So, I've got a(n assumed) normal distribution that is left bound by 0, and I'm trying to extrapolate what would happen if the distribution could go negative.

Given a portion of the right tail of a normal distribution, is there a good method for building out the rest of the distribution?

For instance, in the below image, taking the observations at 0 and estimating the grey unknown distribution:

Observed vs Expected Normal Distribution


To clarify, in most cases we'll know the second and third standard deviations (inferred from the percent of observations on the right side of the curve) from the mean, but not the mean itself - and, we want to find the mean an plot the unknown distribution (grey in the above picture).

Admittedly, I'm rusty in this area, so forgive me if this is a poor question.

Would it make sense to take the distance between the far right 2.1% of observations, call that the standard deviation and infer the mean and variance form that?

  • $\begingroup$ You can fit a normal by choosing the mean and variance. Having a guess of the mean you can produce artificial data by reflecting the data in the axis of the mean. Could be a bad idea, havn't tried it or seen it done. $\endgroup$ Commented Mar 7, 2019 at 22:45
  • $\begingroup$ Along the lines of @JesperHybel's comment - are you trying to estimate the mean and variance of the presumed normal distribution? It's unclear what "estimating the grey unknown distribution" means, and it's probably worth clarifying the question. $\endgroup$
    – Weiwen Ng
    Commented Mar 7, 2019 at 22:48
  • $\begingroup$ @WeiwenNg added some clarification $\endgroup$ Commented Mar 7, 2019 at 23:00
  • $\begingroup$ Maybe use minimum observation as estimate of lower bound and then simply fit a bounded normal by MLE. Reflection seems bad idea ... $\endgroup$ Commented Mar 7, 2019 at 23:11

2 Answers 2


This is called a zero-censored normal distribution. The likelihood function is

P(Y<0 | mu,sigma) for y = 0, and p(Y=y | mu,sigma) for y > 0

You can find the ML estimator using this R code:
y = rnorm(10000,0.5,2) # simulate some sample data 
y[y<0]=0 # censor at zero
minlogL = function (p ) { 
  mu = p[1]
  logsigma=p[2] # must used log-sigma because the second parameter can be negative while sigma must be positive
  return(sum(-log(dnorm(y[y>0],mu,exp(logsigma)))) - sum(y<=0)*log(pnorm(0,mu,exp(logsigma))))

@Helene Hoegsbro Thygesen has the right answer ... I missed that the lower truncation was given and that the observations at 0 were given.

Since I wrote it anyway I post the script for estimating mean and variance

(1) without knowing the lower truncation point

(2) without point probability at zero

the likelihood is then based on the truncated normal density

$$ f(x|\mu,\sigma,\xi) = \frac{1}{1-\Phi\left(\xi\right)} * \frac{1}{\sigma} \phi\left(\frac{x-\mu}{\sigma}\right)$$

The script runs estimation on 1000 samples and plots distribution of estimator

M <- matrix(0,ncol=4,nrow=1000)
for (i in 1:1000)
    N <- 3000
    x <- rnorm(N,sd=2)-1
    x <- x[which(x>0)]
    x_hat <- min(x)

        a <- theta[1]
        b <- exp(theta[2])
        denom <- 1 - pnorm(x_hat,mean=a,sd=sqrt(b))
        ll <- sum( log( dnorm(x,mean=a,sd=sqrt(b))/denom))

model <- solnp(c(0,2),loglik,LB=c(-5,-5),UB=c(5,5))
# variance estimate
# mean estimate


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