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Distribution of big claims in a Lévy insurance risk process: Analytics of a new non-parametric estimator

COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2025年第3期54卷 882-907页

作者： Mozumder, Sharif Hassan, M. Kabir Sorwar, Ghulam Amuedo, Jose Antonio Perez Univ Dhaka Dept Math Dhaka Bangladesh Univ New Orleans Finance & Econ New Orleans LA 70148 USA Keele Univ Keele Business Sch Keele England

In this study, we model aggregate claims using a subordinator, specifically a non-decreasing Levy process. Large positive jumps, exceeding a predetermined threshold, represent significant claims, while frequent but smaller fluctuations capture other sources of non-insurance uncertainty, such as miscellaneous expenses. The primary challenge lies in extracting the necessary mathematical insights to estimate the jump measure from a sample path of truncated aggregate claims. Through a discrete time-point sampling scheme, we conduct an initial comparison between conventional parametric estimators of the Levy measure associated with the subordinator, based on simulated significant claims, and our proposed non-parametric estimator, derived by adapting classical differential processes originally introduced by Rubin and Tucker. The results of this comparison suggest the potential utility of our estimator in the context of real data from the insurance sector. While the primary focus of this work is to uncover the mathematical foundations, a preliminary simulation study, although lacking rigorous numerical analysis, hints at the favorable estimation of the Poisson rate for the number of jumps exceeding the threshold, achieved using our proposed non-parametric estimator of the Levy measure.

关键词： Distribution of big claims Levy insurance risk process non-parametric estimator C65

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First hitting probabilities for semi-Markov chains and estimation

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2017年第5期46卷 2435-2446页

作者： Georgiadis, Stylianos Tech Univ Denmark Dept Appl Math & Comp Sci DK-2800 Lyngby Denmark

We first consider a stochastic system described by an absorbing semi-Markov chain (SMC) with finite state space, and we introduce the absorption probability to a class of recurrent states. Afterwards, we study the first hitting probability to a subset of states for an irreducible SMC. In the latter case, a non-parametric estimator for the first hitting probability is proposed and the asymptotic properties of strong consistency and asymptotic normality are proven. Finally, a numerical application on a five-state system is presented to illustrate the performance of this estimator.

关键词： Absorption probability asymptotic properties first hitting probability non-parametric estimator semi-Markov chains

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non-parametric estimation of the first-order Sobol indices with bootstrap bandwidth

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COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2021年第9期50卷 2497-2512页

作者： Solis, Maikol Univ Costa Rica Escuela Matemat Ctr Invest Matemat Pura & Aplicada San Pedro Costa Rica

Suppose that , where are random inputs, Y is the output, and is an unknown link function. The Sobol indices gauge the sensitivity of each X against Y by estimating the regression curve's variability between them. In this paper, we estimate these curves with a kernel-based method. The method allows to estimate the first order indices when the link between the independent and dependent variables is unknown. The kernel-based methods need a bandwidth to average the observations. For finite samples, the cross-validation method is famous to decide this bandwidth. However, it produces a structural bias. To remedy this, we propose a bootstrap procedure which reconstruct the model residuals and re-estimate the non-parametric regression curve. With the new set of curves, the procedure corrects the bias in the Sobol index. To test the developed method, we implemented simulated numerical examples with complex functions.

关键词： Sobol indices Sensitivity Analysis non-parametric estimator Finite-sample bias Bootstrap bandwidth

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Estimation of convolution in the model with noise

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JOURNAL OF nonparametric STATISTICS 2015年第3期27卷 286-315页

作者： Chesneau, C. Comte, F. Mabon, G. Navarro, F. Univ Caen Basse Normandie Dept Math LMNO UFR Sci Caen France Univ Paris 05 UMR CNRS 8145 MAP5 Sorbonne Paris Cite Paris France CREST Malakoff France Concordia Univ Dept Math & Stat Montreal PQ H4B 1R6 Canada

We investigate the estimation of the l-fold convolution of the density of an unobserved variable X from n i.i.d. observations of the convolution model Y = X + epsilon. We first assume that the density of the noise e is known and define non-adaptive estimators, for which we provide bounds for the mean integrated squared error. In particular, under some smoothness assumptions on the densities of X and e, we prove that the parametric rate of convergence 1/n can be attained. Then, we construct an adaptive estimator using a penalisation approach having similar performances to the non-adaptive one. The price for its adaptivity is a logarithmic term. The results are extended to the case of unknown noise density, under the condition that an independent noise sample is available. Lastly, we report a simulation study to support our theoretical findings.

