The normal probability curve is an even exponential function which is symmetrical about the mean. The left side of the curve may be approximated by an exponential curve close to the original curve, with the domain a subset of the real number system.

I guess you're looking for initial values for the non-linear regression algorithm. For the variance term, you can use the full-width at half the maximum height (FWMH), which is related to the square root of the variance term in the normal density.

For the exact relationship, it depends on how exactly you parameterize the model; I guess it can be found on wikipedia or on the web for the usual parameterizations.

For the exponential family of distributions (normal, gamma, Poisson, bi- and multinominal), there exist closed form solutions for MLE fits that do not require initial guesses / starting values. In R you can use fitdistr (package MASS):

http://www.di.fc.ul.pt/~jpn/r/distributions/fitting.html

There is a package fitdistrplus allows fitting further distribution models and can handle censored data:

https://cran.r-project.org/web/packages/fitdistrplus/index.html

Hi Jochen,

I'm not sure we are speaking of the same fit. If I understand well your answer, you speak about fitting a probability law to a one-dimensional dataset (I mean, a single variable X), which means for the Gaussian finding mean and variance of the dataset, and saying that this dataset comes from iid sampling from a Gaussian [or that X is Gaussian]. In that case, indeed, the problem is (for a Gaussian and several other laws) trivial with point estimates built from the dataset.

I understood (may be wrongly) the question as a problem of having a two-dimensionnal dataset (x, y) which, when drawn as y = f(x), presents a bell-shaped form, and the question is to find the best Gaussian that models this bell-shaped (x,y)-relationship (that is, find µ and sigma such as Y = f((X-µ)/sigma) + epsilon, where f is the distribution function of a centered, reduced Gaussian, and epsilon a centered random variable... expected Gaussian as usual, probably). In this situation of non-linear regression, I'm not sure that closed form exist, hence the initial values problem.

Re-reading the initial post, I's quite possible that I misunderstood the question, but it's not clear what "not a sample" means exactly, so may be Stephen should explain a little bit more why there is no variance and what his problem is exactly...

Hi Emmanuel,

yes, you are right.

I provide a solution in R that is quite robust. The script with explanations/comments in attached, also a plot showing a result (green line: initial guess, red line: fit (blurred read lines indicate the associated uncertainty based on the data; for explanations see script).

The reference to a bell curve is an unintended misdirection. More information is needed for a complete answer.

The first suggestion is try polynomial curve fitting.

What is the effect you are describing with the data? A straight line effect can appear curved if a second effect is detracting or compensating.