This reprint differs from the original in pagination and typographic detail. 1 2 Y. LI ET AL.have strong correlations, which are stationary in time or distance lags. The structure for the observation at a particular time or location within one subject can be very general, for example, a vector or even a function.Our study arises from a colon carcinogenesis experiment. The biomarker that we are interested in is p27, which is a life cycle protein that affects cell apoptosis, proliferation and differentiation. An important goal of the study is to understand the function of p27 in the early stage of the cancer development process. In the experiment, 12 rats were administered azoxymethane (AOM), which is a colon specific carcinogen. After 24 hours, the rats were terminated and a segment of colon tissue was excised from each rat. About 20 colonic crypts were randomly picked along a linear slice on the colon segment. The physical distances between the crypts were measured. Then, within each crypt, we measured cells at different depths within the crypts, and then the expression level of p27 was measured for each cell within the chosen crypts. In this data set, crypts are naturally functional data (Ramsay and Silverman [13]), in that the responses within a crypt are coordinated by cell depths. There is a literature about similar data, for example, Morris et al. [11].However, in this paper we will be focused on a very different perspective. In this application the spatial correlation between crypts is of biological interest, because it helps answer the question: if we observe a crypt with high p27 expression, how likely are the neighboring crypts to have high p27 expression? We will phrase much of our discussion in terms of this example, but as seen in later sections, we have a quite general structure that includes time series as a special case. In that context, the asymptotic theory is as the number of "time series locations," that is, crypts, increases to infinity.Although motivated by a very specific problem, nonparametric covariance/correlation estimators are worth being investigated in their own right. They can be used in a statistical analysis as: (a) an exploratory device to help formulate a parametric model, (b) an intermediate tool to do spatial prediction (kriging), (c) a diagnostic for parametric models and (d) a robust tool to test correlation. Understanding the theoretical properties of the nonparametric estimator is important under any of these situations. A limiting distribution theory would be especially valuable for purpose (d).There is previous work on the subject of nonparametric covariance estimation. Hall, Fisher and Hoffmann [7] developed an asymptotic convergence rate of a kernel covariance estimator in a time series setting. They required not only an increasing time domain, but increasingly denser observations. Diggle and Verbyla [5] suggested a kernel-weighted local linear regression estimator for estimating the nonstationary variogram in longitudinal data, without developing asymptotic theory. Gua...

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Objectives Childhood obesity increases risk factors related to metabolic diseases and watermelon's bioactive components can help reduce these risk factors. However, no study has investigated the effects of watermelon juice containing rind in children with obesity or overweight. The objective of this study was to examine the effects of blenderized watermelon with rind on anthropometric and clinical markers of BMI, body fat, glucose, insulin, A1C, inflammation, lipid profile, liver function enzymes, and satiety hormones. Methods A randomized, cross over clinical design was implemented where children (n = 17, 8 females/9 males, age 12.9 ± 2.0 years) consumed one cup (240 mL, 70 kcal) of blenderized watermelon juice with rind or isocaloric sugar juice (control) every day for eight weeks with a four-week washout period. Results Significantly lower BMI (p = 0.032) and BMI percentile (p = 0.038), were observed when comparing eight weeks of watermelon juice intake to eight weeks of sugar juice intake. Sugar juice consumption increased BMI percentile (p = 0.014) compared to baseline. A decrease in body fat, measured with Bod Pod within watermelon juice consumption was observed, but not in sugar juice (p = 0.047). Body fat was lower in watermelon juice than sugar juice intake at week eight (p = 0.036). A decrease in A1C was observed within watermelon juice intake (p = 0.008) but not with sugar juice intake. A1C watermelon juice week eight was lower than sugar juice week eight (p = 0.012). No significant differences between trials were observed for leptin, ghrelin, c-reactive protein, glucose, insulin, lipid profiles, and liver function enzymes. Conclusions The results support that blenderized watermelon juice consumption improved cardiometabolic risk factors including BMI, BMI percentile, body fat, and A1C in desirable directions. To our knowledge, this is the first study examining the effects of watermelon juice with flesh and rind consumption on cardiometabolic risk factors in overweight/obese children. Our study shows that watermelon is a potential alternative to unhealthful snacks for reducing the risk factors related to obesity. Funding Sources The National Watermelon Promotion Board [NWPB 19-20].