Take the 2-minute tour ×
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It's 100% free, no registration required.

Suppose I have a black box that generates data following a normal distribution with mean m and standard deviation s. Suppose, however, that whenever it outputs a value < 0 it does not record anything (can't even tell that it's outputted such a value). We have a truncated gaussian distribution without a spike.

How can I estimate these parameters?

share|improve this question
    
I changed the tag from "truncated-gaussian" to "truncation" because most answers will be potentially useful in situations involving other distributions. –  whuber Aug 23 '10 at 14:21
add comment

2 Answers 2

The model for your data would be:

$y_i \sim N(\mu,\sigma^2) I(y_i > 0)$

Thus, the density function is:

$$f(y_i|-) = \frac{exp(-\frac{(y_i-\mu)^2}{2 \sigma^2})}{\sqrt{2 \pi \sigma}\ (1 - \phi(-\frac{\mu}{\sigma}))}$$

where,

$\phi(.)$ is the standard normal cdf.

You can then estimate the parameters $\mu$ and $\sigma$ using either maximum likelihood or bayesian methods.

share|improve this answer
add comment

As Srikant Vadali has suggested, Cohen and Hald solved this problem using ML (with a Newton-Raphson root finder) around 1950. Another paper is Max Halperin's "Estimation in the Truncated Normal Distribution" available on JSTOR (for those with access). Googling "truncated gaussian estimation" produces lots of useful-looking hits.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.