Suppose I know the random variables follow normal distribution, i.e., $X\sim\mathcal(\mu,\sigma^2)$, but I can only observe the square of each sample $X^2_i$. How could I extract the parameters of $\mu$ and $\sigma$. I am thinking of moment matching, but it is unclear to me how to proceed. Anybody can help?
How to extract parameters of normal distribution from squared observations?
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statistics
random-variables
machine-learning
estimation
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1Does it help that your distribution has a name? Well, square root of your sample. https://en.wikipedia.org/wiki/Folded_normal_distribution – 2017-02-20
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0Multiple of a non-central chi-squared distribution. See Wikipedia if you're not familiar with it. That may help. – 2017-02-20
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0There is a table of the non central moments of the normal distribution in https://en.wikipedia.org/wiki/Normal_distribution . You can estimate the 2nd, 4th, 6th, etc n.c. moments from the data by using $\sum_i X_i^2/n$, and so on. Equating the formulas to the empirical moments gives you simultaneous polynomial equations in $\mu^2$ and $\sigma^2$. Note you will be unable to ever learn the sign of $\mu$ from such data, but by using the 2nd and 4th moment formulas you should be able to estimate $\mu^2$ and $\sigma^2$. – 2017-06-30