# Get log-normal distribution parameters by min, max, mean

Assuming there are three values existing for a dataset, min=100, mean=1000, max=10000. Is it possible to derive the mu and sigma value of assuming the data fit to lognormal distribution?

And if yes, how to implement in python?

• I am pretty sure that knowing the sample size would be crucial for this problem, since with a higher sample size min will be smaller and max bigger – rep_ho Nov 18 '19 at 12:13
• Your question is equivalent to "Assuming there are three values existing for a dataset, min$=2$, mean$=3$, max$=4$. Is it possible to derive the $\mu$ and $\sigma$ value of assuming the data fit to a normal distribution? though with a scaling factor of $\log_e 10$ – Henry Nov 18 '19 at 17:00

Given a random sample $$X_1,X_2,\dots,X_n$$ from a density $$f(x)$$ and cdf $$F(x)$$, the joint density of the sample minimum and maximum is $$f_{X_{(1)},X_{(n)}}(x_1,x_n)=\frac{n!}{(n-2)!}f(x_1)f(x_n)[F(x_n)-F(x_1)]^{n-2}.$$ Based on the central limit theorem, unless $$n$$ is small or $$\sigma$$ large, the distribution of the sample mean $$\bar X$$ conditional on $$X_{(1)}=x_1$$ and $$X_{(n)}=x_n$$ should be well approximated by a normal distribution with the appropriate mean and variance (derived from the mean and variance of the truncated lognormal distribution of the observations in-between the minimum and maximum $$x_1$$ and $$x_n$$). The likelihood based on observations $$x_{1},x_{n},\bar x$$ is then $$L(\mu,\sigma) = f_{X_{(1)},X_{(n)}}(x_1,x_n)f_{\bar X|X_{(1)}=x_1,X_{(n)}=x_n}(\bar x)$$ which you can maximise numerically with respect to $$\mu$$ and $$\sigma$$.

R implementation:

lnormpar <- function(x1, xn, xbar, n, start=c(0,1)) {
# negative log likelihood
nll <- function(theta) {
mu <- theta[1]
sigma <- theta[2]
z1 <- (log(x1)-mu)/sigma
z2 <- (log(xn)-mu)/sigma
# mean and variance of (x_1,x_n)-truncated lognormal
mu1.trunc <- exp(mu + sigma^2/2)*
(pnorm(z2 - sigma) - pnorm(z1 - sigma))/
(pnorm(z2) - pnorm(z1))
mu2.trunc <- exp(2*mu + 2*sigma^2)*
(pnorm(z2 - 2*sigma) - pnorm(z1 - 2*sigma))/
(pnorm(z2) - pnorm(z1))
var.trunc <- mu2.trunc - mu1.trunc^2
# joint density of x1, xn, xbar
ll <-
sum(dlnorm(c(x1,xn), mu, sigma, log=TRUE)) +
(n-2)*log(plnorm(xn, mu, sigma) - plnorm(x1, mu,sigma)) +
dnorm(xbar, (x1 + xn + (n-2)*mu1.trunc)/n, sqrt(var.trunc/(n-2)), log=TRUE)
-ll
}
# maximise the log likelihood
opt <- optim(start, nll, hessian=TRUE)
# extract parameter estimates
res <- cbind(opt$par, sqrt(diag(solve(opt$hessian))))
rownames(res) <- c("mu","sigma")
colnames(res) <- c("Estimate","Std. Error")
res
}


The result assuming a sample size of $$n=10$$:

> lnormpar(x1=100,xn=10000,xbar=1000,n=10)
Estimate Std. Error
mu    6.489252  0.5747346
sigma 1.409383  0.3306496

• I think the same logic applies to the log-normal distribution in question, though one has to omit the part of the argument involving the central limit theorem. – Vadim Nov 18 '19 at 12:07
• I know that the answer writer is aware of this, but the way OP phrased the question compels me to point out: this is a way of estimating $\mu$ and $\sigma$. We cannot derive them in the sense of finding their true values, from the information given. – Ceph Nov 18 '19 at 20:11

Strictly speaking: no. The log-normal distribution is defined on the real half-axis $$(0, +\infty)$$, so its min is $$0$$, while its max is $$+\infty$$. That is: the min and max values that you gave are meaningless.

However, log-normal is a distribution with only two parameters ($$\mu$$ and $$\sigma$$), so you could determine them, if you had values of two meaningful parameters. E.g., if you were given the mean and the median, you could determine the parameters from the relations: \begin{align} \text{mean} & = e^{\mu + \frac{\sigma^2}{2}},\\ \text{median} & = e^\mu. \end{align}

• but surely some parameters will be more likely than others, even tho any lognormal is (0,inf) – rep_ho Nov 18 '19 at 12:01
• One can do a probabilistic estimate, as suggested here the other answer. However in this case it would be reasonable to include more data than given in the question. – Vadim Nov 18 '19 at 12:09
• The min and max values of the question are explicitly about a "dataset," not about the underlying distribution. In light of that, this discussion--although correct--is irrelevant. – whuber Nov 18 '19 at 17:53