EDF and EQf orthogonal component decompositions and tests of uniformity

Abstract

Orthogonal expansions of a probability density function and the corresponding quantile density function are employed to motivate new as well as existing omnibus quadratic tests for uniformity. Utilizing Legendre polynomial component decompositions, the proposed goodness-of-fit criteria are based on Fourier analytic techniques applied to the empirical distribution function (EDF) and the empirical quantile function (EQF). Individual EDF components involve the continuous Legendre polynomials and component aggregates provide the Neyman smooth statistics. The EQF components are spacings statistics with Hahn polynomial vector weight functions. Aggregates of these spacings components provide natural EQF analogues of the Neyman smooth statistics, an orthogonal decomposition of the Greenwood spacings statistic, and a discrete spacings analogue of the Anderson-Darling statistic. Asymptotic distribution theory is obtained under uniformity as well as fixed and local alternatives. Results from Monte Carlo studies indicate adequacy of asymptotics for small samples and suggest a hybrid EDF-EQF quadratic statistic as an omnibus test for uniformity.

Keywords:

• Goodness of fit
• L‐moments
• LU‐statistics
• orthogonal components
• O‐statistics
• quantile
• spacings