Generating generalized inverse Gaussian random variates

Abstract

The generalized inverse Gaussian distribution has become quite popular in financial engineering. The most popular random variate generator is due to Dagpunar (Commun. Stat., Simul. Comput. 18:703–710, 1989). It is an acceptance-rejection algorithm method based on the Ratio-of-Uniforms method. However, it is not uniformly fast as it has a prohibitive large rejection constant when the distribution is close to the gamma distribution. Recently some papers have discussed universal methods that are suitable for this distribution. However, these methods require an expensive setup and are therefore not suitable for the varying parameter case which occurs in, e.g., Gibbs sampling. In this paper we analyze the performance of Dagpunar’s algorithm and combine it with a new rejection method which ensures a uniformly fast generator. As its setup is rather short it is in particular suitable for the varying parameter case.

Appendix

1.1 Proof of Lemma 11

Let λ∈[0,1) be fixed. Recall that . Notice that does not contain any point left of the line u=−mv since we have used μ=m in (3), i.e., we have shifted quasi-density g by the mode m to the left, see Fig. 5. Consequently, −mv ⁺≤u ⁻ and . Let x ⁺ defined as in Sect. 2, i.e., it is the unique root of (5) greater than m. Thus $(x-m)\sqrt{g(x)}$ is monotonically increasing in [m,x ⁺].

Fig. 5

, , (−mv ⁺,u+)×(0,v ⁺), and (u ₁,v ₁) in proof of Lemma 11

Now choose x ₁∈(0,x ⁺−m) and let (u ₁,v ₁) be the point on the boundary of corresponding to x ₁, i.e., $v_{1}=\sqrt{g(x_{1}+m)}$ and u ₁=xv ₁. Then does not intersect the open rectangle (u ₁,u ⁺)×(v ₁,v ⁺) and thus

We therefore find

(10)

Now let

$$x_0^+(\beta) = \frac{1}{\beta} \bigl((1+\lambda)+\sqrt{(1+ \lambda)^2+\beta^2} \bigr). $$

It is the unique maximum of $x\sqrt{g(x)}$, see Sect. 5. Since $((x-m)\sqrt{g(x)} )'\geq (x\sqrt{g(x)} )'$ for all x≥m, we find $x_{0}^{+}\leq x^{+}$. Now define

$$u^*=\bigl(x_0^+-m\bigr)\sqrt{g\bigl(x_0^+ \bigr)}. $$

Clearly $u^{*}\leq u^{+}=\sup (x-m)\sqrt{g(x)}$, and thus ε=u ⁺−u ^∗≥0. From (10) we then obtain

(11)

Now set

We first have to check whether the condition x ₁∈(0,x ⁺−m) is fulfilled. For the limits β→0 we find

An immediate consequence is that for sufficiently small β>0, $x_{1}(\beta) < x_{0}^{+}(\beta)-m(\beta) \leq x^{+}(\beta)-m(\beta)$ which shows that x ₁(β)∈(0,x ⁺−m) when β is close enough to zero. Thus inequality (11) holds. Moreover,

$$\lim_{\beta\to 0} u_1(\beta) = \lim_{\beta\to 0} v_1(\beta) \cdot \lim_{\beta\to 0} x_1( \beta) = 1. $$

For the denominators in (11) we find

Finally,

Collecting all limits we find that all fractions on the right hand side of inequality (11) converge to 0 and thus as claimed.