Showing papers on "Mixed model published in 1976"
TL;DR: The restricted maximum likelihood (REML) estimators as discussed by the authors have the property of invariance under translation and the additional property of reducing to the analysis variance estimators for many, if not all, cases of balanced data (equal subclass numbers).
Abstract: The maximum likelihood (ML) procedure of Hartley aud Rao [2] is modified by adapting a transformation from Pattersou and Thompson [7] which partitions the likelihood render normality into two parts, one being free of the fixed effects. Maximizing this part yields what are called restricted maximum likelihood (REML) estimators. As well as retaining the property of invariance under translation that ML estimators have, the REML estimators have the additional property of reducing to the analysis variance (ANOVA) estimators for many, if not all, cases of balanced data (equal subclass numbers). A computing algorithm is developed, adapting a transformation from Hemmerle and Hartley [6], which reduces computing requirements to dealing with matrices having order equal to the dimension of the parameter space rather than that of the sample space. These same matrices also occur in the asymptotic sampling variances of the estimators.
401 citations
TL;DR: In this article, the authors compared the properties of autoregressive, moving average, and mixed models of increasing complexity for annual, monthly, weekly, and daily series, and compared the merits of disaggregation model and its level of complexity.
Abstract: The fundamental issue of what level of complexity is needed in stochastic models for generating or forecasting hydrologic events is studied by comparing the properties of autoregressive, moving average, and mixed models of increasing complexity for annual, monthly, weekly, and daily series. Several methods of removing seasonal nonhomogeneity of monthly series are compared, and the seasonality of monthly model coefficients is investigated. For small time increments a harmonic analysis of the coefficients is necessary to limit the number of parameters. The relative goodness of fit of several models of monthly and daily series are determined by diagnostic checks applied to the residuals, and their probability distributions are investigated. The issue of long-range versus short-range dependence is analyzed by comparing the statistics describing the span of dependence of several time series models. The need for compatible models for different time resolutions is discussed. The merits of the disaggregation model and its level of complexity are discussed and compared with those of other models.
66 citations