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 Incoming links: Truncated gamma rejection sampling with a Pareto envelope

Write

Z = \int_{a}^{\infty} x^{α - 1} e^{- β x} d x

. A proper target requires

β > 0

, and a proper proposal requires

b > 0

. Normalization gives

f (x) = x^{α - 1} e^{- β x} / Z

g (x) = b a^{b} x^{- b - 1}

x > a

. Their ratio is

\frac{f ( x )}{g ( x )} = \frac{x ^{α + b} e ^{- β x}}{b a ^{b} Z} .

(α + b) / x - β

. It is positive before

x_{*} = (α + b) / β

and negative after it. Since

a < α / β < x_{*}

, the maximum occurs within the proposal support. Consequently the tight envelope for truncated gamma rejection sampling with a Pareto envelope is

M = (\frac{α + b}{β})^{α + b} e^{- (α + b)} (b a^{b} \int_{a}^{\infty} x^{α - 1} e^{- β x} d x)^{- 1} .

The accompanying acceptance rule can be evaluated without

Z

\frac{f ( x )}{M g ( x )} = (\frac{x}{x _{*}})^{α + b} exp [- β (x - x_{*})] \leq 1.