FIM for Arnold and Strauss's bivariate gamma distribution

Abstract

The Fisher information matrix (FIM) as well as tools for its numerical computation is derived for Arnold and Strauss’ [1988. Bivariate distributions with exponential conditionals, J. Amer. Statist. Assoc. 83 (1988) 522–527] bivariate gamma distribution. Numerical tabulations of the matrix are also provided for practical purposes.

Introduction

Arnold and Strauss’ (1988) bivariate gamma distribution is specified by the joint probability density function (pdf):

$$f(x,y;a,b,c,\alpha ,\beta )={\mathit{Kx}}^{\alpha -1}{y}^{\beta -1}\exp \{-(\mathit{ax}+\mathit{by}+\mathit{cxy})\}$$

for $x>0$, $y>0$, $\alpha >0$, $\beta >0$, $a>0$, $b>0$, and $c>0$, where $K=K(a,b,c,\alpha ,\beta )$ denotes the normalizing constant. This is also known as the conditionally specified bivariate gamma distribution (see Kotz et al., 2000). It has received applications in areas such as cross-over trials, flood frequency analysis, life testing, and precipitation modeling. Using the definition of the Kummer function

$$\Psi (a,b,z)=\frac{1}{\Gamma (a)}{\int }_0^{\infty }{t}^{a-1}(1+t{)}^{b-a-1}\exp (-\mathit{zt})\mathrm{d}t\text{,}$$

one can show that

$$\frac{1}{K}=b^{\alpha -\beta }{c}^{-\alpha }\Gamma (\alpha )\Gamma (\beta )\Psi \left(\alpha ,\alpha -\beta +1,\frac{\mathit{ab}}{c}\right)\text{.}$$

The aim of this note is to calculate the Fisher information matrix (FIM) corresponding to (1) and to provide useful numerical tabulations of the FIM. For a given observation $(x,y)$, the FIM is defined by

$$\left(I_{\mathit{jk}}\right)=\left\{E\left(-{\displaystyle \frac{\mathrm{\partial }\log L(\mathit{\theta })}{\mathrm{\partial }{\theta }_j}}\frac{\mathrm{\partial }\log L(\mathit{\theta })}{\mathrm{\partial }{\theta }_k}\right)\right\}$$

for $j=1,2,\dots ,p$ and $k=1,2,\dots ,p$, where $L(\mathit{\theta })=f(x,y)$ and $\mathit{\theta }=\left({\theta }_1,{\theta }_2,\dots ,{\theta }_p\right)$ are the parameters of the pdf f. It has the meaning “information about the parameters $\mathit{\theta }$ contained in the observation x.” It is related to the covariance matrix of the estimate of $\mathit{\theta }$ (being its inverse under certain conditions). The FIM plays a significant role in statistics and many other areas of science. Some recent references on applications of the FIM include:

• The FIM plays a key role in the analysis and applications of statistical image reconstruction methods based on Poisson data models. The elements of the FIM are a function of the reciprocal of the mean values of sinogram elements (see, for example, Li et al., 2004).

• The calculation of the FIM is of central importance in many practical systems which can be described as the output of a multidimensional linear separable-denominator system with Gaussian measurement noise, e.g. nuclear magnetic resonance (NMR) spectroscopy (see Ober et al., 2003).

• Retout et al. (2002) address the problem of the choice and the evaluation of designs in population pharmacokinetic studies that use non-linear mixed-effect models. Criteria, based on the FIM, are developed to optimize designs and adapted to such models.

• Sanchez-Montanes and Pearce (2001) discuss how the FIM may be used as part of an optimization procedure for selecting odor sensors within a population so as to maximize the accuracy with which the overall sensory system may estimate the stimulus.

• Johnson et al. (2000) use the FIM for the analysis of passive-ranging systems based on wave-front coding.

• Versyck et al. (1997) and Emery and Nenarokomov (1998) describethe application of the FIM in optimal experiment designs (the definition of the conditions under which an experiment is to be conducted in order to maximize the accuracy with which the results are obtained).

• Barrett et al. (1995) describe applications of the FIM to objective assessment of image quality.

• Penny et al. (1994) use the FIM to examine the problem of choosing an optimum, or near-optimum, set of measurement locations for experimental modal testing.

For details on the theory of the FIM, see Cox and Hinkley (1974).

The exact form of the FIM is derived in Section 2. The calculations use properties of the Kummer function (see, for example, Gradshteyn and Ryzhik, 2000). Some tools for the numerical computation of the FIM are developed in Section 3. Finally, Section 4 provides some tabulations of the FIM.