Research Article | Open Access
Sidi Ali Ould Abdi, Sophie Dabo-Niang, Aliou Diop, Ahmedoune Ould Abdi, "Asymptotic Normality of a Nonparametric Conditional Quantile Estimator for Random Fields", Advances in Decision Sciences, vol. 2011, Article ID 462157, 35 pages, 2011. https://doi.org/10.1155/2011/462157
Asymptotic Normality of a Nonparametric Conditional Quantile Estimator for Random Fields
Given a stationary multidimensional spatial process , we investigate a kernel estimate of the spatial conditional quantile function of the response variable given the explicative variable . Asymptotic normality of the kernel estimate is obtained when the sample considered is an -mixing sequence.
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 in , the integer lattice points in the -dimensional Euclidean space (). 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 , Guyon , Anselin and Florax , Cressie , or Ripley .
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 for independent samples and time-dependent observations; see, for example, Stute , Samanta , Portnoy , Koul and Mukherjee , Honda , Cai , Gannoun et al. , and Yu et al. . 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 , Hallin et al. , Abdi et al. , and Dabo-Niang and Thiam  have paid attention to study these regression methods. Hallin et al.  gave a Bahadur representation and an asymptotic normality results of a local linear conditional quantile estimator. The method of Koenker and Mizera  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.  deals with —mean () and almost complete consistencies of a kernel estimate of conditional quantiles. The paper of Dabo-Niang and Thiam  gives the 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 , Lu and Chen ([21, 22]), Hallin et al. ([23, 24]), Lahiri and Zhu , Carbon et al. , and Dabo-Niang and Yao .
The rest of the paper is organized as follows. In Section 2, 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 be an -valued measurable strictly stationary spatial process , with same distribution as the vector of variables and defined on a probability space . A point in will be referred to as a site and may also include a time component.
We assume that the process under study is observed over a rectangular domain , . We will write if and for a constant such that for all such that . In the sequel, all the limits are considered when . For , we set .
Let be a set of sites. will denote in what follows, the Borel -field generated by . We assume that the regular version of the conditional probability of given exists and has a bounded density with respect to Lebesgue’s measure over . For all , we denote by (resp., ) the conditional distribution function (resp., the conditional density) of given . In the following, is a fixed point in and we denote by a neighborhood of this point. For , we denote by the Euclidian norm of . We suppose that the marginal and joint densities and of, respectively, and exist with respect to Lebesgue measures on and .
For , the conditional quantile of order of , denoted by , can be written as a solution of the equation .
To insure that exists and is unique, we assume that is strictly increasing. Let and . The conditional distribution and the corresponding density can be estimated by the following respective estimators: where is a kernel density, is a distribution function, is the first derivative of , and (resp., ) are sequences of positive real numbers tending to 0 when . Remark that we can write , where and are, respectively, the estimates of and . The kernel estimate of the conditional quantile is related to the conditional distribution estimator. A natural estimator of is defined such that
Remark 2.1. Another alternative characterization of the conditional quantile (see, e.g., Gannoun et al. ) is
Then, one can consider the alternative local constant estimator (see Hallin et al. ) defined by
Let us mention that it can be shown that (2.4) is equivalent to
where is the same estimator of as except that .
In this paper, we will focus on the study of the asymptotic behavior of , 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  and Gannoun et al. .
We denote by the derivative of order of a function . In what follows, and will denote any positive constant.
3.1. General Assumptions
If , then one can use a condition like is of class , for , and as in Ferraty et al. ., We assume that the conditional density of given exists and is uniformly bounded in .
For simplicity, we assume the following condition on the kernel (see, e.g., Devroye ).There exist and , such that is of class with a symmetric, Lipschitz, bounded, and compact support density . In addition, we assume that the restriction of is a strictly increasing function.
Assumption is classical in nonparametric estimation and is satisfied by usual kernels such as Epanechnikov and Biweight, whereas the Gaussian density is also possible; it suffices to replace the compact support assumption by . Assumption ensures the existence and the uniqueness of the quantile estimate . with . with ,
where and are defined in Section 3.2.1, and Hypotheses or on the bandwidths are similar to that of Carbon et al.  and imply the classical condition in nonparametric estimation, that is, . and . Let be a given nonnegative real number. One assume that there exits a sequence of integers tending to infinity and such that(i),(ii),(iii), (iv),
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 of -mixing. Then, we consider the -mixing coefficients of the field , defined by the following: there exists a function as , such that subsets of with finite cardinals are where Card (resp., Card) is the cardinality of (resp., ), dist the Euclidean distance between and , and is a symmetric positive function nondecreasing in each variable. Throughout the paper, it will be assumed that satisfies either or for some and some . We assume also that the process satisfies a polynomial mixing condition:
If , then 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  and Carbon et al. [18, 19]. See Doukhan  for discussion on mixing and examples.
3.2.2. Local Dependence Condition
Since we aim to get the same rate of convergence as in the i.i.d. case, we need some local dependency assumptions. Then, we assume the following local dependency condition used in Tran .
The joint probability density of exists and satisfies for some constant and for all .
In addition, let the density of exist and where is a positive constant.
In the following, the notations and 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.
The proof of this theorem is based on the following three lemmas.
Lemma 4.3. Under assumptions , and , one has:
Lemma 4.4. Under assumptions of Theorem 4.1, one has: where is an element of the interval of extremities and .
