# Distance Metrics for Comparing Theoretical Sampling Distributions

I have a sampling probability density function (denoted $f(x|M,N)$ that becomes hard to calculate for large degrees of freedom (i.e. as $M$ and $N$ get big). If I have another pdf (say $f^{*}(x|M,N)$ such that,

$$\lim_{M\to\infty} \lim_{N\to\infty} f(x|M,N)=f^{*}(x|M,N)$$

what distance metrics would be best for determining the values of $M$ and $N$ such that $f^{*}$ closely approximates $f$? Part of my problem is that I am not sure how to define "closely approximates"... I have tried calculating the total variation distance between the two pdf's while varying the sample sizes but am unsure if this is a good metric to use. I am not very familiar with distance metrics and would appreciate any thoughts.