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**Small sample inference for fixed effects from restricted maximum likelihood.**
*(English)*
Zbl 0890.62042

Summary: Restricted maximum likelihood (REML) is now well established as a method for estimating the parameters of the general Gaussian linear model with a structured covariance matrix, in particular for mixed linear models. Conventionally, estimates of precision and inference for fixed effects are based on their asymptotic distribution, which is known to be inadequate for some small-sample problems.

We present a scaled Wald statistic, together with an \(F\) approximation to its sampling distribution, that is shown to perform well in a range of small sample settings. The statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias. This approach has the advantage that it reproduces both the statistics and \(F\) distributions in those settings where the latter is exact, namely for Hotelling \(T^2\) type statistics and for analysis of variance \(F\)-ratios. The performance of the modified statistics is assessed through simulation studies of four different REML analyses and the methods are illustrated using three examples.

We present a scaled Wald statistic, together with an \(F\) approximation to its sampling distribution, that is shown to perform well in a range of small sample settings. The statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias. This approach has the advantage that it reproduces both the statistics and \(F\) distributions in those settings where the latter is exact, namely for Hotelling \(T^2\) type statistics and for analysis of variance \(F\)-ratios. The performance of the modified statistics is assessed through simulation studies of four different REML analyses and the methods are illustrated using three examples.

### MSC:

62H12 | Estimation in multivariate analysis |

62J05 | Linear regression; mixed models |

62P10 | Applications of statistics to biology and medical sciences; meta analysis |