A Direct Derivation of the Exact Fisher Information Matrix for Bivariate Bessel Distribution of Type I ()
1. Introduction
The role of the Bessel functions in probability distributions can be trace back to [1,2]. The application of the univariate Bessel distribution in creating a robust alternative to the normal distribution is investigated by [3]. Univariate Bessel function distributions have been also used to the signal processing, [4]. Basic properties of these distributions with their links with some well-known distributions are described in [5]. More discussion on Bessel Distributions can be found in [6]. The bivariate Bessel distribution of type one (BB1) is specified by the following joint density function
(1.1)
for
where the constant
is given by

The density (1.1) is introduced by [7], by using a characterization of Bessel distribution due to the [8]. Reference [8] showed that if U and V are independent chi-squared random variables with common degrees of freedom v, then the distribution of
subject to
is a Bessel distribution. In a nice generalization, Reference [7] have proved that the joint pdf of
and
is given by (1.1). They have called this distribution as bivariate Bessel distribution of type I. For more information about Bessel distribution, see [7,8]. It is well-known that under certain condition, the inverse of Fisher Information Matrix (FIM) is the covariance matrix of the estimate of the parameters. The FIM has many application in statistics and other sciences. For an excellent recent references on applications of the FIM see [9]. The aim of this paper is to compute FIM for the bivariate density function (1.1) and is organized as follow: an explicit expression for the FIM for Bivariate Bessel distribution of type I is given in Section 2. Computing FIM for a special case of bivariate density function is given in Section 3. In Section 4, we provide some tools for the numerical computation of the FIM. Some tables of the FIM are also given.
2. An Explicit Expression for the FIM
For a given observation
the FIM has the form
(2.1)
where
and
are the parameters of the density. According to (1.1), the unknown vector of parameters is
The log likelihood of (1.1) for observation
is Finally,
(2.2)
Take
and
. In the following, we compute all the second derivatives of log likelihood (2.2) subject to parameters.










,




where
is the derivative of digamma function.
For computing the elements of FIM, following [7], let
and
then,
and 
We have found these identities take the computations easy.
At first note that if U has chi-squred distribution with
degree of freedom then,








Similarly, we have











and

Therefore, the elements of FIM can be calculated as below.














and finally

3. Special Case
Since above experessions are very tedios, we also compute FIM for some assumtions as below. We assume that,
and
for some fixed k. Then the elements of FIM can be computed as below. As we see the experessions are easier than the former experessoins and the dimension of the parameters is shorter than the peremier parameters.





and finally,

4. Numerical Computation and Tables of the FIM
In this section, some numerical tabulations of the FIM are given to illustrate the computations in the last section.
Take,




and

In the Tables 1-6, we have computed the elements of FIM for the Bivariate Bessel distribution of type I, for

Table 1. Elements of the FIM for (a1, b1, a2, b2) = (2, 1, 3, 1).

Table 2. Elements of the FIM for (a1, b1, a2, b2) = (2, 1, 3, 2).

Table 3. Elements of the FIM for (a1, b1, a2, b2) = (3, 1, 3, 2).

Table 4. Elements of the FIM for (a1, b1, a2, b2) = (3, 2, 3, 1).

Table 5. Elements of the FIM for (a1, b1, a2, b2) = (3, 2, 2, 1).

Table 6. Elements of the FIM for (a1, b1, a2, b2) = (3, 1, 2, 1).
different values of

and 
5. Acknowledgements
The authors would like to thank respectful editor in chief and a referee for their helpful comments.