1) Ideally, one should estimate confidence set of parameters (in your case, it would be a set in 3-dimentional space of real numbers). Approximating it with individual confidence intervals for each of the parameters (i.e. approximating it with the direct product of three CI's) in many cases is not a good move.

If you have as few as 3 parameters you may be able to estimate the confidence set as is.

2) To estimate the range of predicted outcomes you should randomize them within the confidence set.

3) When you get fit with R=0.995 you should be EXTREMELY careful in checking whether you have overfitting or not. Are your data really so smooth and nice and follow the model so precisely? And do you have enough data points?

Konstantin, yes, the data are fitted very well by the model. I am fitting a frequency histogram which follows an asymptotic pattern with the formula y = a + 2.718 ^ (b - c * x). Such histograms which denote patterns of commonness (few species) and rarity (many species) usually have very similar shapes, so I am confident that the model is a good fit. The data points are the number of frequency categories observed, in this case 18. I would assume that the approach to separately randomize each parameter is conservative, since the parameters are in reality not independent, and so randomizing them separately will result in a broader range of variation (more conservative) for the model prediction for each x. Is that correct? I would then define the range of the outcome not as confidence interval in itself but being based on independent confidence intervals of each parameter. I am not sure how I would randomize all three parameters together in context to obtain a true 95% confidence interval for the outcome. Would that not narrow down the confidence interval for each parameter?

1) Are that frequency histograms composed of data measured in real life? I am pretty much sure that the histograms are the result of fitting such an exponential frequency model to the really-measured data on species prevalence (or something else).

If my assumption is true then you should study the statistical properties of that first model fit to real data, not the second fit of a model to model output.

2) The "conservativeness" of the approach with independent parameter randomization depends upon the method of calculating the individual CI's. If the CI's are taken as min and max values of respective parameters over the entire confidence set then yes, it is correct that the approach is conservative. But if the CI's are calculated using univariate approach (freeze all params besides the studied one, compute the 1-dimentional CI) then there is a chance for the approach to be "overoptimistic" (if the true confidence set is significantly skewed).

3) To construct a 3D 95% parameter parallelepiped you have to take (0.95)^(1/3)=98.3% 1D parameter CI's.

Konstantin, no, the model is fitted to real data (number of occurrences for each frequency class). I used a two-step approach to fit the model, first by using a log-scale for the occurrences and fitting a linear model and then by transforming the linear model back and fitting the corresponding non-linear model to the same data. I compared the 95% and 98.3% CIs for the three parameters and there is practically no difference except after the second decimal. So it seems both approaches would yield similar results. I am running the model fitting in STATISTICA and it appears the parameter CIs are estimated simultaneously, not in univariate fashion. So based on your advice, it appears that I am on the right track. Thanks!

Robert, Your approach would be correct if the parameter estimates were uncorrelated. However, this is rarely the case, so you need to draw random vectors from the multivariate (d-3) normal distribution with mean (a,b,c)' and the parameter covariance matrix.

If the covariance matrix is C=WL^0.5 L^0.5W' = UU' then the parameter vectors are drawn as follows.

1) select Z to be a vector of three random normals with mean a,b,c and variance 1.0

2) Calculate V=U*Z

3) V is a multivariate vector with properly correlated values and means a, b, c.

5) Now drqw a random set of predictors and calculate the predicted value .

6) Repeat

I would do this in a matrix language such as R or Matlab

Robert, multi-step approaches to estimating model parameters are often just a simplifying approximation. If you have only 3 parameters in your model I would recommend to estimate them simultaneously (using a standard numerical residual-minimizing procedure).

John, why can you suppose that the parameters of the model follow normal joint distribution? There is absolutely no "physical" argument to support this.

Robert is using least squares to estimate parameters after log transformation of the dependent variable. Generally this leads to approximately normally distributed dependent variables (Otherwise why transform?) which leads to an asymptotically normal distribution for parameters. If Robert were to take a more modern approach estimating parameters directly by maximum likelihood, then he would need to simulate from the distribution underlying the likelihood.

Although perhaps imperfect in selection of an underlying probability model, this would be better than assuming that the parameters are independent. Particularly for a non-linear model where parameters are notoriously redundant.

Konstantin, You might also note that the asymptotic distribution of the parameter vector under maximum likelihood estimation for non-linear models is the multivariate normal distribution. This also provides a basis for assuming normality in Robert's problem, assuming he has a modestly large sample size.

For a very short summary, see Appendix A of McCullagh and Nelder (1989) discussing asymptotic normality of maximum likelihood estimates. Also a great source for more general nonlinear model formulations.

http://www.amazon.com/Generalized-Edition-Monographs-Statistics-Probability/dp/0412317605