^{1}

^{2}

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In this paper, we establish asymptotically optimal simultaneous confidence bands for the copula function based on the local linear kernel estimator proposed by Chen and Huang [1]. For this, we prove under smoothness conditions on the derivatives of the copula a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation. We also show that the bias term converges uniformly to zero with a precise rate. The performance of these bands is illustrated by a simulation study. An application based on pseudo-panel data is also provided for modeling the dependence structure of Senegalese households’ expense data in 2001 and 2006.

Let us consider a random vector

The function C is called a copula associated with the random vector

where

From these facts, estimating bivariate distribution function can be achieved in two steps: 1) estimating the margins F and G; 2) estimating the copula C.

In this paper, we are dealing with nonparametric copula estimation. We consider a copula function C with uniform margins U and V defined on

The aim of this paper is to construct asymptotic optimal confidence bands, for the copula C, from the local linear kernel estimator proposed by Chen and Huang [

There are two main methods for estimating copula functions: parametric and nonparametric methods. The Maximum likelihood estimation method (MLE) and the moment method are popular parametric approaches. It happens that one may use a nonparametric approach like the MLE-method and, at the same time, estimates margins by using parametric methods. Such an approach is called a semi-parametric estimation method (see [

A pure nonparametric estimation of copulas treats both the copula and the margins in a parameter-free way and thus offers the greatest generality. Nonparametric estimation of copulas goes back to Deheuvels [

Omelka, Gijbels and Veraverbeke [

In parallel, powerful technologies have been developed for density and distribution function kernel estimation. We refer to Mason [

Let

where

unknown copula C. To prevent boundary bias, Chen and Huang suggested using a local linear version of the kernel k given by

where

where

Our best achievement is the construction of asymptotic confidence bands from a uniform in bandwidth law of the iterated logarithm (LIL) for the maximal deviation of the local linear estimator (1), and the uniform convergence of the bias to zero with the same speed of convergence.

The paper is organized as follows. In Section 2, we expose our main results in Theorems 1, 2 and 3. Simulation studies and applications to real data sets are also made in this section to illustrate these results. In Section 3, we report the proofs of our assertions. The paper is ended by Appendix in which we postpone some technical results and numerical computations.

Here, we state our theoretical results. Theorem 1 gives a uniform in bandwidth LIL for the maximal deviation of the estimator (1). Theorem 2 handles the bias, while Theorem 3 provides asymptotic optimal simultaneous confidence bands for the copula function

Theorem 1. Suppose that the copula function

where

Remark 1. Theorem 1 represents a uniform in bandwidth law of the iterated logarithm for the maximal deviation of the estimator (1). As in [

Theorem 2. Suppose that the copula function

Because a number of copula families do not possess bounded second-order partial derivatives, the application of these results is limited by a corner bias problem. To overcome this difficulty and apply these results to a wide family of copulas, we adopt the shrinkage method of Omelka et al. [

where

For such a bandwidth

By condition (H_{1}), (5) is equivalent for

This latter estimator (6) is exactly the improved shrinked version proposed by Omelka et al. [

If conditions (H_{1}) and (H_{2}) hold, then we can infer from Theorem 1 that

This is still equivalent to

To make use of (7) for forming confidence bands, we must ensure that the bandwidth

This would be the case if condition H_{2}) holds and

Theorem 3. Suppose that the assumptions of Theorem 1 and Theorem 2 hold. Then for any local data-driven bandwidth _{1}) and (H_{2}), and any

and,

where

Remark 2. Whenever (9) and (10) hold jointly for each

provide asymptotic simultaneous optimal confidence bands (at an asymptotic confidence level of 100%) for the copula function

We make some simulation studies to evaluate the performance of our asymptotic confidence bands. To this end, we compute the confidence bands given in (11) for some classical parametric copulas, and check for whether the true copula is lying in these bands. For simplicity, we consider for example two families of copulas: Frank and Clayton, defined respectively as follows:

and

We fix values for the parameter

・ step 1: Generate two values u and v from

・ step 2: Set

・ step 3: Compute

Then _{1})

and (H_{2}) are fulfilled. That is the case for

In

We can also remark some simulitudes between

As we cannot visualize all the information in the above figures, we provide in Appendix some numerical computations to best appreciate the performance of our bands. To this end, we generate 10 couples

In this subsection, we apply our theoretical results to select graphically, among various copula families, the one that best fits sample data. Towards this end, we shall represent in a same 2-dimensional graphic the confidence bands established in Theorem 3 and the curves corresponding to the different copulas considered. To illustrate this, we use data expenses of Senegalese households, available in databases managed by the National Agency of Statistics and Demography (ANSD) of the Republic of Senegal (www.ansd.sn). The data were obtained from two sample surveys: ESAM2 (Senegalese Survey of Households, 2nd edition, 2001-2002) and ESPS (Monitoring Survey of Poverty in Senegal, 2005-2006). Because of the not availability of recent data, we deal with the pseudo-panel data utilized in [