关键词： adaptive estimation convolution of densities measurement errors oracle inequality non-parametric estimator

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The rate of occurrence of failures for semi-Markov processes and estimation

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STATISTICS & PROBABILITY LETTERS 2002年第3期59卷 245-255页

作者： Ouhbi, B Limnios, N Ecole Natl Super Arts & Metiers Meknes Morocco Univ Technol Compiegne Equipe Math Appl F-60205 Compiegne France

This paper investigates the rate of occurrence of failures (ROCOF) of finite state semi-Markov systems. Firstly, a formula for evaluating the ROCOF for semi-Markov systems is derived. As a consequence of this result, we derive the ROCOF of Markov systems as well as the ROCOF of the alternating renewal processes and we give their asymptotic values. From the semi-Markov ROCOF formula, we propose a statistical estimator of this function. Finally, the uniform strong consistency and the asymptotic normality of this estimator are obtained. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： rate of occurrence of failures semi-Markov process non-parametric estimator asymptotic properties

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DISTRIBUTION-FREE CONFIDENCE-INTERVALS FOR QUANTILES IN SMALL SAMPLES

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 1985年第4期14卷 979-990页

作者： STEINBERG, SM DAVIS, CE UNIV N CAROLINA DEPT BIOSTATCHAPEL HILLNC 27514

estimators for quantiles based on linear combinations of order statistics have been proposed by Harrell and Davis(1982) and kaigh and Lachenbruch (1982). Both estimators have been demonstrated to be at least as efficient for small sample point estimation as an ordinary sample quantile estimator based on one or two order statistics: Distribution-free confidence intervals for quantiles can be constructed using either of the two approaches. By means of a simulation study, these confidence intervals have been compared with several other methods of constructing confidence intervals for quantiles in small samples. For the median, the Kaigh and Lachenbruch method performed fairly well. For other quantiles, no method performed better than the method which uses pairs of order statistics.

关键词： median quantile non-parametric estimator order statistic

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Optimal Composite Markers for Time-Dependent Receiver Operating Characteristic Curves with Censored Survival Data

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SCANDINAVIAN JOURNAL OF STATISTICS 2010年第4期37卷 664-679页

作者： Hung, Hung Chiang, Chin-Tsang Natl Taiwan Univ Dept Math Taipei 10617 Taiwan

To increase the predictive abilities of several plasma biomarkers on the coronary artery disease (CAD)-related vital statuses over time, our research interest mainly focuses on seeking combinations of these biomarkers with the highest time-dependent receiver operating characteristic curves. An extended generalized linear model (EGLM) with time-varying coefficients and an unknown bivariate link function is used to characterize the conditional distribution of time to CAD-related death. Based on censored survival data, two non-parametric procedures are proposed to estimate the optimal composite markers, linear predictors in the EGLM model. Estimation methods for the classification accuracies of the optimal composite markers are also proposed. In the article we establish theoretical results of the estimators and examine the corresponding finite-sample properties through a series of simulations with different sample sizes, censoring rates and censoring mechanisms. Our optimization procedures and estimators are further shown to be useful through an application to a prospective cohort study of patients undergoing angiography.

