Ui=(1/n̂)∑j∈Inexp(-∥i-j∥/2). The choice of Ui in the models is motivated by a reinforcement of the spatial local dependency. Set(5.8)In={i=(i,j)∈(N*)2,1≤i≤61,1≤j≤61}.
We supposed that the field (ξi,i∈ℕ2) is observable over the rectangular region In, observed over a subset 𝒪n and nonobserved in a subset Vn. A sample of size n̂=61*61=3721 obtained from each model is plotted in Figure 1.The observation sites and the random fields(ξi) of Models  1, 2, and 3 in, respectively, Panels (a), (b), (c), and (d).
(a)(b)(c)(d)For the prediction purpose, subsets𝒪n of size M=441 and Vn (different with respect to the model) and the quantiles of order 0.5,0.025,0.975 have been considered.We want to predict the valuesξi1,…,ξim at given fixed sites i1,…,im in Vn, with m=10. We provide the plots of the kernel densities estimators of the field of each model in Figure 2. The distributions of the models look asymmetric and highly heteroskedastic in Model  3. These graphics exhibit a strongly bimodal profile of Model  3. That means that a simple study of conditional mean or conditional median can miss some of the essential features of the dataset.The densities of the random fields(ξi) of Models  1–3 in, respectively, Panels (a), (b), and (c).
(a)(b)(c)As explained above, for anyk∈{1,…,m}, we take the conditional median estimate ξ̂ik, as a predictor of ξik. We compute these predictors with the vicinity 𝒰={-1,0,1}×{-1,0,1}∖{(0,0)} and select the standard normal density as kernel K1 and the Epanechnikov kernel as K. For the bandwidth selection, we use two bandwidth choices.(1)
We consider first the rule developed in Yu and Jones [32] and take h1=h2=hn:
(5.9)hn=hmean(α(1-α)ϕ(Φ-1(α))2)1/(4+d),
where hmean is the bandwidth for kernel smoothing estimation of the regression mean obtained by cross-validation, and ϕ and Φ are, respectively, the standard normal density and distribution function.(2)
The second rule is to choose firsth1=hn, and once h1 is computed, use it and compute h2 by cross-validation using the conditional distribution function estimate F̂x(·).Remark 5.2.
The selection of the appropriate bandwidths in double-kernel smoothing conditional quantile is an important task as in classical smoothing techniques.
As we see above, one can choose eitherh1=h2 or different bandwidths. The first choice can be motivated by the paper of Gannoun et al. [12] who dealt with a double kernel estimator of the conditional quantile function with a single bandwidth. They compute the common value of the bandwidth by the use of the rule developed in Yu and Jones [32] for local linear quantile regression. One can also consider the rule where (see Gannoun et al., 2003) h1=h2=hn where hmean, instead of being the bandwidth obtained by cross-validation of the regression mean estimate, is
(5.10)hmean=(CR1(K1)σ(x)n̂R2(K1)2(m′′(x))2g(x))1/(4+d),
where
(5.11)R1(K1)=∫-∞∞K12(u)du,R2(K1)=∫-∞∞u2K1(u)du,m(x) and σ(x) are, respectively, the conditional mean and conditional variance, and C is a constant. One can also choose the bandwidths using the following mean square error result of the conditional quantile estimate:
(5.12)MSE(q̂α(x))≈R1(K1)α(1-α)n̂h1dg(x)fx(qα(x))2+o(h12b1+h22b2+h1b1h2b2).
In the case of different bandwidthsh1andh2, we may still use the same rule as above for h1 (h1=hn). Therefore, only the choice of h2 seems to deserve more future theoretical investigation since the simulations (see Tables 4 and 5) suggest that different choices of these two bandwidths are appreciable. This theoretical choice of h2 (by, e.g., cross-validation) is beyond the scope of this paper and deserves future investigations.To evaluate the performance of the quantile predictorξ̂ik (with h1=h2) or ξ̂ik2 (with h1≠h2) and compare it to the mean regression predictor (see Biau and Cadre [20]):(5.13)ξ̆ik=∑i∈OnξiK((ξ̃ik-ξ̃i)/hK)∑i∈OnK((ξ̃ik-ξ̃i)/hK),
we compute the mean absolute errors:(5.14)MAE=110∑k=110|ξik-ξikn|,whereξikn=ξ̂ik,ξ̂ik2,orξ̆ik.Tables1–6 give the quantile estimates for ξik,k=1,…,m,p∈{0.025,0.5,0.975}, 𝒰={-1,0,1}×{-1,0,1}∖{(0,0)}, and the quantile and mean regression prediction errors of the predictors ξ̂ik and ξ̃ik. In each table, Inf (resp., Sup) is the lower (resp., upper) bound of the confidence interval (see above) of the median estimator. The bandwidth choice considered in these first three tables is the first choice h1=h2=hn. Clearly, the predictors give good results. The median prediction errors are rather smaller than those of mean regression for the first two models (less than 0.038 (resp., 0.012) for Model  1 (resp., Model  2)). Notice that the quantile and mean regression prediction results of Model 3 are similar and rather worse than those of Models  1 and 2. This can be explained by the fact that multimodality of the field of Model  3 cannot be captured by conditional median and the conditional mean.Table 1
