## Nonlinear estimation

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.

## Questions

• 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.
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.