Instead of smoothing these observations denoted by

to define the kernel estimator of the true copula. Here,

This application is limited to Archimedean copulas. We will consider for example three parametric families of copulas: Frank, Gumbel and Clayton. Our aim is to find graphically, using our confidence bands, the family that best fits these pseudo-panel data. The unknown parameter

We now apply the maximum likelihood method for fitting copulas and compare it with our graphical method described in

Copula | ||
---|---|---|

Clayton | 1.38 | |

Gumbel | 1.69 | |

Frank | −0.57 |

Copula | Estimation of q | Log-likelihood |
---|---|---|

Clayton | 1.38 | 547.61 |

Gumbel | 1.69 | 28.57 |

Frank | −0.57 | 299.78 |

where

From

This paper presented a nonparametric method to estimate the copula function by providing asymptotic confidence bands based on the local linear kernel estimator. The results are applied to select graphically the best copula function that fits the dependence structure of the Senegalese households pseudo-panel data.

In perspective, similar results can be obtained with other kernel-type estimators of copula function like the mirror-reflection and transformation estimators.

In this section, we first expose technical details allowing us to use the methodology of Mason [

We begin by decomposing the difference

The probabilistic term

is called the deviation of the estimator from its expectation. We’ll study its behavior by making use of the methodology described in [

is the so-called bias of the estimator. It is deterministic and its behavior will depend upon the smoothness conditions on the copula C and the bandwidth h.

Recall the estimator proposed by Deheuvels in [

where

with

To study the behavior of the deviation

be the uniform bivariate empirical distribution function based on a sample

Then one can observe that

For all

where g belongs to a class of measurable functions

Since

To make use of Mason’s Theorem in [

(G.ii)There exists some constant

(F.i)

(F.ii)

The checking of these conditions constitutes the proof of the following proposition which will be done in Appendix.

Proposition 1. Suppose that the copula function C has bounded first order partial derivatives on

where

Corollary 1. Under the assumptions of Proposition 1, for any sequence of constants

Proof. (Corollary 1)

First, observe that the condition

Next, by the monotonicity of the function

Combining this and Proposition 1, we obtain

Thus the Corollary 1 follows from (16).

Proof. (Theorem 1)

The proof is based upon an approximation of the empirical copula process

By Theorem 3.2 in [

where

This yields

By the works of Wichura on the iterated law of logarithm (see [

which readily implies

Since

The proof is then finished by applying Corollary 1 which yields

Thus, there exists a constant

Proof. (Theorem 2)

For all

and

with

Hence

By continuity of F and G, we have for n large enough,

and

Thus,

By applying a 2-order Taylor expansion and taking account of the symmetry of the kernels

we obtain, by Fubini, that for all

Since the second order partial derivatives are assumed to be bounded, then we can infer that

and hence,

Proof. (Theorem 3)

From (8), we can infer that for any given

That is

On the other hand we deduce from (7) that for all

Case 1. If

then (23) becomes

Thus, for any given

Case 2. If

then, analogously to Case 1, we can infer from (23) that, for any given

Letting

or

Now, by observing that

we can write, for any

and

That is, (9) and (10) hold.

The authors are very grateful to anonymous referees for their valuable comments and suggestions.

DiamBâ,Cheikh TidianeSeck,Gane SambLô, (2015) Asymptotic Confidence Bands for Copulas Based on the Local Linear Kernel Estimator. Applied Mathematics,06,2077-2095. doi: 10.4236/am.2015.612183

Proof. It suffices to check the conditions (G.i), (G.ii), (F.i) and (F.ii) given in Section 3.

Checking for (G.i). For

Then,

This implies

Checking for (G.ii).

We have to show that

where

Now we express A and B as integrals of the copula function

Since because

We can also notice that

Thus

For n enough large, we have by continuity of F and G,

and

By splitting the integrals, we obtain after simple calculus that

All these six terms can be bounded up by applying Taylor expansion. Precisely, we have

From this, we can conclude that

and

with

Checking for (F.i). We have to check that

Consider the following classes of functions:

It is clear that by applying the lemmas 2.6.15 and 2.6.18 in van der Vaart and Wellner (see [

Checking for (F.ii).

