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Truncated normal distribution

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Let X be normally distributed with mean μ and variance σ 2 and consider the conditional distribution of X in the interval [a,b]⊂ℝ. The distribution of X conditional on X∈[a,b] is the truncated normal distribution. The conditional density is

f(x∣x∈[a,b])=1 σϕ(x−μσ)Φ(b−μσ)−Φ(a−μσ)
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, where ϕ and Φ 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

E[X∣X∈[a,b]] =μ+∫ a b(x−μ)1 σϕ(x−μσ)Φ(b−μσ)−Φ(a−μσ)dx =μ+1 Φ(b−μσ)−Φ(a−μσ)∫ a bx−μσ1 2 πe −1 2 (x−μσ) 2 dx =μ+1 Φ(b−μσ)−Φ(a−μσ)∫ a bσ∂∂x[−1 2 πe −1 2 (x−μσ) 2 ]dx =μ+1 Φ(b−μσ)−Φ(a−μσ)σ[−1 2 πe −1 2 (x−μσ) 2 ] a b =μ−σϕ(b−μσ)−ϕ(a−μσ)Φ(b−μσ)−Φ(a−μσ).
\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

category: Statistics, Econometrics

Revised on January 8, 2008 21:47:17 by Jason Blevins (76.182.57.114)
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