1. 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;

(1)

Now, the following function can be considered:

(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. 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 , 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 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 θ.

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.

2. 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);

(4)

At first, since the information matrix should be calculated for one observation, (for one observation) was obtained by;

(5)

where the elements of the symmetry information matrix (5) were as follows;

such that;

See Appendix A1.

Especially, suppose as design space and . Now, for all the values of, the following can be written;

• Two amounts of and are equal to zero andincrease for and decrease for . There exists a similar position for , where;

• increases for and decreases for. There exists a similar position for , where;

In this case, increases as increases and After proposing the two above properties (items 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];

(7)

where 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;

(8)

where ,

(p denotes the number of parameters)[5]. The design is called the saturated design when m=p.

In Table 1, an optimal design is shown for model (3) based on the information matrix (5). In this case, for different amounts of parameter and every value of in the interval [-5, +5] and also the equivalence Theorem[6] (See Appendix A2), the following can be written[9];

(10)

Table 1. Locally D- and A-optimal design for some values of λ and any values of the other parameters

Figure 1.

According to Equation (10), Figure 1 shows that in the two points, is equal to 3; but, for this quantity is less than 3 (the number of parameters). Then, it can be said that the following locally D-optimal design exists with two support points -1, +1;

In this case, it can be also seen that the maximum of the determinant of the information matrix exists when in Table 1.

3. 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 and for different values of . This result is shown in Table 1, where 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 minimizes (Table 1).

Appendix A1:

To calculate the elements of the information matrix (5), the derivatives of the log-likelihood function (4) with respect to three parameters and λ is needed as follows;

Appendix A2:

Theorem (Equivalence Theorem)[9]: Based on the design the following three items are equivalent:

• is the locally D-optimal design if: , where denotes the true value of parameters.

• (G-optimality criterion), p denotes the number of parameters.

• , where is the support points.