Proof of Theorem 4.1. By assumption is of class . Then a Taylor expansion on a neighborhood of gives
where is an element of the interval of extremities and .
It follows that Then, we have
Lemmas 4.2 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 . In addition, Lemma 4.3 permits to write This last tends to 0 by . 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, choose and , where is a real number such that is verified. That is, . Thus, it suffices to choose , where is a real number arbitrarily small.
The hypothesis holds if . Then, it suffices to have . That is, . Similarly holds if . That is, .
Remark 4.6. Let and be defined in the previous remark. Condition (i) of is equivalent to . Then, it suffices to have . As is arbitrarily small, it is enough to have . That is, . In this case, for any positive real number such that , the choice gives an example where is satisfied.
Let . We have . Thus, to ensure that condition holds, it suffices to have . That is, .
Lastly, the condition is . It can be written equivalently as or . Under the condition , we have . Then, to satisfy the condition , it is enough to have . That is, .
Thus, , and are satisfied when and
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 stationary valued random field at a given fixed point when observations are taken from a subset of , not containing , see Biau and Cadre  or Dabo-Niang and Yao .
Assume that the value of the field at a given location depends on the values taken by the field in a vicinity of (); then the random variables whose components are the are an valued spatial process, where is the cardinal of . In other words, we expect that the process satisfies a Markov property; see, for example, Biau and Cadre  or Dabo-Niang and Yao . Moreover, we assume that , where is a fixed bounded set of sites that does not contain .
Suppose that is bounded and observed over a subset of . The aim of this section is to predict , at a given fixed point not in . It is well known that the best predictor of given the data in in the sense of mean-square error is Let for each . To define a predictor of , let us consider the -valued random variables . The notation of the previous sections is used by setting .
As a predictor of , one can take the conditional median estimate .
We deduce from the previous consistency results the following corollary that gives the convergence of the predictor .
Corollary 5.1. Under the conditions of Theorem 4.1 where
This consistency result permits to have an approximation of a confidence interval and predictive intervals that consists of the confidence intervals with bounds and , ().
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 by a Gaussian random field with mean and covariance function defined by
We consider a random field from the three following models.
Model 1. One has
Model 2. One has
Model 3. One has
where is a , is a independent of , and . The choice of in the models is motivated by a reinforcement of the spatial local dependency. Set We supposed that the field is observable over the rectangular region , observed over a subset and nonobserved in a subset . A sample of size obtained from each model is plotted in Figure 1.
For the prediction purpose, subsets of size and (different with respect to the model) and the quantiles of order have been considered.
We want to predict the values at given fixed sites in , with . 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.
As explained above, for any , we take the conditional median estimate , as a predictor of . We compute these predictors with the vicinity and select the standard normal density as kernel and the Epanechnikov kernel as . For the bandwidth selection, we use two bandwidth choices. We consider first the rule developed in Yu and Jones  and take : where is the bandwidth for kernel smoothing estimation of the regression mean obtained by cross-validation, and and are, respectively, the standard normal density and distribution function. The second rule is to choose first , and once is computed, use it and compute by cross-validation using the conditional distribution function estimate .
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 either or different bandwidths. The first choice can be motivated by the paper of Gannoun et al.  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  for local linear quantile regression. One can also consider the rule where (see Gannoun et al., 2003) where , instead of being the bandwidth obtained by cross-validation of the regression mean estimate, is where and are, respectively, the conditional mean and conditional variance, and is a constant. One can also choose the bandwidths using the following mean square error result of the conditional quantile estimate:
In the case of different bandwidths , we may still use the same rule as above for (). Therefore, only the choice of 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 (by, e.g., cross-validation) is beyond the scope of this paper and deserves future investigations.
To evaluate the performance of the quantile predictor (with ) or (with ) and compare it to the mean regression predictor (see Biau and Cadre ): we compute the mean absolute errors:
Tables 1–6 give the quantile estimates for , , and the quantile and mean regression prediction errors of the predictors and . 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 . 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.
We derive from the results of the tables confidence and predictive intervals where the extremities are the and quantiles estimates, for each of the 10 prediction sites. Note that the confidence interval is generally more precise than the predictive one.
Tables 4–6 give the same estimates (for Models 1–3) as the first three tables but using both the same and the different choices of and 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.
This section is devoted to proofs of lemmas used to establish the main result of Section 4. To this end, some needed additional lemmas and results will be stated and proved in this section.
Before giving the proof of Lemma 4.2, we introduce the following notation and establish the following preliminary lemma.
Let be the random variable defined by
Remark A.2. If one takes in , then .
Proof of Lemma A.1. Note that we can deduce easily from assumption the existence of two positive constants and such that
for all and any given integer .
Now, let us calculate the variance term. We have
Let us first consider . By taking the conditional expectation with respect to , we get From an integration by parts and an usual change of variables, we obtain Then, using, respectively, , , and , we have Thus, we can write Using (A.2) and the fact that the bandwidths tend to 0, we get .
Concerning , we have Then, we can write The conditional expectation with respect to permits to write For the second term of the right-hand side of (A.10), the same argument as that used above and (A.2) permit to obtain For the first term, an integration by parts, assumptions , , and lead to Thus, we have From (A.2), we have . It is then clear that Therefore, the term has the same limit as the last term of the right-hand side of (A.10) which goes directly (by Bochner’s lemma) to .
Consequently, we have