A NEW TRIPLE OPTIMALITY CRITERION CDKL

In this paper, three optimality criteria will be compounded to give a new criterion, namely, CDKL-optimality for parameter estimation, estimating the area under curve and model discrimination for any kind of regression models, with homoscedastic or herteroscedastic errors, which may be Gaussian or not. CDKL-compound criterion is proposed for copula models.


1-Introduction
The optimal design theory was firstly introduced by Smith [1], some optimality criteria are proposed by Atkinson et al. [2].C-optimality is presented by Elfving [3] which supplied a geometrical interpretation to obtain C-optimal designs.Dette and Holland-Letz [4] proposed Coptimal designs for heteroscedastic regression by using a geometric characterization.The most important criterion for parameter estimation is D-optimality, announced by Wald [5], numerous publications on D-optimality can be seen in Atkinson et al. [2].Atkinson [6,7] proposed Toptimality for discriminating between two rival models and between several models.In order to act towards any distribution for the random errors see, López-Fidalgo et al. [8,9], which proposed KL-optimality criterion based on the Kullback-Liebler distance.A generalization for the KLoptimality criterion demonstrated by Tommasi [10], to deal with discrimination amongst different non-normal models.
In this paper, the aim is to associate the three-optimality criterion in one design criterion for model discrimination, parameter estimation and estimating the area under curve, namely, CDKL-optimality criterion.CDT-optimality criterion introduced by Abd El-monsef and Seyam [11] which is only for regression models, but the proposed criterion named by CDKL-optimality can be applied for nested regression models or not.Moreover, it has suggested for copula models.
During a long time, statisticians have been concerned with the connection between a multivariate distribution function and its lower dimensional margins.Sklar [12] gave the answer to this situation for univariate marginal case by forming a new class of functions is called copulas.In many areas of applied statistics copula models becomes a standard tool, such as, medicine Nikoloulopoulos and Karlis [13], marketing Danaher and Smith [14] and time series analysis Patton [15].This paper is prepared as follows: C-, D-and KL-optimum designs are presented in Section 2 respectively.DKL -and CD-optimum designs is reviewed in Section 3. A new criterion namely CDKL-optimality is derived in Section 4. In Section 5, CDKL-criterion is proposed for copula models.

KLoptimum design
Lόpez-Fidalog et al. [9] proposed the KL-criterion which discriminate between models with normally distributed observations.It was applied to discriminate between the popular Michaelis-Menten model besides an extension of it under the lognormal and gamma distributions.
The set is a singleton then the design is called a regular design, otherwise is called singular design.Lόpez-Fidalog et al. [9] is the directional derivative of Eq. ( 2).
For goodness of a design  in the sense of discrimination purposes, the Kl-efficiency of a design  relative to the optimum design  21 * is

C-optimum design
Elfving [3] considered the geometric characterization of locally C-optimal designs.Obtaining C-optimal designs in linear and non-linear regression models have studied by many authors see for example, Studden [16].C-optimality is based on estimating the linear combination of the parameters    with minimum variance, where c is a known vector of constants for more details see, Atkinson et al. [2] and Pazman and Pronzato [17].
For goodness of a design , the C-efficiency of  is defined as

D-optimum design
D-optimal designs are introduced to minimize the generalized variance of the estimated regression coefficients by maximizing the determinant of the Fisher information matrix.
D-optimal design has been central to work on optimum experimental designs and is studied widely; see Wang et al. [18].

3-Compound criteria
A compound design criterion is defined as a weighted product of the efficiencies that is maximized to give an exact or a continuous design.As a beginning, two compound criteria will be proposed, which are DKL-optimality that combines D-optimality with KL-optimality for discriminating between models and CD-optimality that combines parameter estimation with Coptimality for feature of interest, respectively.

DKL-optimum designs
Tommasi [20] suggested DKL-optimality criterion for model discrimination and parameter estimation, which defined as a weighted geometric mean of KL-and D-efficiencies.The DKLcriterion can be used for any kind of regression models, nested or not.Hence, DKL-optimality criterion which is general applicability than DT-optimality criterion which introduced by Atkinson [21] applies only for nested regression models.When the assumed true model is  2 (, ,  2 ) instead of  1 (, ,  1 ), then the KL-optimum design is denoted by  12 * .

CD-optimum design
CD-optimality proposed by Atkinson et al. [2] which combining C-and D-optimality's criteria.CD-optimum design was applied for parameter estimation and estimation of the area under curve, which maximize a weighted product of the efficiencies .(13) Taking logarthim in Eq. ( 13), the right hand side becomes Designs maximizing Eq. ( 14) are called CD-optimum and are denoted by   * .The derivative function for Eq. ( 14) is given by Atkinson et al. [2] showed that the upper bound of   (,   * ) over  ∈  is one, achieved at the points of the optimum design.

4-CDKL-optimum designs
A new compound criterion CDKL-optimality will be constructed to estimate a parametric function such as the area under the curve, parameter estimation, and model discrimination for any kind of regression models.

Proof.
Since the terms in Equation ( 18) have been scaled, therefore the upper bound of   over  ∈  is one, achieved at the points of the optimum design.Moreover,   is the linear combination of the directional derivatives for C-optimality, D-optimality and KL-optimality criteria.Thus, the new criterion CDKL-optimality satisfies the conditions of convex optimum design and the theorem has been proved.
A measure of efficiency of a design  relative to a CDKL-optimum design is given by

5-CDKL-optimum design for copula models
Frѐchet [22] introduced a question about the relationship between a multidimensional probability function and its lower dimensional margins, and Sklar [12] answered this question by using the copula function.The copula function ties the joint and the margins together.Thus, the instead of estimating the joint distribution we could be replaced by estimating the margins and constructing a copula.
In particular, the  ̂ lie inside the unit square.The maximum  ̂ of   (. ) is called the maximum pseudo-likelihood estimator, see GrØnneberg and Hjort [23].