This paper introduces a method of blockwise empirical Euclidean likelihood for weakly dependent processes. The strong consistency andasymptotic normality of the blockwise empirical Euclidean likelihoodestimation are proved. It is deduced that the blockwise empirical Euclidean likelihood ratio statistic is asymptotically a chi-square statistic. These results show that the blockwise empirical Euclidean likelihood estimation is more asymptotically effcient than the original empirical (Euclidean) likelihood estimation and it is useful for weakly dependent processes.

It is very powerful for constructing nearly saturated factorial designs to character-iza fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary worldength pattern of a regular symmetrical fractional factorial split-plot (FFSTP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial dwsign theory and coding theory, we obtain some general and unified combinato-rial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to theses identites, we further estab-lish some general and unified ruleds for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.

It is known by Zhang and Park (J. Statist. Plann. Inference 91 (2000) 107) that there are no minimum aberration (MA) designs with respect to both treatments and blocks for blocked regular mixed-level factorial designs. So it should be compromised between the block wordlength pattern and treatment wordlength pattern. Two methods are considered in this article. The first isMAblocking scheme of an MA design. The other is to combine the components of the two wordlength pattern vectors into one combined wordlength pattern according to the modified hierarchical assumptions and an appropriate ordering of the numbers of alias or confounding relations. The relationship between the two types of optimal blocked designs is investigated. A complete catalogue of optimal blocked regular mixed factorial designs of the above two types with 16 or 32 runs is given.

By considering the so-called algebraic orthogonality among experimental runs, this paper introduces a new criterion, minimum inner-product moment (MIPM), for general asymmetrical designs, and shows that MIPM is equivalent to the minimum moment aberration (MMA) criterion for the natural weights. Furthermore, the relationship between the generalized minimum aberration (GMA) and some model-dependent efficiency criteria is investigated by using the complex contrasts. Thus, two new justifications of GMA criterion is given from the points of view of orthogonality among experimental runs and design efficiency. They are generalizations of the related results of Butler (2003) and Cheng, Deng, and Tang (2002) for twolevel factorial designs.