Various mixed models that are relevant for analyzing Army test data are described, along with several hypothesis testing and interval estimation problems. The problems come up in the context of investigating gun tube accuracy of an M1 Series tank; in particular, for the study of tube-to-tube dispersion. Factors that affect tube-to-tube variability might include the tanks, ammunition lot, ammunition temperature, firing occasions, and so on. Some of these are fixed factors, and others are random factors. The inference problems that arise in the study of tube-to-tube dispersion are somewhat different from those usually encountered in typical ANOVA situations. A unified approach to solving these problems is presented using the concepts of generalized p values and generalized confidence intervals. The performance of the resulting tests and confidence intervals is numerically investigated and is found to be quite satisfactory. Analysis of some Army test data is presented to illustrate the results.

Abstract. Two-weight L p norm inequalities are proved for Cesàro means of Laguerre polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and "unweighted" cases, by including all values of p ≥ 1 for all positive orders of the Cesàro summation and all values of the Laguerre parameter α > −1. Almost everywhere convergence results are obtained as a corollary. For the Cesàro means the hypothesized conditions are shown to be necessary for the norm inequalities. Necessity results are also obtained for the norm inequalities with the supremum of the Cesàro means; in particular, for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth.

Abstract. An accurate estimate is obtained of the Cesàro kernel for Hermite expansions. This is used to prove two-weight norm inequalities for Cesàro means of Hermite polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and "unweighted" cases. An almost everywhere convergence result is obtained as a corollary. It is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the Cesàro supremum of the means are necessary.

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