1. Introduction

Linear models are generally of the form

(1)

(where y is an observation vector, X is an design matrix of fixed constants having rank r , is an vector of unknown parameters, is an vector of unknown random errors having zero means) and . The Ordinary Least Square (OLS) solution of (1) is , a unique solution.

In practice, not all linear models of the form in (1) are of full rank. When X is not of full rank, then is singular and the normal equations do not have a unique solution. However, there are various approaches of obtaining the inverse of singular matrices, for which the row echelon form given by Elswick et al (1991), Moore Penrose and the generalized inverse, Searle (1977) are popular in the literature. The generalized inverse is the approach we apply in this paper.

2. Form of Estimability

With X less than full rank and singular, i.e. , there is an infinite number of solutions of to the normal equations. Attention is therefore directed not to the solutions themselves but to linear functions of their elements. Consider a linear function of the parameters in , where is a known vector. This linear function is defined

as being an estimable function if there exists some linear combinations of the observations whose expected value is , i.e. if there exists a vector such that the expected value of is , then is said to be estimable. Consider the following theorem given in Graybill, 1976:

Theorem 1. (Graybill, 1976)

Assuming a linear model in (1), is an estimable function if and only if there exist an vector such that

Proof. If there exist a vector such that , then,

.

Only if: Conversely, if is estimable, then,

Thus

In addition, Elswick et al (1991) argues that if X is of full rank, exists and the rows of matrix serve as the necessary set of vectors because

3. Illustration

We demonstrate this discussion by considering the data from a study to compare classical and nearest neighbour methods in the analysis of varietal trials (See, e.g. Nwobi, 2000). In the experiment, nine (9) different varieties of cassava crop were tried, six at a time over a maximum of five years in such a way that these varieties were not replicated equally. The model (without interaction) is given by

(3)

where is the yield from the j^th trial of the i^th variety, is the general mean, is the effect of the i^th variety, is the random error associated with .

Equation (3) is written in matrix form as

(4)

Based on the model in (2), the parameter vector is given by

The components of the model (4) are

from where we obtain

A generalized inverse of written as such that see, e.g. Searle (1977) is

with

and .

The function

is estimable for any given values to the .

With this we obtain the solution to the normal equation as

Therefore, the Best Linear Unbiased Estimator (BLUE) of is

To see if where is estimable, we write the parameter as where, in this case, we define , a matrix; is of dimension , so that

Since , is not estimable. However, considering , this function may be written for and as where so that . This implies that is estimable.

Similarly, since there are 9 (nine) parameters, taking two (contrast) at a time gives estimable functions

Therefore, . Thus, we can say that a linear combination of estimable functions is estimable.

4. Conclusions

We have shown that for any arbitrary vector, is estimable with BLUE . The solution of the normal equation, , confirms that this approach is equivalent to the Nearest Neighbour method of analysis of designed experiments. Both methods agree on the selection of varieties though the value of these estimates are not unique due to the application of generalized inverses. Furthermore, we verified that the linear combination of estimable functions is estimable.