Testing a differential condition and local normality of densities

In this paper, we consider testing if a density satisfies a differential equation. This result can be applied to see if a density belongs to a particular family of distributions. For example, the standard normal density f(t) satisfies the differential equation f'(t)+tf(t)=0. If a density satisfies this equation at that point t, then it is called locally standard normal at that point. Thus, there is a practical need to test whether a density satisfies a certain differential equation. We consider a more general differential equation F(x)=0 involving f(x). We can test the null hypothesis H0: f satisfies the equation F(x)=0 against the alternative hypothesis Ha: F(x)≠0. The testing procedure is accompanied by an asymptotic normality statement.