Non-linear Differential Equations For Quantile Functions
The non-linear ordinary differential equation given for normal distribution is a special case of that available for any quantile function whose second derivative exists. In general the equation for a quantile, Q(p), may be given. It is
augmented by suitable boundary conditions, where
and ƒ(x) is the probability density function. The forms of this equation, and its classical analysis by series and asymptotic solutions, for the cases of the normal, Student, gamma and beta distributions has been elucidated by Steinbrecher and Shaw (2008). Such solutions provide accurate benchmarks, and in the case of the Student, suitable series for live Monte Carlo use.
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