关键词： area under the ROC curve (AUC) censoring extended generalized linear model Gaussian process non-parametric estimator optimal composite marker receiver operating characteristic (ROC) curve survival time-varying coefficient vital status

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ASYMPTOTIC NORMALITY OF GENERALIZED FUNCTIONAL estimatorS DEPENDENT ON COVARIABLES

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 1990年第3期24卷 377-390页

作者： MANTEIGA, WG Departamento de Estadística e Investigación Operativa Facultad de Matemáticas Universidad de Santiago de Compostela Spain

Given a random vector ( X,Y ) with distribution function H ( x,y ) such that the marginal random variable X has support S , different statistical parameters θ( x ) associated with Y conditioned by X = x can be defined by means of functionals T:R×Θ× F →R (Θ a parameter space, F a space of distribution functions) as those for which ʃ T(y,θ(x),F(·|x))dF(y|x)=0 , where F (·| x ) represents the conditional distribution of the random variable Y given X = x . This paper shows the asymptotic normality of the general class of estimators θ n ( x ) ( x∈S ) defined as the solutions of ʃT(y, θ ̂ n (x), F ̂ n (·|x))d F ̂ n (y|x)=0 where F̂ n ( y | x ) is a non-parametric estimator of F ( y | x ). This result is applied to several particular cases.

关键词： 62G05 Functional estimator non-parametric estimator covariables

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non-parametric estimation for time-dependent AUC

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2010年第5期140卷 1162-1174页

作者： Chiang, Chin-Tsang Hung, Hung Natl Taiwan Univ Dept Math Taipei 10617 Taiwan

The area under the receiver operating characteristic (ROC) curve (AUC) is one of the commonly used measure to evaluate or compare the predictive ability of markers to the disease status. Motivated by an angiographic coronary artery disease (CAD) study, our objective is mainly to evaluate and compare the performance of several baseline plasma levels in the prediction of CAD-related vital status over time. Based on censored survival data, the non-parametric estimators are proposed for the time-dependent AUC. The limiting Gaussian processes of the estimators and the estimated asymptotic variance-covariance functions enable us to further construct confidence bands and develop testing procedures. Applications and finite sample properties of the proposed estimation methods and inference procedures are demonstrated through the CAD-related death data from the British Columbia Vital Statistics Agency and Monte Carlo simulations. (C) 2009 Elsevier B.V. All rights reserved.

关键词： AUC Bivariate estimation Bootstrap Gaussian process Kaplan-Meier estimator non-parametric estimator ROC Smoothing parameter Survival data

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Sampling strategies for the assessment of tree species diversity

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JOURNAL OF VEGETATION SCIENCE 1998年第2期9卷 161-172页

作者： Gimaret-Carpentier, C Pelissier, R Pascal, JP Houllier, F Univ Lyon 1 Lab Biometrie Genet & Biol Populat CNRS UMR 5558 F-69622 Villeurbanne France Inst Francais Pondichery Pondicherry 605001 India

This paper aims at proposing efficient vegetation sampling strategies. It describes how the estimation of species richness and diversity of moist evergreen forest is affected by (1) sampling design (simple random sampling, random cluster sampling, systematic cluster sampling, stratified cluster sampling);(2) choice of species richness estimators (number of observed species vs, non-parametric estimators) and (3) choice of diversity index (Simpson vs. Shannon). Two sites are studied: a 28-ha area situated in the Western Ghats of India and a 25-ha area located at Pasoh in Peninsular Malaysia. The results show that: (1) whatever the sampling strategy, estimates of species richness depend on sample size in these very diverse forest ecosystems which contain many rare species;(2) Simpson's diversity index reaches a stable value at low sample sizes while Shannon's index is affected more by the addition of rare species with increasing sample size;(3) cluster sampling strategies provide a good compromise between cost and statistical efficiency;(4) 300-400 sample trees grouped in small clusters (10-50 individuals) are enough to obtain unbiased and precise estimates of Simpson's index;(5) the local topography of the Western Ghats has a major influence on forest composition, the steep slopes being richer and more diverse than the ridges and gentle slopes;(6) stratified duster sampling is thus an interesting alternative to systematic cluster sampling.

关键词： alpha-diversity beta-diversity non-parametric estimator rare species Simpson's index topography tropical forest

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