Optimal Designs in a Simple Linear Regression with Skew-Normal Distribution for Error Term

The locally D-optimal design was derived for simple linear regression with the error term of Skew-Normal distribution. In this paper, to obtain a D-optimal design, the locally D-optimal criterion was considered, because of depending the information matrix on unknown parameters.


Introduction
There are many papers which discuss the optimal design for simple linear regression when the error terms have normal distribution. In this paper, the Skew-Normal distribution was considered for error term. The central role of general random variables in probability and statistics is well-known and can be traced to the simplicity of the functional forms, basic symmetry properties of the probability density function (pdf) and cumulative function (cdf) of the standard normal random variable, Z; Now, the following function can be considered: ( ) = 2 ( )Φ( ) (2) is a bona fide pdf of a random variable X which inherits a few features of the normal random variables. Some of these features happen to be the ones which make the normal distribution the darling of statistical inference.
The class of distribution (2) was introduced in[2] and Christened Skew-Normal distribution with the skewness parameter , in symbol X SN ( ). The right-tails of these distributions are virtually indistinguishable for > 2; thus, in this paper, only optimal design was discussed for ∈ [−2, +2]. Some properties of this kind of distribution can be found in [7].
In this paper, a simple linear regression model was considered as well as = 0 + 1 + , where the error term had skew-normal distribution with-parameter ; meaning that, ~( ). In this situation, Y has also skew-normal distribution with the following pdf; (3) where = ( 0 , 1 , ) . As was already written, is obtaining of this paper was to obtain the locally D-optimal design of this model based on the unknown parameter vector .
In this paper, there was concentration on the criterion dependence on the variance of parameter estimator. As is known, the variance of parameter estimator (ML) is inversely proportional to the information matrix [2]. Thus, there have been searched designs maximizing the information on the estimates as represented in the Fisher information matrix in ( ; ), where denotes a design. The outline of the paper is as follows. In Section 2, the information matrix, the locally D-optimal criterion which is a function of the information matrix and the locally D-optimal design for model (3) are introduced. At last, conclusion is made in Section 3.

Locally D-optimal Design
To obtain the D-optimal deign, the information matrix should be known, which was calculated using the derivative degree two of the log-likelihood function. In this paper, the information matrix was obtained based on the following log-likelihood function according to model (3); At first, since the information matrix should be calculated for one observation, (for one observation) was obtained by; where the elements of the symmetry information matrix 66 H. Jafari, R. Hashemi: Optimal Designs in a Simple Linear Regression with Skew-Normal Distribution for Error Term (5) were as follows;  1 and 2), an optimal design should be obtained for model (3). As is known, there are many optimality criteria for obtaining an optimal design such that D-and A-optimality criteria which are functions of the information matrix (5) and shown by the following notations [1]; denotes a design with two components; the first components are some values of design space and the weight of them are the second components, so that design can be defined as follows; }, ( ; ) = ∑ =1 . ( ; ) and ≤ ≤ ( +1) 2 (p denotes the number of parameters) [5]. The design is called the saturated design when m=p.

Conclusions
In this paper, the Skew-normal distribution was considered for error term in simple linear regression. In this kind of model, there are three parameters, two of which are related to the regression model and one is the parameter of Skew-normal distribution. Then, based on these three parameters and Caratheodory's theorem [3], a design with three support points assumed. To obtain an optimal design, the D-optimal criterion was considered. In this case, due to the dependence of the information matrix on unknown parameters, the locally D-optimal design was obtained [8].
In this situation, there was only one locally D-optimal design for every value of the parameters 0 , 1 ∈ [−5, +5] and for different values of ∈ [−2, +2]. This result is shown in Table 1, where = 1.2 maximizes the determinate of the information matrix. Also, it was shown that locally A-optimal design was the same as locally D-optimal design, where = 0.99 minimizes � −1 (ξ * )� (Table 1).