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## Kernel Estimation (kernel + estimation)
## Selected Abstracts## Kernel estimation of quantile sensitivities NAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 6 2009Guangwu LiuAbstract Quantiles, also known as value-at-risks in the financial industry, are important measures of random performances. Quantile sensitivities provide information on how changes in input parameters affect output quantiles. They are very useful in risk management. In this article, we study the estimation of quantile sensitivities using stochastic simulation. We propose a kernel estimator and prove that it is consistent and asymptotically normally distributed for outputs from both terminating and steady-state simulations. The theoretical analysis and numerical experiments both show that the kernel estimator is more efficient than the batching estimator of Hong 9. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009 [source] ## Detecting spatial hot spots in landscape ecology ECOGRAPHY, Issue 5 2008Trisalyn A. NelsonHot spots are typically locations of abundant phenomena. In ecology, hot spots are often detected with a spatially global threshold, where a value for a given observation is compared with all values in a data set. When spatial relationships are important, spatially local definitions , those that compare the value for a given observation with locations in the vicinity, or the neighbourhood of the observation , provide a more explicit consideration of space. Here we outline spatial methods for hot spot detection: kernel estimation and local measures of spatial autocorrelation. To demonstrate these approaches, hot spots are detected in landscape level data on the magnitude of mountain pine beetle infestations. Using kernel estimators, we explore how selection of the neighbourhood size (,) and hot spot threshold impact hot spot detection. We found that as , increases, hot spots are larger and fewer; as the hot spot threshold increases, hot spots become larger and more plentiful and hot spots will reflect coarser scale spatial processes. The impact of spatial neighbourhood definitions on the delineation of hot spots identified with local measures of spatial autocorrelation was also investigated. In general, the larger the spatial neighbourhood used for analysis, the larger the area, or greater the number of areas, identified as hot spots. [source] ## Quantifying habitat use in satellite-tracked pelagic seabirds: application of kernel estimation to albatross locations JOURNAL OF AVIAN BIOLOGY, Issue 3 2000A. G. WoodWe develop a new approach to quantifying habitat use within the foraging ranges of satellite-tracked seabirds. We applied kernel estimation techniques to 167 days (3738 locations) of data from Black-browed and Grey-headed albatrosses Diomedea melanophris and D. chrysostoma during the chick-rearing period of the breeding cycle at South Georgia. At this time the activity range of these two species covers an estimated 440,000 and 640,000 km2, respectively, with very substantial overlap. In contrast, kernel estimation reveals that the main foraging areas of these two sympatric, congeneric species are very distinct. Based on location density categories accounting for 50% of locations, the foraging areas cover c. 81,500 and c. 119,700 km2, respectively, with 42% and 50% of the range of one species overlapping with that of the other. [source] ## Variable kernel density estimation AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 3 2003Martin L. HazeltonSummary This paper considers the problem of selecting optimal bandwidths for variable (sample-point adaptive) kernel density estimation. A data-driven variable bandwidth selector is proposed, based on the idea of approximating the log-bandwidth function by a cubic spline. This cubic spline is optimized with respect to a cross-validation criterion. The proposed method can be interpreted as a selector for either integrated squared error (ISE) or mean integrated squared error (MISE) optimal bandwidths. This leads to reflection upon some of the differences between ISE and MISE as error criteria for variable kernel estimation. Results from simulation studies indicate that the proposed method outperforms a fixed kernel estimator (in terms of ISE) when the target density has a combination of sharp modes and regions of smooth undulation. Moreover, some detailed data analyses suggest that the gains in ISE may understate the improvements in visual appeal obtained using the proposed variable kernel estimator. These numerical studies also show that the proposed estimator outperforms existing variable kernel density estimators implemented using piecewise constant bandwidth functions. [source] ## Legendre polynomial kernel estimation of a density function with censored observations and an application to clinical trials COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 8 2007Simeon M. BermanLet f(x), x , ,M, M , 1, be a density function on ,M, and X1, ,., Xn a sample of independent random vectors with this common density. For a rectangle B in ,M, suppose that the X's are censored outside B, that is, the value Xk is observed only if Xk , B. The restriction of f(x) to x , B is clearly estimable by established methods on the basis of the censored observations. The purpose of this paper is to show how to extrapolate a particular estimator, based on the censored sample, from the rectangle B to a specified rectangle C containing B. The results are stated explicitly for M = 1, 2, and are directly extendible to M , 3. For M = 2, the extrapolation from the rectangle B to the rectangle C is extended to the case where B and C are triangles. This is done by means of an elementary mapping of the positive quarter-plane onto the strip {(u, v): 0 , u , 1, v > 0}. This particular extrapolation is applied to the estimation of the survival distribution based on censored observations in clinical trials. It represents a generalization of a method proposed in 2001 by the author [2]. The extrapolator has the following form: For m , 1 and n , 1, let Km, n(x) be the classical kernel estimator of f(x), x , B, based on the orthonormal Legendre polynomial kernel of degree m and a sample of n observed vectors censored outside B. The main result, stated in the cases M = 1, 2, is an explicit bound for E|Km, n(x) , f(x)| for x , C, which represents the expected absolute error of extrapolation to C. It is shown that the extrapolator is a consistent estimator of f(x), x , C, if f is sufficiently smooth and if m and n both tend to , in a way that n increases sufficiently rapidly relative to m. © 2006 Wiley Periodicals, Inc. [source] |