Optimal Choice of Nonparametric Estimates of a Density and of Its Derivatives

Horová, Ivanka; Vieu, Philippe; Zelinka, Jiří

doi:10.1524/strm.2002.20.14.355

Cited by 5 publications

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“…There exists a class of optimal kernels minimizing MISE(λ h,K ) with respect to K (see e.g. Müller (1988), Marron and Nolan (1989), Horová et al (2002)). Here we recommend to use kernels of order two, namely the Epanechnikov kernel…”

Section: Kernel Estimates Of the Hazard Functionmentioning

confidence: 99%

Hazard function for cancer patients and cancer cell dynamics

Horová¹,

Pospíšil²,

Zelinka³

2009

Journal of Theoretical Biology

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Abstract. The aim of the paper is to develop a procedure for an estimate of an analytical form of a hazard function for cancer patients. Although a deterministic approach based on cancer cell population dynamics yields the analytical expression, it depends on several parameters which should be estimated. On the other hand a kernel estimate is an effective nonparametric method for estimating of hazard functions. This method provides the pointwise estimate of the hazard function. Our procedure consists in two steps: in the first step we find the kernel estimate of the hazard function and in the second step the parameters in the deterministic model are obtained by the least squares method. A simulation study with different types of censorship is carried out and the developed procedure is applied to real data.

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Section: Kernel Estimates Of the Hazard Functionmentioning

confidence: 99%

Hazard function for cancer patients and cancer cell dynamics

Horová¹,

Pospíšil²,

Zelinka³

2009

Journal of Theoretical Biology

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“…Such a procedure is called a factor method (see, for example, Müller et al (1987) and Horová et al (2002)). Letĥ h opt;;k be an estimate of h opt;;k .…”

Section: Kernel Estimate Of the Hazard Function And Its Derivativesmentioning

confidence: 99%

“…Formula (13) provides the basis for the automatic procedure for simultaneous choice of the bandwidth, the kernel and the order of the kernel. This procedure (Horová et al, 2002) will be briefly reviewed in the next section.…”

Section: Kernel Estimate Of the Hazard Function And Its Derivativesmentioning

confidence: 99%

“…Kernels K 2 N kÀ2 \ S k minimizing the functional TðKÞ are called optimal kernels and are denoted K opt;;k . These kernels can be expressed by means of the Gegenbauer polynomials (see, for example, Granovsky and Müller, 1991;Granovsky et al, 1995;Horová et al, 2002). Some of these kernels K opt;;k for different values of and k 2 I ðk 0 Þ as well as the corresponding factors ;k are given in Tables 1-3. In view of the fact that in the following we will only focus on the estimate…”

Section: Choice Of Parametersmentioning

confidence: 99%

“…The kernel estimates depend on the bandwidth, the kernel and the order of the kernel. The paper by Horová et al (2002) dealt with simultaneous choice of these parameters for density estimates. The similar procedure proposed in this article allows us to chooses these three parameters in some optimal way.…”

Section: Introductionmentioning

confidence: 99%

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Kernel estimates of hazard functions for carcinoma data sets

2006

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SUMMARYThe present article focuses on kernel estimates of hazard functions and their derivatives. Our approach is based on the model introduced by Müller and Wang (1990). In order to estimate the hazard function in an effective manner an automatic procedure in a paper by Horová et al. (2002) is applied. The procedure chooses a bandwidth, a kernel and an order of a kernel. As a by-product we propose a special procedure for the estimation of the optimal bandwidth. This is applied to the carcinoma data sets kindly provided by the Masaryk Memorial Cancer Institute in Brno. Attention is also paid to the points of the most rapid change of the hazard function.

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Density estimate and its application to analysis of temperature series

et al. 2003

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SUMMARYNonparametric density estimates attempt to reconstruct the probability density from which a random sample has come, using the sample values and as few assumptions as possible about the density. These methods are smoothing operations on the sample distribution. Methods of kernel estimates represent one of the most effective nonparametric methods. These methods are simple to understand, easy to implement and they have very good mathematical properties. We employed the automatic procedure for the selection of the bandwidth, the kernel and the order of the kernel. This procedure is used for analysis of air temperature fluctuations for series of Central England and Prague-Klementinum in the periods 1661-2000 and 1771-2000, respectively. Graphical representation of the family of estimated densities in three dimensional space provide a better explanation of the long-term trends in temperature distribution of both series.

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Optimal Choice of Nonparametric Estimates of a Density and of Its Derivatives

Cited by 5 publications

References 0 publications

Hazard function for cancer patients and cancer cell dynamics

Hazard function for cancer patients and cancer cell dynamics

Kernel estimates of hazard functions for carcinoma data sets

Density estimate and its application to analysis of temperature series

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