xbeta
Truncated normal distribution
Let be normally distributed with mean and variance and consider the conditional distribution of in the interval . The distribution of conditional on is the truncated normal distribution. The conditional density is
f(x \vert x \in [a,\,b]) =
\frac{\frac{1}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right)}
{ \Phi\left(\frac{b - \mu}{\sigma}\right)
- \Phi\left(\frac{a - \mu}{\sigma}\right) }
for , where and denote respectively the standard normal density and CDF.
In Econometrics, this distribution is used in the Censored regression model (also commonly called the Tobit model).
Derivation of the Mean
\begin{aligned}
\mathop{E}\left[X \vert X \in [a,\,b]\right]
&= \mu + \int_a^b (x - \mu)
\frac{\frac{1}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right)}
{ \Phi\left(\frac{b - \mu}{\sigma}\right)
- \Phi\left(\frac{a - \mu}{\sigma}\right) }\,dx \\
&= \mu + \frac{1}{ \Phi\left(\frac{b - \mu}{\sigma}\right)
- \Phi\left(\frac{a - \mu}{\sigma}\right) }
\int_a^b \frac{x - \mu}{\sigma}
\frac{1}{\sqrt{2\pi}}
e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\,dx \\
&= \mu + \frac{1}{ \Phi\left(\frac{b - \mu}{\sigma}\right)
- \Phi\left(\frac{a - \mu}{\sigma}\right) }
\int_a^b \sigma \frac{\partial}{\partial x} \left[ -\frac{1}{\sqrt{2\pi}}
e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \right]\,dx \\
&= \mu + \frac{1}{ \Phi\left(\frac{b - \mu}{\sigma}\right)
- \Phi\left(\frac{a - \mu}{\sigma}\right) }
\sigma \left[ -\frac{1}{\sqrt{2\pi}}
e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \right]_a^b \\
&= \mu - \sigma \frac{\phi\left(\frac{b - \mu}{\sigma}\right)
- \phi\left(\frac{a - \mu}{\sigma}\right)}
{ \Phi\left(\frac{b - \mu}{\sigma}\right)
- \Phi\left(\frac{a - \mu}{\sigma}\right) }.
\end{aligned}
Special Cases
- Left censoring of standard normal: , , .
\mathop{E}\left[X \vert X \gt a \right] = \frac{\phi(a)}{1 - \Phi(a)}.
This expression is called the inverse Mills ratio and is denoted
\lambda(a) \equiv \frac{\phi(a)}{1 - \Phi(a)}.
It is the hazard function of the Normal distribution.
References
Revised on January 8, 2008 21:47:17
by
Jason Blevins
(76.182.57.114)