Define the class of functions

It’s clear that

and, for

where

Let

and

Then, one can easily see that

This implies, for all large m, that

By right-continuity of

and conclude that

Different values of | ||||
---|---|---|---|---|

(0.58, 0.12) | −0.320 | 0.097 | 0.701 | |

(0.54, 0.47) | −0.139 | 0.302 | 0.887 | |

(0.07, 0.50) | −0.455 | 0.056 | 0.857 | |

(0.21, 0.59) | −0.355 | 0.162 | 0.672 | |

(0.18, 0.42) | −0.441 | 0.118 | 0.585 | |

(0.42, 0.52) | −0.204 | 0.268 | 1.032 | |

(0.73, 0.61) | 0.005 | 0.475 | 0.887 | |

(0.07, 0.96) | −0.271 | 0.069 | 0.755 | |

(0.69, 0.85) | 0.090 | 0.602 | 1.118 | |

(0.28, 0.72) | −0.298 | 0.233 | 0.729 | |

(0.96, 0.29) | 0.081 | 0.289 | 1.100 | |

(0.16, 0.45) | −0.362 | 0.152 | 0.664 | |

(0.67, 0.75) | 0.066 | 0.576 | 1.093 | |

(0.22, 0.47) | −0.320 | 0.203 | 0.706 | |

(0.12, 0.55) | −0.240 | 0.118 | 0.786 | |

(0.85, 0.80) | 0.082 | 0.716 | 1.109 | |

(0.46, 0.42) | −0.174 | 0.326 | 0.852 | |

(0.83, 0.22) | −0.327 | 0.217 | 0.700 | |

(0.65, 0.26) | −0.223 | 0.248 | 0.804 | |

(0.11, 0.30) | −0.251 | 0.103 | 0.776 | |

(0.32, 0.40) | −0.163 | 0.307 | 0.863 | |

(0.85, 0.65) | 0.073 | 0.638 | 1.100 | |

(0.31, 0.70) | −0.143 | 0.309 | 0.884 | |

(0.51, 0.60) | 0.015 | 0.484 | 1.012 | |

(0.89, 0.14) | 0.139 | 0.139 | 1.267 | |

(0.80, 0.66) | 0.131 | 0.637 | 1.158 | |

(0.24, 0.87) | −0.244 | 0.239 | 0.782 | |

(0.10, 0.13) | −0.283 | 0.096 | 0.744 | |

(0.31, 0.08) | −0.333 | 0.079 | 0.694 | |

(0.65, 0.53) | 0.009 | 0.509 | 1.018 |

Different values of | ||||
---|---|---|---|---|

(0.48, 0.25) | −0.398 | 0.075 | 0.628 | |

(0.12, 0.80) | −0.338 | 0.077 | 0.689 | |

(0.35, 0.68) | −0.337 | 0.189 | 0.688 | |

(0.73, 0.21) | −0.317 | 0.119 | 0.710 | |

(0.74, 0.44) | 0.203 | 0.279 | 0.823 | |

(0.14, 0.67) | −0.131 | 0.066 | 1.158 | |

(0.56, 0.81) | 0.070 | 0.418 | 0.950 | |

(0.98, 0.21) | 0.077 | 0.202 | 1.036 | |

(0.29, 0.25) | −0.461 | 0.038 | 0.566 | |

(0.77, 0.22) | −0.316 | 0.137 | 0.714 | |

(0.47, 0.38) | −0.294 | 0.297 | 0.733 | |

(0.13, 0.86) | −0.398 | 0.128 | 0.628 | |

(0.68, 0.02) | −0.478 | 0.019 | 0.548 | |

(0.34, 0.20) | −0.373 | 0.146 | 0.654 | |

(0.41, 0.47) | −0.254 | 0.315 | 0.773 | |

(0.53, 0.24) | −0.354 | 0.212 | 0.673 | |

(0.38, 0.33) | −0.343 | 0.235 | 0.683 | |

(0.31, 0.60) | 0.233 | 0.280 | 0.793 | |

(0.21, 0.97) | 0.032 | 0.209 | 0.994 | |

(0.07, 0.48) | −0.356 | 0.063 | 0.670 | |

(0.87, 0.45) | −0.234 | 0.449 | 0.793 | |

(0.78, 0.44) | −0.240 | 0.439 | 0.786 | |

(0.60, 0.72) | 0.044 | 0.593 | 0.983 | |

(0.43, 0.57) | −0.214 | 0.425 | 0.812 | |

(0.26, 0.33) | −0.298 | 0.246 | 0.729 | |

(0.10, 0.90) | 0.052 | 0.100 | 0.975 | |

(0.86, 0.40) | −0.285 | 0.399 | 0.741 | |

(0.46, 0.62) | 0.194 | 0.456 | 0.833 | |

(0.05, 0.52) | −0.449 | 0.049 | 0.578 | |

(0.45, 0.04) | −0.416 | 0.039 | 0.611 |