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) and mean regression predictor of Model  1 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ikSupα=0.975ξ̆ik−0.341−0.267−0.204−0.170−0.0740.056−0.160−0.701−0.556−0.560−0.483−0.410−0.150−0.481−0.558−0.317−0.232−0.246−0.1750.017−0.278−0.369−0.118−0.107−0.072−0.0260.196−0.056−0.649−0.452−0.225−0.319−0.185−0.002−0.4810.0080.1920.2920.2710.3490.5130.270−0.402−0.273−0.235−0.185−0.0970.068−0.140−0.701−0.603−0.489−0.483−0.363−0.088−0.507−0.613−0.486−0.353−0.382−0.278−0.130−0.457−0.613−0.529−0.463−0.437−0.345−0.193−0.444MAE0.0380.073Table 2
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) and mean regression predictor of Model  1 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ikSupα=0.975ξ̆ik−0.486−0.311−0.228−0.248−0.1850.033−0.248−0.190−0.165−0.107−0.100−0.0360.080−0.095−0.138−0.314−0.077−0.0730.168−0.006−0.079−0.508−0.615−0.366−0.367−0.119−0.285−0.368−0.266−0.349−0.162−0.1710.008−0.011−0.1760.1540.1960.2510.2580.3200.3720.258−0.285−0.267−0.118−0.136−0.004−0.057−0.144−0.432−0.494−0.338−0.336−0.177−0.032−0.3160.046−0.3510.2120.2150.7810.3580.213−0.0150.0430.1620.1540.2660.3720.148MAE0.0080.012Table 3
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) and mean regression predictor of Model  2 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ikSupα=0.975ξ̆ik−0.508−0.331−0.269−0.276−0.222−0.140−0.303−0.0570.0690.1230.1200.1720.2910.140−0.487−0.428−0.357−0.367−0.306−0.175−0.399−0.550−0.484−0.397−0.402−0.321−0.267−0.373−0.0110.1360.1770.1770.2180.3730.1530.0110.0930.1320.1320.1710.2910.125−0.0690.1110.1500.1430.1750.3240.1340.1540.2660.3140.3160.3650.4100.3090.0680.1600.2270.2270.2940.3660.237−0.402−0.328−0.276−0.278−0.228−0.128−0.296MAE0.0040.02Table 4
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) for different bandwidth choices and the conditional median ξ̂ik obtained with the same bandwidths h1 and h2 of Model 1 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ik2Supα=0.975ξ̂ik−1.13−1.27−0.71−0.73−0.2−0.19−1.13−0.67−3.780.02−0.23.370.18−0.67−1.13−2.73−0.44−0.711.32−0.03−0.920.330.740.930.830.931.220.33−0.44−0.470.09−0.060.360.18−0.160.180.410.280.561.350.970.180.350.680.890.841.011.410.460.110.540.620.690.841.380.330.180.280.240.530.790.790.180.640.820.950.951.091.250.64MAE0.150.37Table 5
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) for different bandwidth choices and the conditional median ξ̂ik obtained with the same bandwidths h1 and h2 of Model 2 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ik2Supα=0.975ξ̂ik−2.18−1.97−1.4−1.69−1.4−1.17−1.97−1.02−0.48−0.96−0.39−0.310.38−1.02−0.98−0.88−0.81−0.76−0.63−0.39−1.12−0.88−0.84−0.44−0.59−0.35−0.04−0.88−0.160.30.280.410.510.91−0.25−0.88−0.55−0.52−0.43−0.30.23−0.880.641.211.521.291.371.940.57−0.76−0.41−0.09−0.140.140.54−0.590.030.420.730.540.671.29−0.04−0.76−0.55−0.45−0.39−0.22−0.04−0.76MAE0.180.48Table 6
Quantile estimates (q̂α(ξ̃i0),α∈{0.025,0.5,0.975}) for different bandwidth choices and the conditional median ξ̂ik obtained with the same bandwidths h1 and h2 of Model 3 and confidence interval for 𝒰 and n̂=441.
n̂=441α=0.025InfTrueξ̂ik2Supα=0.975ξ̂ik0.190.70.50.790.881.330.79−0.180.240.070.350.471.190.35−0.48−1.7−0.48−0.141.420.29−0.14−2.35−3.02−2.54−2.17−1.31−1.68−2.17−2.35−2.29−2.1−2.17−2.05−1.41−2.170.280.650.770.750.851.650.77−2.35−6.14−1.44−1.686.11−0.47−1.68−0.66−0.060.230.210.491.120.21−0.040.320.620.420.531.190.42−0.66−0.63−0.3−0.10.430.69−0.1MAE0.20.2We derive from the results of the tables95% confidence and predictive intervals where the extremities are the 2.5% and 97.5% quantiles estimates, for each of the 10 prediction sites. Note that the confidence interval is generally more precise than the predictive one.Tables4–6 give the same estimates (for Models  1–3) as the first three tables but using both the same and the different choices of h1 and h2 described the aftermentioned rules. As proved by the numerical experiments (all the results are not presented here for seek of simplicity), the prediction results are improved when the sample size increases but notice that the improvement depends on the number of sites which are dependent on the prediction site in interest. Depending on the position of the sites, the conditional mean regression gives sometimes better prediction than the conditional median and vice versa. A next step would be to apply the predictor to a spatial real data that deserves future investigations.



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*Source: 462157-2012-02-22.xml*