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for the parametrization of a model from experimental data I want to estimate the variance of samples to weight them correctly. Because of restrictions of the underlying process (a chemical reaction), some of these samples may only contain only one data point. Even though these values are quite uncertain, they may contain important information about the process.

Let me explain my conundrum in more detail: For the prametrization I use a Least-Squares type algorithm that requires gridded data and accepts weights of the data points. I have the data of a variety of experiments where the reaction rate $r$ and the temperature $t$ has been measured continuously. I select a temperature grid $[t_1, \ldots t_n]$ and calculate the corresponding $[r_1, \ldots, r_n]$ by assigning the experimental values to "bins", i.e. temperature intervals, and calculating the mean. Now I also want to estimate the variance of these samples to weight the data points correctly, but because of the underlying process some of these bins (typically at very low and very high temperatures) will contain very few - even only one - value.

What would be the statistically sound method to estimate the variances?

I have considered using pooled variance, but I cannot assume that all the sample variances are equal.

Thank you for your help!

Felix

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