xbeta
Let X normally distributed with mean μ σ 2 X [ a , b ] ⊂ ℝ X X ∈ [ a , b ] truncated normal distribution . The conditional density is
f ( x ∣ x ∈ [ a , b ] ) = 1 σ ϕ ( x − μ σ ) Φ ( b − μ σ ) − Φ ( a − μ σ ) 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 a ≤ x ≤ b ϕ Φ 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
E [ X ∣ X ∈ [ a , b ] ] = μ + ∫ a b ( x − μ ) 1 σ ϕ ( x − μ σ ) Φ ( b − μ σ ) − Φ ( a − μ σ ) dx = μ + 1 Φ ( b − μ σ ) − Φ ( a − μ σ ) ∫ a b x − μ σ 1 2 π e − 1 2 ( x − μ σ ) 2 dx = μ + 1 Φ ( b − μ σ ) − Φ ( a − μ σ ) ∫ a b σ ∂ ∂ x [ − 1 2 π e − 1 2 ( x − μ σ ) 2 ] dx = μ + 1 Φ ( b − μ σ ) − Φ ( a − μ σ ) σ [ − 1 2 π e − 1 2 ( x − μ σ ) 2 ] a b = μ − σ ϕ ( b − μ σ ) − ϕ ( a − μ σ ) Φ ( b − μ σ ) − Φ ( a − μ σ ) . \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: [ a , b ] = [ a , ∞ ] μ = 0 σ = 1
E [ X ∣ X > a ] = ϕ ( a ) 1 − Φ ( a ) . \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
λ ( a ) ≡ ϕ ( a ) 1 − Φ ( a ) . \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)