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Block designs have been widely used in almost all areas of human investigations. It is commonly assumed that all blocks have the same size which may not, however, always be available because of practical restrictions. In this paper, the problem of main-effect plans with blocks of unequal sizes is addressed. The sufficient condition for each pair of factors to be orthogonal through the block factor is indicated. Useful properties on ANOVA (analysis of variance) of such designs are discussed. Construction procedures via augmenting new blocks and combining main-effect plans with unequal block sizes are also provided. The resulting designs have flexible sizes of blocks and factors.

When the data contain outliers or come from population with heavy-tailed distributions,which appear very often in spatio-temporal data, the estimation methods based on the least square method will not perform well.More robust estimation methods are required. We propose the local linear estimation for the spatio-temporal model based on the local modal method. Asymptotic theory properties and data analysis results show that the proposed estimator is more efficient than the ordinary least square-based estimation in the case of outliers or heavy-tailed error distributions, and as asymptotically efficient as the least square estimator when there are no outliers and the error is a normal distribution. The modal expectation-maximization algorithm is adopted and the asymptotic distributions of estimators are driven when the data are mixing correlation.