Having math - statistics edu it might be reasonable first to look at https://en.wikipedia.org/wiki/Generaliz ... of_moments
. Actually the basis of estimation parameters is its closeness to observed data.To refresh your knowledge you can look at some simple statistics handbook how one estimate mean and variance for different distributions say Poisson and Gaussian.
On the other hand you should be aware that financial observations are represented in the form of time series. As far as market changes in time daily observation could not be modeled as identical distribution independent random variable. Hence you should be quite critical with respect to what you will do. On the other hand it might be the best what could be done.Which method is better to estimate parameters sounds somewhat informal. If there is no similar statement in Statistics then the answer will be alway quite subjective. Hence it might be better to use two methods.
One of financial advances is also relates to volatility estimation phenomena. They have assumed that underlying of an option belong to the class of diffusion processes From real world observations they could find its volatility. Then they use option perfect BS model of option pricing to observed option prices and estimate volatility of underlying asset. And practice shows that volatility of Put and Call options on the same underlying asset are different an for the same maturity and volatility estimate is also different on the options say Call with different maturity. It might be something wrong with parameters estimation too.