# Two-sample t-test / ANOVA on functions, with unequal variances

Suppose $N$ experiments can be made in varying conditions. Each of them yields an estimate $f_i$ of a continuous (and, if necessary, positive) function of x over some interval. Experiment $i$ is repeated a number of times $n_i$, and each time it yields a function $f_i^j$. I am given the following data $$\forall x \quad \forall i \quad {n_i,f_i(x),\sigma_i(x)}$$ where $$f_i(x) = \frac{1}{n_i}\sum_{j=1}^{n_i} f_i^j(x)$$ $$\sigma_i^2(x) = \frac{1}{n_i}\sum_{j=1}^{n_i} (f_i^j(x)-f_i(x))^2$$

Suppose experiment $1$ is the reference experiment. I would like to know whether some or all of the $f_i$ agree with function $f_1$. If they do not agree, I would like to know the region in which there is disagreement. Note however that since the experiments were made in different conditions, it is expected that the variance functions do not agree, and have to be treated as different for each experiment.

For now, I was planning on doing a two-way ANOVA on each $x$ value followed by a two-sample $t$-test if necessary to determine the disagreeing experiments. However, I was looking for a method that treats the function as a whole, heteroscedastic, and that is bayesian if possible. I could not find anything. Literature references appreciated.

• This seems like a question from functional data analysis. Did you look at the standard FDA reference? – mpiktas Aug 2 '11 at 6:08
• What about checking whether the variable $\|f_i-f_1\|$ is different from zero, where $\|\cdot\|$ is some norm, $\sup_{x}|f(x)|$ for example? – mpiktas Aug 2 '11 at 6:12
• thanks for the book, didn't know it! It is definitely something I'll look up! And the sup norm is adequate also. What about finding discrepancy regions? – yannick Aug 2 '11 at 20:22