The BS density with unit scale parameter is typically expressed in the following form
where
$\gamma >0$. We will reparametrize the model, defining
$\theta =\frac{1}{2{\gamma}^{2}}$, so the density becomes
where
$S\left(x\right)={x}^{1/2}+{x}^{3/2}$ and
$T\left(x\right)=x+{x}^{1}2$. Note that the parametrization in Equation (
1) belongs to the natural exponential family of distributions. This guarantees that the parameter space for
$\theta $ is convex. If a random variable
X has (
1) as its density, we write
$X\sim BS(1,\theta )$; here the 1 refers to the fact that the scale parameter is set equal to 1. If this random variable is multiplied by a positive scale parameter
$\beta $, we denote the resulting distribution by
$BS(\beta ,\theta )$. Clearly, with
$\beta =1$, this is a one parameter exponential family. The corresponding parameter space is
$\{\theta :0<\theta <\infty \}$. Using a result in Arnold and Strauss [
9], the most general family of bivariate densities with conditionals in this one parameter family (
1) (i.e., belonging to the natural exponential family of distributions) is of the form
where
${\theta}_{10}>0,{\theta}_{01}>0$ and
${\theta}_{11}\ge 0.$ The corresponding marginal density of
X is
which is clearly not of the BS form unless
${\theta}_{11}=0$, which corresponds to the case in which
X and
Y are independent. The conditional densities are indeed of the BS form. Thus, specifically
and
For added flexibility, we introduce scale parameters
${\beta}_{X}$ and
${\beta}_{Y}$ in this bivariate density. Thus, if
$({X}^{*},{Y}^{*})$ has density (
2) and if we define
$X={\beta}_{X}{X}^{*}$ and
$Y={\beta}_{Y}{Y}^{*}$, then we can say that
$(X,Y)$ has a general bivariate BS conditionals (BVBSC) distribution and we will write
$(X,Y)\sim BVBSC({\beta}_{X},{\beta}_{Y},{\theta}_{10},{\theta}_{01},{\theta}_{11}).$ Figure 1 shows the scatterplot for the
$BVBSC(10,10,{\theta}_{01},{\theta}_{10},{\theta}_{11})$ distribution with different values for the vector
$({\theta}_{01},{\theta}_{10},{\theta}_{11})$. Note that the model can assume different shapes for the bivariate scatterplot depending on the values for the vector
$({\theta}_{01},{\theta}_{10},{\theta}_{11})$.
2.1. Drawing Values from the Model
One advantage of the model is that both conditional distributions are specified in relatively simple forms. There is a wellknown formula expressing a
$BS(\beta ,\theta )$ variable denoted by
U as a function of a standard normal variable
Z, thus
This can used whenever a BS draw is to be simulated.
There are two approaches that can be used to simulate draws from a general BS conditionals distribution. One approach involves simulation of a draw from the
X marginal followed by use of the conditional density of
Y given
X to find the corresponding second coordinate of
$(X,Y)$. For this approach, we wish to avoid drawing values directly from the marginal distribution because of possible problems in the computation of the normalizing constant. For this reason, to draw values from the marginal distribution of
X, we propose the following steps based on the Metropolis–Hastings algorithm (Algorithm 1):
Algorithm 1 Metropolis–Hastings algorithm. 
Set a start value ${z}_{0}$. A proposal can be selected by considering ${z}_{0}\sim BS(1,{\theta}_{10})$. Given a last value, say ${z}_{k1}$, propose a new value as ${z}_{k}^{*}={x}_{k1}+\u03f5$, where $\u03f5\sim N(0,\tau )$. A possible choice for $\tau $ can be $\tau =1+{\theta}_{10}$, which corresponds to the variance of X for the independence case ${\theta}_{11}=0$. If ${z}_{k}>0$, set ${z}_{k}={z}_{k}^{*}$ with probability $\pi =\frac{S\left({x}_{k}\right){e}^{{\theta}_{10}T\left({z}_{k}\right)}}{S\left({z}_{k1}\right){e}^{{\theta}_{10}T\left({z}_{k1}\right)}}\sqrt{\frac{{\theta}_{10}+{\theta}_{11}T\left({z}_{k1}\right)}{{\theta}_{10}+{\theta}_{11}T\left({z}_{k}\right)}}$. Otherwise, set ${z}_{k}={z}_{k1}$. Repeat steps 2 and 3 as many times as necessary.

To eliminate the dependence among the drawn values, it is possible to consider a burnin period and to thin the resulting series. Having drawn a final sample for
X, say
${x}_{1}^{*},{x}_{2}^{*},\dots ,{x}_{n}^{*}$, we can simulate corresponding values for
Y (using (
5)) such that
${y}_{i}^{*}\sim BS(1,{\theta}_{01}+{\theta}_{11}T\left({x}_{i}^{*}\right))$. Finally, the scale parameters can be introduced by setting
${x}_{i}={\beta}_{X}{x}_{i}^{*}$ and
${y}_{i}={\beta}_{Y}{y}_{i}^{*}$ for
$i=1,\dots ,n$. The pairs
$({x}_{1},{y}_{1}),({x}_{2},{y}_{2}),\dots ,({x}_{n},{y}_{n})$ then constitute a simulated random sample from the model.
An alternative approach is one involving Gibbs sampler simulation utilizing the relative simplicity of the two corresponding conditional distributions (and the ease of drawing from univariate BS distributions). For it, we begin with an arbitrary value for X, say
${x}_{0}^{*}$. Then, we use the conditional distribution of
Y given
$X={x}_{0}^{*}$ ( as in (
5)) to simulate a corresponding
${y}_{0}^{*}$. Next, we generate a simulated value of
X, say
${x}_{1}^{*}$, using the conditional distribution of
X given
$Y={y}_{0}^{*}$ ( as in (
4)), and continue in this fashion using (
5) and (
4) alternately. Use of burnin and thinning is also recommended for this approach.