I have a set of points $\{x_i,y_i\}$ with 20 elements on the scatter plot, the distribution looks like Gaussian. My goal is to estimate the Gaussian center $x_{center}=\mu$ of this distribution. I'm trying to apply the bootstrapping method by sampling 20 elements with replacement each time and find the average of each sample. I'll repeat this procedure many times to get a small distribution of average.
My question is since my original dataset might not be a normal distribution, should I normalize the probability $y_i$ and use that as weight in my resampling of $x_i$? If so, in the output distribution of average, how can I determine its peak as my estimation? Is it just the highest bar?
Thanks:)
The standard practice is taking the (unweighted) average of the BS sample, i.e., let $w_i$ be the weight of the $x_i$, hence for each sample you have $$ \mu_{b} = \sum_{i=1}^{20} w_i x_i, $$ for $b=1,...,B$, where $B$ is the number of BS samples. Then the BS point estimator is $$ \mu_B = \frac{1}{B}\sum_{b=1}^B \mu_b. $$ In your case it should correspond to the "peak" of the BS-based sample distribution.