A Wiener Germ approximation of the noncentral chi square distribution and of its quantiles (Q5943411)

The authors present some approximation formulas for the distribution function of noncentral \(\chi^2\) random variables with \(\nu\) degrees of freedom and a noncentrality parameter \(\delta^2.\) For example, \[ F(x, \nu, \delta^2) \sim \Phi \biggl(\pm \biggl[ \nu(s-1)^2 \biggl( {1\over 2s} + \mu^2 -{1\over s}h (1-s)\biggl) - \ln\biggl({1\over s} -{2\over s}{h(1-s) \over 1+2\mu^2 s}\biggr) +{2\over \nu}B(s)\biggr]^{1\over 2}\biggr), \] where \(\mu^2 = \delta^2/\nu\), \(h(1-y) ={1\over y^2}\bigl[ (1-y) \ln(1-y) +y - {1\over 2} y^2\bigr]\) for \(y<1\), \(h(1-y) = -h \bigl( 1-{1\over y}\bigr)\), \(B(s)\) is some function. The authors use these formulas to construct an approximation algorithm of numerical evaluation of \(F(x, \nu, \delta^2)\). It is also applied for a reliable approximation of the quantiles of the distribution for large values of noncentrality and degrees of freedom. Comparison of approximations and ``exact'' results are presented too.