Discrete triangular distributions are introduced, in order to serve as kernels in the non-parametric estimation for probability mass function. They are locally symmetric around every point of estimation. Their variances depend on the smoothing bandwidth and establish a bridge between Dirac and discrete uniform distributions. The boundary bias related to the discrete triangular kernel estimator is solved through a modification of the kernel near the boundary. The mean integrated squared errors and then the optimal bandwidth are investigated. We also study the adequate bandwidth for excess zeros. The performance of the discrete triangular kernel estimator is illustrated using simulated count data. An application to count data from football is described and compared with a binomial kernel estimator.
One of the most intriguing questions in plant ecology is which evolutionary strategy allows widely distributed species to increase their ecological range and grow in changing environmental conditions. Phenotypic plasticity and local adaptations are major processes governing species range margins, but little is known about their relative contribution for tree species distribution in tropical forest regions. We investigated the relative role of phenotypic plasticity and local adaptation in the ecological distribution of the widespread palm Euterpe edulis in the Brazilian Atlantic Forest. Genetic sampling and experiments were performed in old‐growth remnants of two forest types with higher (Seasonal Semideciduous Forests vs. Submontane Rainforest) and lower biogeographic association and environmental similarities (Submontane Rainforest vs. Restinga Forest). We first assessed the molecular genetic differentiation among populations, focusing on the group of loci potentially under selection in each forest, using single‐nucleotide polymorphism (SNPs) outliers. Further, we looked for potential adaptive divergence among populations in a common garden experiment and in reciprocal transplants for two plant development phases: seedling establishment and sapling growth. Analysis with outlier loci indicated that all individuals from the Semideciduous Forest formed a single group, while another group was formed by overlapping individuals from Submontane Rainforest and Restinga Forest. Molecular differentiation was corroborated by reciprocal transplants, which yielded strong evidence of local adaptations for seedling establishment in the biogeographically divergent Rainforest and Semideciduous Forest, but not for Restinga Forest and Submontane Rainforest. Phenotypic plasticity for palm seedling establishment favors range expansion to biogeographically related or recently colonized forest types, while persistence in the newly colonized ecosystem may be favored by local adaptations if climatic conditions diverge over time, reducing gene flow between populations. SNPs obtained by next‐generation sequencing can help exploring adaptive genetic variation in tropical trees, which impose several challenges to the use of reciprocal transplants.
Asymmetric discrete triangular distributions are introduced in order to extend the symmetric ones serving for discrete associated-kernels in the nonparametric estimation for discrete functions. The extension from one to two orders around the mode provides a large family of discrete distributions having a finite support. Establishing a bridge between Dirac and discrete uniform distributions, some different shapes are also obtained and their properties are investigated. In particular, the mean and variance are pointed out. Applications to discrete kernel estimators are given with solution to a boundary bias problem.
Papaya (Carica papaya L.) is a tropical and climacteric fruit with antioxidant, anticarcinogenic and anti-mutagenic properties, containing carotenoids and high nutritional value.• Brazil is ranked second in the world for papaya production with 17.5% of global production.• One of the most important criteria for determining the ripeness stage over time is the peel color.
Multinomial responses frequently occur in dose level experiments. For example, in a study of the influence of gamma radiation on the emergence of house flies (Musca domestica L., 1758), three disjoint outcomes occurred: death before the pupae opened, death during emergence, and life after emergence. Although the flies are easy to breed, this sort of bioassay is, in general, very expensive since it requires the use of a gamma radiation source. Experiments therefore need to be designed to involve the minimum number of different doses. Here the theory of optimum experimental design is applied to provide efficient experiments to estimate the parameters of those multinomial logistic models that are a special case of the multivariate logistic models of Glonek and McCullagh (1995, Journal of the Royal Statistical Society, Series B 57, 533-546). The purpose is to reduce the overall experimental cost. The general equivalence theorem (Fedorov, 1972, Theory of Optimal Experiments) is adapted to this class of models, providing an effective method of generating and checking the optimality of designs. One example on flies demonstrates the method, which can be easily implemented.
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