The QLIM Procedure

Standard Distributions

Table 29.3 through Table 29.10 show all the distribution density functions that PROC QLIM recognizes. You specify these distribution densities in the PRIOR statement.

Table 29.3: Beta Distribution

PRIOR statement	BETA(SHAPE1=a, SHAPE2=b, MIN=m, MAX=M)
	Note: Commonly $m=0$ and $M=1$ .
Density	$\frac{(\theta -m)^{a-1} (M-\theta )^{b-1}}{B(a,b)(M-m)^{a+b-1}}$
Parameter restriction	$a>0$ , $b>0$ , $-\infty <m<M<\infty$
Range	$\left\{ \begin{array}{ll} \left[ m, M \right] & \mbox{when } a = 1, b = 1 \\ \left[ m, M \right) & \mbox{when } a = 1, b \neq 1 \\ \left( m, M \right] & \mbox{when } a \neq 1, b = 1 \\ \left( m, M \right) & \mbox{otherwise} \end{array} \right.$
Mean	$\frac{a}{a+b}\times (M-m)+m$
Variance	$\frac{ab}{(a+b)^2(a+b+1)}\times (M-m)^2$
Mode	$\left\{ \begin{array}{ll} \frac{a-1}{a+b-2}\times M+\frac{b-1}{a+b-2}\times m & a > 1, b > 1 \\ m \mbox{ and } M & a < 1, b < 1 \\ m & \left\{ \begin{array}{l} a < 1, b \geq 1 \\ a = 1, b > 1 \\ \end{array} \right. \\ M & \left\{ \begin{array}{l} a \geq 1, b < 1 \\ a > 1, b = 1 \\ \end{array} \right. \\ \mbox{not unique} & a = b = 1 \end{array} \right.$
Defaults	SHAPE1=SHAPE2=1, $\Variable{MIN}\rightarrow -\infty$ , $\Variable{MAX}\rightarrow \infty$

Table 29.4: Gamma Distribution

PRIOR statement	GAMMA(SHAPE=a, SCALE=b )
Density	$\frac{1}{b^ a\Gamma (a)} \theta ^{a-1} e^{-\theta /b}$
Parameter restriction	$a > 0, b > 0$
Range	$[0,\infty )$
Mean	$ab$
Variance	$ab^2$
Mode	$(a-1)b$
Defaults	SHAPE=SCALE=1

Table 29.5: Square Root Gamma Distribution

PRIOR statement	SQGAMMA(SHAPE=a, SCALE=b )
Density	$\frac{2}{b^ a\Gamma (a)} \theta ^{2a-1} e^{-\theta ^{2}/b}$
Parameter restriction	$a > 0, b > 0$
Range	$[0,\infty )$
Mean	$\frac{\Gamma (a+\frac{1}{2})}{\Gamma (a)}\sqrt {b}$
Variance	$\left\{ a-\left[\frac{\Gamma (a+\frac{1}{2})}{\Gamma (a)}\right]^{2}\right\} b$
Mode	$\sqrt {(a-\frac{1}{2})b}, \qquad a\geq \frac{1}{2}$
Defaults	SHAPE=SCALE=1
See Stacy (1962) for more details.

Table 29.6: Inverse-Gamma Distribution

PRIOR statement	IGAMMA(SHAPE=a, SCALE=b)
Density	$\frac{b^ a}{\Gamma (a)} \theta ^{-(a+1)}e^{-b/\theta }$
Parameter restriction	$a > 0, b > 0$
Range	$0<\theta <\infty$
Mean	$\frac{b}{a-1},\qquad a > 1$
Variance	$\frac{b^2}{(a-1)^2(a-2)},\qquad a>2$
Mode	$\frac{b}{a+1}$
Defaults	SHAPE=2.000001, SCALE=1

Table 29.7: Square Root Inverse-Gamma Distribution

PRIOR statement	SQIGAMMA(SHAPE=a, SCALE=b)
Density	$\frac{2b^ a}{\Gamma (a)} \theta ^{-(2a+1)}e^{-b/\theta ^{2}}$
Parameter restriction	$a > 0, b > 0$
Range	$0<\theta <\infty$
Mean	$\frac{\Gamma (a-\frac{1}{2})}{\Gamma (a)}\sqrt {b},\qquad a > \frac{1}{2}$
Variance	$\left\{ \frac{1}{a-1}-\left[\frac{\Gamma (a-\frac{1}{2})}{\Gamma (a)}\right]^{2}\right\} b,\qquad a>1$
Mode	$\sqrt {\frac{b}{a+\frac{1}{2}}}$
Defaults	SHAPE=2.000001, SCALE=1
See Stacy (1962) for more details.

Table 29.8: Normal Distribution

PRIOR statement	NORMAL(MEAN= $\mu$ , VAR= $\sigma ^2$ )
Density	$\frac{1}{\sigma \sqrt {2\pi }} \exp \left( - \frac{(\theta - \mu )^2}{2\sigma ^2}\right)$
Parameter restriction	$\sigma ^2 > 0$
Range	$-\infty <\theta <\infty$
Mean	$\mu$
Variance	$\sigma ^2$
Mode	$\mu$
Defaults	MEAN=0, VAR=1000000

Table 29.9: t Distribution

PRIOR statement	T(LOCATION= $\mu$ , DF= $\nu$ )
Density	$\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)\sqrt {\pi \nu }}\left[1+\frac{(\theta -\mu )^2}{\nu }\right]^{-\frac{\nu +1}{2}}$
Parameter restriction	$\nu > 0$
Range	$-\infty <\theta <\infty$
Mean	$\mu , \text { for }\nu >1$
Variance	$\frac{\nu }{\nu -2}, \text { for }\nu >2$
Mode	$\mu$
Defaults	LOCATION=0, DF=3

Table 29.10: Uniform Distribution

PRIOR statement	UNIFORM(MIN=m, MAX=M)
Density	$\frac{1}{M-m}$
Parameter restriction	$-\infty <m<M<\infty$
Range	$\theta \in [m, M]$
Mean	$\frac{m+M}{2}$
Variance	$\frac{(M-m)^2}{12}$
Mode	Not unique
Defaults	MIN $\rightarrow -\infty$ , MAX $\rightarrow \infty$