A note on recursive smoothers for semimartingales

The kernel function method developed by Yamato (1971) to estimate a probability density function essentially is a way of smoothing the empirical distribution function. This paper shows how one can generalize this method to estimate signals for a semimartingale model. A recursive convolution smoothed estimate is used to obtain an absolutely continuous estimate for an absolutely continuous signal of a semimartingale model. It is also shown that the estimator obtained has a smaller asymptotic variance than the one obtained in Thavaneswaran (1988).     

Simulation‐based sample‐sizing and power calculations in logistic regression with partial prior information

There have been many approximations developed for sample sizing of a logistic regression model with a single normally‐distributed stimulus. Despite this, it has been recognised that there is no consensus as to the best method. In pharmaceutical drug development, simulation provides a powerful tool to characterise the operating characteristics of complex adaptive designs and is an ideal method for determining the sample size for such a problem. In this paper, we address some issues associated with applying simulation to determine the sample size for a given power in the context of logistic regression. These include efficient methods for evaluating the convolution of a logistic function and a normal density and an efficient heuristic approach to searching for the appropriate sample size. We illustrate our approach with three case studies. Copyright © 2016 John Wiley & Sons, Ltd.     

Berechnung der verteilungsfunktion quadratischer formen von normalverteilen zufallsgröben and deren anwendung in der statistik

Zusammenfassung: Für die Bestimmung der Verteilungsfunktion F(x) von quadrastischen Formen normalverteilter Zufallsgröen werden Verfahren angegeben.Diese Verfahren werden mit den aus der Literatur bekannten Ergebnissen verglichen.Im Staz 2 wird eine quadratische Faltungsgleichung hergeleitet, aus der F(x) berechenbar ist. Weiterhin werden Abschätzungen nach oben und unten für F(x) hergeleitet.Die für endliche quadratischer Formen übertragen.Anwendungsbeispiele aus der Statistik zufälliger Prozesse (Identifikation und Konfidenzintervalle für die Mittelwerfunktion eines GAUSSPROZESSES) und aus der parameterfreien Statistik werden angegeben.Da zur Berechnung von F(x) eine quadratische Faltungsgleichung zu lösen ist, werden im letzten Teil der Arbeit Lösungs möglichkeiten fü diese Gleichung diskutiert und numerische Ergebnisse angegeben.     

A Method for Estimating Time Dependent Intervention Benefits under Arbitrarily Varying Age and Exogenous Components of Hazard

A method for estimating the dependence of intrinsic intervention benefits on time elapsed since the intervention took place is proposed. The method is aimed at intervention programs against diseases where one or all of the following components of hazard intensity may undergo important and unknown variations: 1) the intervention benefits to a subject are a function of the time elapsed since the intervention took place, or since inception for a continuing treatment, 2) the subjects vulnerability is an unknown function of their age, 3) the exogenous or environmental baseline intensity, to which all are assumed subjected, fluctuates arbitrarily with calendar time. During the time span of a study, these variables interact in a complex way, possibly masking the real contribution of the intervention. However, with very general assumptions about how hazard components interact, the cumulative hazards of subpopulations treated at different times in the past are shown to be described mathematically by a convolution of the time elapsed dependent intervention benefit function with the age and calendar time dependent baseline intensity. Starting from the cumulative hazards of untreated and treated subpopulations that had the intervention at different times in the past, a method of deconvolution through regularization is proposed to reconstruct the time elapsed dependence of the intervention benefit function. The regularization technique used is of the penalized least square smoothing type, it is applied to the solution of Volterra integral equations of the first kind under noisy inputs. Simulations, to test for the reconstruction of different modes of time elapsed variation of the intervention benefits, are carried out on realistically noisy data sets taken to be available at a limited number of time points. The stability of the estimated reconstructions, to measurement errors, is examined through repeated simulations with random noise added to inputs. The method is applied to a Brazilian data set where BCG vaccination resulted in a small reduction in the cumulated risk of leprosy infection.     

基于投影降维技术的期权组合非线性VaR模型

高维期权组合VaR值的计算时间和计算工作量随着市场风险因子维数的增加而迅速增加.为此,引入投影降维技术,用少数几个风险因子来解释高维期权组合总的风险,并结合快速卷积方法,建立了基于投影降维技术的市场风险因子呈厚尾分布情形下的期权组合非线性VaR模型,达到减少计算时间和计算工作量的目的,同时期权组合价值变化的信息又没有太大的损失.数值结果表明,投影降维技术能够达到与快速卷积方法、Monte-Carlo方法差不多的估算精度,而计算效率明显优于快速卷积方法、Monte-Carlo方法,计算时间和计算工作量明显减少.     

Exact distribution of the convolution of negative binomial random variables

In this note, we derive the exact distribution of S by using the method of generating function and BELL polynomials, where S = X₁ + X₂ + ??? + X_n, and each X_i follows the negative binomial distribution with arbitrary parameters. As a particular case, we also obtain the exact distribution of the convolution of geometric random variables.     

卷积型积分方程的求解方法

利用 Laplace变换法建立了求解卷积型积分方程的有效方法 .通过例题演示了具体的求解过程 ,并得到了两个有用的定理     

Stability of characterization of Weibull distribution

If assumptions of the theorem are satisfied not exactly but only approximately, then may we state that the conclusion of the theorem is also fulfilled approximately? Theorems, in which the problems of this kind are considered, are called stability theorems. The present paper presents some comments on characterization of the Weibull distribution by the lack of memory property and stability estimation in this characterization.     

On the convolution of normal and t random variables

The need for the convolution of normal and Student's t random variables arises in many areas. Since the 1930s, various authors have attempted to derive closed-form expressions for the probability density function (pdf) of the convolution, but with little success. Here, general closed-form expressions are derived for the pdf.     

Multiscale Gaussian convolution algorithm for estimate of Gaussian mixture model

Abstract

This paper introduces a multiscale Gaussian convolution model of Gaussian mixture (MGC-GMM) via the convolution of the GMM and a multiscale Gaussian window function. It is found that the MGC-GMM is still a Gaussian mixture model, and its parameters can be mapped back to the parameters of the GMM. Meanwhile, the multiscale probability density function (MPDF) of the MGC-GMM can be viewed as the mathematical expectation of a random process induced by the Gaussian window function and the GMM, which can be directly estimated by the use of sample data. Based on the estimated MPDF, a novel algorithm denoted by the MGC is proposed for the selection of model and the parameter estimates of the GMM, where the component number and the means of the GMM are respectively determined by the number and the locations of the maximum points of the MPDF, and the numerical algorithms for the weight and variance parameters of the GMM are derived. The MGC is suitable for the GMM with diagonal covariance matrices. A MGC-EM algorithm is also presented for the generalized GMM, where the GMM is estimated using the EM algorithm by taking the estimates from the MGC as initial parameters of the GMM model. The proposed algorithms are tested via a series of simulated sample sets from the given GMM models, and the results show that the proposed algorithms can effectively estimate the GMM model.