We have data points $Y_i$ that are normally distributed (?) with a common variance and mean that is a non-linear function of an explanatory variable $X_i$ and a parameter $\lambda$. The value of $\lambda$ results in a particular derivative of the mean of $Y$ as a function of $X$. In particular the value $\lambda=1$ is of interest. Thus the client is interested in estimating the derivative of the mean function.
- Are there any bounds on $\lambda$?
- Is a normality assumption reasonable?
- Is a constant variance assumption reasonable?
Although the client indicated they were interested in estimating the derivative, it may be sufficient (or better) to directly estimate $\lambda$.
Bayesian approach for posterior on $\lambda$
A Bayesian approach via MCMC could be employed to jointly estimate the variance and $\lambda$ where the full conditional for $\lambda$ would not be available in closed form. Thus, the $\lambda$ step could be a Metropolis-Hastings step or slice sampling step or something else.
Derivative based on splines
One way to estimate the derivatives is to use splines. In this case it may be worthwhile to create a centered version of the data by subtracting the mean when $\lambda=1$. Now, if $\lambda$ is truly 1, the derivative should be zero. Thus, when we construct confidence/credible intervals at any $X$ value, we can determine if these intervals contain 0.