Rates of Convergence in CLT for Two Sample U-Statistics in Non iid Case and Multiphasic Growth Curve

Abstract

We obtain nonuniform rates of convergence in central limit theorem for two sample U-statistics in non iid case when moment generating function of the kernel ϕ necessarily exists, but the kernel may not be bounded. The rates are sharp when the kernel is bounded, like in the case of Wilcoxon two sample U statistics. Precision of these results motivates to explore data analysis of plant growth in the set-up of U-statistics. Growth patterns of Sisal plants, having high economic return for extracted leaf fibres, are tested for two different growth environment by two sample Wilcoxon U statistic. In the Indian Statistical Institute (ISI) Giridih farm these plants are grown in two different types of land viz., a high land with rock layer below topsoil having scarcity of irrigation, and the other with sandy soil structure near a hilly rivulet occasionally flooded in rainy seasons for a few days. The latter environment turns out to be more conducive for growth. We study plant growth viz., growth in number of leaves and plant height from field experiments. These variables are further studied for a subgroup of randomly sampled plants. Length and mid width of sisal leaves are studied for overall growth. Proliferation rates and second derivatives are also calculated. Almost sure confidence bands for sisal growth curves are computed in the set-up of U-statistics. These reveal multiphasic growth patterns. The study is of interest in assessing economic potential of sisal plantation in Jharkhand.