Moments of the log non-central chi-square distribution

AbstractThecumulants and momentsof thelog of thenon-centralchi-squaredistribution are derived. For example, the expected log of a chi-squarerandomvariablewithvdegreesoffreedomislog2+ψ v2 . Applicationstomodelingprobabilitydistributionsarediscussed. 1 Introduction The Edgeworth and Cornish-Fisher expansions allow one to approximate thedensity, distribution, and quantile functions of probability distributions whosecumulants are known. It is often remarked that these expansions are inaccuratefor one-sided and highly skewed probability distributions. [2, 3]For example, consider a random variable which is the product of severalchi-square random variables. The k th raw moment of a chi-square randomvariable with vdegrees of freedom is 2 k Γ(k+v/2)/Γ(v/2). The raw momentsof the product of multiple independent chi-squares is simply the product ofthe moments. The moments can then be converted to the cumulants. Thecumulants arethen used in the Edgeworthexpansion to approximatethe density.For even a small number of factors, the Edgeworth expansion is inaccurate,sometimes yielding negative estimates, as illustrated in Figure 1.