Robust Continuous Quadratic Distance Estimation Using Quantiles for Fitting Continuous Distributions

Quadratic distance estimation making use of the sample quantile function over a continuous range is introduced. It extends previous methods which are based only on a few sample quantiles and it parallels the continuous GMM method. Asymptotic properties are established for the continuous quadratic distance estimators (CQDE) and the implementation of the methods are discussed. The methods appear to be useful for balancing robustness and efficiency and useful for fitting distribution with model quantile function being simpler than its density function or distribution function.


Introduction
For estimation in a classical setup, we often assume to have n independent, iden-ity conditions so that ML estimators attain the lower bound as given by the information matrix. In many applications, this condition is not met. We can consider the following example which gives the Generalized Pareto distribution (GPD) and draw the attention on the properties of the model quantile function which appears to have nicer properties than the density function and hence motivate us to develop continuous quadratic distance (CQD) estimation using quantiles on a continuum range which generalizes the quadratic distance (QD) methods based on few quantiles as proposed by LaRiccia and Wehrly [1] which can be viewed as based on a discrete range and hence CQD estimation might overcome the arbitrary choice of quantiles of QD as CQD will essentially make use of all the quantiles over the range with 0 The method is robust but not very efficient as only two points are used here to obtain moment type of equations and there is also arbitrariness on the choice of these two points. Castillo and Hadi [9] have improved this method by first selecting a set of two points, θ , LaRiccia and Wehrly [1] proposed to construct quadratic distance based on the discrepancy of ( ) ( ) their results to establish consistency and asymptotic normality of continuous quadratic distance estimators and since the paper aims at providing results for practitioners for their applied works, the presentation will emphasize methodologies with less technicalities so that it might be more suitable for applied researchers for their works. First, we shall briefly outline how to form the quadratic distance to obtain the CQD estimators and postpone the details for later sections of the paper.
CQD estimators can be viewed as estimators based on minimizing a continuous quadratic form as given by with: 1)

( )
, o k s t is an optimum symmetric positive definite kernel assumed to be fully specified.
2) a and b are chosen values with a being close to 0 and b close to 1 and 0 1 a b < < < .
In practice, we work with an asymptotic equivalent objective function  i.e., Since the kernel ( ) which is similar to the expression used to obtain continuous GMM estimators as given by Carrasco to obtain CQD estimators. Unless otherwise stated, by CQD estimators we mean estimators using the objective function of the form as defined by expression (5 The paper is organized as follows. Section 2 gives some preliminary results such as statistical functional and its influence function from which the sample quantiles can be viewed as robust statistics with bounded influence functions.
CQD estimation using quantiles will inherit the same robustness property. Some of the standard notions for the study of kernel functions will also be reviewed.
By linking a kernel to a linear operator in the Hilbert space of functions which are square integrable over the range ( ) , a b with an inner product, it allows a norm . to be introduced. Also, the notion of self adjoint linear operator which can be viewed as analogous to a symmetric matrix in Euclidean space is also introduced in Section 2. Section 3 gives asymptotic properties of the CQD estimators based on an estimate optimum kernel. An estimate of the covariance matrix is also given in Section 3.
Finally, we shall mention that simulation studies are not discussed in this paper as numerical quadrature methods are involved for evaluating the integrals over the range [ ] , a b for computing the objective function, we prefer to gather numerical aspects and simulation aspects together for further works and include these type of results in a separate paper leaving this paper focusing only on the methodologies.

Some Preliminaries
In this section we shall review the notion of statistical functional and its influence function and view a sample quantile as a statistical functional. Using its influence function, we can see that the sample quantile is a robust statistic and using the influence functions of two sample quantiles, we can also obtain the asymptotic covariance of the two sample quantiles.

Statistical Functional and Its Influence Function
Often, a statistic can be represented as a functional of the sample distribution from which we can obtain the asymptotic variance of ( ) We can see that for a suitable functional space, it is natural to consider the Hilbert space of functions which are square integrable so that a norm and linear operators can be defined in this space. This will facilitate the studies of kernels which are function of ( ) , s t . The necessary notions are introduced in the following section.

Linear Operators Associated with Kernels in a Hilbert Space
The functional space that we are interested is the space of integrable function with the range [ ] Just as a matrix A has its transpose * A matrix and if A is symmetric then * = A A , these notions can be extended to a functional space as a linear operator A has its adjoint * A and if the kernel defining A is symmetric then * = A A , A is called self adjoint. More precisely, given * , A A is found using the following equality, see Definition 6 given by Carrasco and Florens [12] (page 823), In this paper we focus on positive definite symmetric kernel ( ) , k s t which can be viewed as the covariance of ( ) For our purpose, we shall focus on a linear operator K with its kernel defined by Equation (7) for the rest of the paper. Since K and 1 − K are related and if we can construct an estimator for K , we can construct an estimator for 1 − K and the construction of these estimators will be discussed in the next sub-section.

Estimation of K and K −1
The methods used to construct an estimator for K follows from the techniques proposed by Carrasco and Florens [12]. The steps are given below: 1) We need a preliminary consistent estimate ( ) The notion of influence function was not mentioned in Carrasco and Florens [12].
3) Since It turns out that ( ) This also means that the kernel for In Section 3 we shall turn our attention to asymptotic properties of CQD estimators using the objective function ( ) n n Q α θ an using the norm . , it can also be expressed neatly as is the linear operator as defined by expression (9).
For consistency, we shall make use the basic consistency Theorem, i.e., Theorem 2.1 as given by Newey and McFadden [12] (page 2121). For establishing asymptotic normality for the CQD estimators, the procedures are similar to those used for establishing asymptotic normality of continuous GMM estimators as given by Theorem 8 given by Carrasco and Florens [12] (page 811, page 825).

Consistency
Assuming ∈ Ω θ and Ω is compact and observe that

Asymptotic Normality
The basic assumption used to establish asymptotic normality for the CQD esti- Before considering the Taylor expansion, we also need the following notation and the notion of a random element with zero mean and covariance given by the kernel of the associated linear operator K, i.e.,