Chisquared and related inducing pivot variables: an application to orthogonal mixed models

Abstract

We use chi-squared and related pivot variables to induce probability measures for model parameters, obtaining some results that will be useful on the induced densities. As illustration we considered mixed models with balanced cross nesting and used the algebraic structure to derive confidence intervals for the variance components. A numerical application is presented. ARTICLE HISTORY

Estimation and incommutativity in mixed models

“…w,p,u of these samples strongly converge to the corresponding exact quantiles of the distribution induced for the variance components when N → ∞; see again [10]. Then we get the induced…”

“…If normality is assumed, we may use inducing pivot variables to obtain confidence intervals; see [10]. For each j ∈ D, let χ 2 j,1 , .…”

“…These estimators enable us to obtain generalized least squares, GLS, estimators for estimable vectors. In Section 3 we assume the model to be normal and use inducing pivot variables to obtain confidence intervals for the variance components; see [10]. These confidence intervals may be used, through duality, to test hypotheses about them.…”

Confidence intervals and ellipsoids for estimable functions and estimable vectors in models with orthogonal block structure

“…The empirical quantiles z q,v (c d ) of the samples converge almost surely, again according to the reverse Glivenko-Cantelli theorem, to the exact quantile z q (c d ), when v → ∞; see Ferreira et al 21 We now have the (approximate)1 − q level confidence intervals…”

“…The empirical quantiles $$$ {z}_{q,v}\left({\boldsymbol{c}}_d\right) $$$ of the samples converge almost surely, again according to the reverse Glivenko‐Cantelli theorem, to the exact quantile $$$ {z}_q\left({\boldsymbol{c}}_d\right) $$$, when $$$ v\to \infty $$$; see Ferreira et al 21 …”