scholarly journals Nonparametric Estimation of Extreme Quantiles with an Application to Longevity Risk

Risks ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 77
Author(s):  
Catalina Bolancé ◽  
Montserrat Guillen
Keyword(s):  
Nonparametric Estimation ◽  
Simulation Study ◽  
Kernel Estimation ◽  
Heavy Tail ◽  
Accurate Estimation ◽  
Longevity Risk ◽  
Age At Death ◽  
Exponential Tail ◽  
Extreme Quantiles ◽  
Tail Shape

A new method to estimate longevity risk based on the kernel estimation of the extreme quantiles of truncated age-at-death distributions is proposed. Its theoretical properties are presented and a simulation study is reported. The flexible yet accurate estimation of extreme quantiles of age-at-death conditional on having survived a certain age is fundamental for evaluating the risk of lifetime insurance. Our proposal combines a parametric distributions with nonparametric sample information, leading to obtain an asymptotic unbiased estimator of extreme quantiles for alternative distributions with different right tail shape, i.e., heavy tail or exponential tail. A method for estimating the longevity risk of a continuous temporary annuity is also shown. We illustrate our proposal with an application to the official age-at-death statistics of the population in Spain.

Kernel Based Estimation for a Non-Homogeneous Poisson Processes

2013 ◽  
Vol 805-806 ◽  
pp. 1948-1951
Author(s):  
Tian Jin
Keyword(s):  
Air Pollution ◽  
Nonparametric Estimation ◽  
Simulation Study ◽  
Poisson Model ◽  
Poisson Processes ◽  
Finite Sample ◽  
Process Data ◽  
Homogeneous Poisson Process ◽  
Finite Sample Properties ◽  
Non Homogeneous Poisson Process

The non-homogeneous Poisson model has been applied to various situations, including air pollution data. In this paper, we propose a kernel based nonparametric estimation for fitting the non-homogeneous Poisson process data. We show that our proposed estimator is-consistent and asymptotically normally distributed. We also study the finite-sample properties with a simulation study.

scholarly journals The Reciprocal Generalized Inverse Gaussian Frailty with Application in Life Annuity Business

2020 ◽  
pp. 112-131
Author(s):  
Walter Onchere ◽  
Richard Tinega ◽  
Patrick Weke ◽  
Jam Otieno
Keyword(s):  
Simulation Study ◽  
Unobserved Heterogeneity ◽  
Survival Curve ◽  
Gaussian Model ◽  
Frailty Models ◽  
Residual Lifetime ◽  
Longevity Risk ◽  
Inverse Gaussian ◽  
Life Annuity ◽  
Valuation Of Life

Aims: As shown in literature, several authors have adopted various individual frailty mixing distributions as a way of dealing with possible heterogeneity due to unobserved covariates in a group of insurers. This research contribution is to generalize the frailty mixing distribution to nest other classes of frailty distributions not in literature and apply the proposed distributions in valuation of life annuity business. Methodology: A simulation study is done to assess the performance of the aforementioned models. The baseline parameters is estimated using Bayesian Inference and a better model is suggested for valuation of life annuity business. Results: As a result of generalizing the frailty some new classes of frailty distributions are constructed such as; the Reciprocal Inverse Gaussian Frailty, the Inverse Gamma Frailty, the Harmonic Frailty and the Positive Hyperbolic Frailty. From the simulation study, the proposed new frailty models shows that ignoring frailty leads to an underestimation of future residual lifetime since the survival curve shifts to the right when heterogeneity is accounted for. This is consistent with frailty literature. The Reciprocal Inverse Gaussian model closely represents the Association of Kenya Insurers graduated rates with a slight increase in survival due to longevity risk. Conclusion: The proposed new frailty models show an increase in the insurers expected liability when unobserved heterogeneity is accounted for. This is consistent with frailty literature and thus can be applied to avoid underestimating the insurer’s liability in the context of life annuity business. The RIG model as proposed in estimating future liability by directly adjusting the AKI mortality rates shows an increase in longevity risk. The extent of heterogeneity of the insured group determines the level of risk. The RIG frailties should be considered for multivariate cases where the insureds are clustered in groups.

scholarly journals Optimal heavy tail estimation – Part 1: Order selection

2017 ◽  
Vol 24 (4) ◽  
pp. 737-744 ◽  
Author(s):  
Manfred Mudelsee ◽  
Miguel A. Bermejo
Keyword(s):  
Time Series ◽  
Heavy Tails ◽  
Characteristic Exponent ◽  
Tail Probability ◽  
Heavy Tail ◽  
Infinite Variance ◽  
Accurate Estimation ◽  
Monte Carlo Experiment ◽  
River Elbe ◽  
Tail Estimation

Abstract. The tail probability, P, of the distribution of a variable is important for risk analysis of extremes. Many variables in complex geophysical systems show heavy tails, where P decreases with the value, x, of a variable as a power law with a characteristic exponent, α. Accurate estimation of α on the basis of data is currently hindered by the problem of the selection of the order, that is, the number of largest x values to utilize for the estimation. This paper presents a new, widely applicable, data-adaptive order selector, which is based on computer simulations and brute force search. It is the first in a set of papers on optimal heavy tail estimation. The new selector outperforms competitors in a Monte Carlo experiment, where simulated data are generated from stable distributions and AR(1) serial dependence. We calculate error bars for the estimated α by means of simulations. We illustrate the method on an artificial time series. We apply it to an observed, hydrological time series from the River Elbe and find an estimated characteristic exponent of 1.48 ± 0.13. This result indicates finite mean but infinite variance of the statistical distribution of river runoff.

NONPARAMETRIC ESTIMATION OF VOLATILITY FUNCTIONS: THE LOCAL EXPONENTIAL ESTIMATOR

2002 ◽  
Vol 18 (4) ◽  
pp. 985-991 ◽  
Author(s):  
Flavio A. Ziegelmann
Keyword(s):  
Nonparametric Estimation ◽  
Simulation Study ◽  
Kernel Smoothing ◽  
Asymptotic Properties ◽  
Superior Performance ◽  
Linear Estimator ◽  
Smoothing Techniques ◽  
Conditional Mean ◽  
Fully Adaptive ◽  
Local Linear Estimator

Kernel smoothing techniques free the traditional parametric estimators of volatility from the constraints related to their specific models. In this paper the nonparametric local exponential estimator is applied to estimate conditional volatility functions, ensuring its nonnegativity. Its asymptotic properties are established and compared with those for the local linear estimator. It theoretically enables us to determine when the exponential is expected to be superior to the linear estimator. A very strong and novel result is achieved: the exponential estimator is asymptotically fully adaptive to unknown conditional mean functions. Also, our simulation study shows superior performance of the exponential estimator.

scholarly journals Longevity Risk-Sharing Annuities: Partial Indexation in Mortality Experience

2017 ◽  
Vol 11 (1) ◽  
Author(s):  
Saisai Zhang ◽  
Johnny Siu-Hang Li
Keyword(s):  
Simulation Study ◽  
Risk Sharing ◽  
Idiosyncratic Risk ◽  
The Other ◽  
Longevity Risk ◽  
Partial Mortality ◽  
Mortality Experience ◽  
Other Hand

AbstractIn a conventional fixed annuity, idiosyncratic risk is diversified away while systematic longevity risk is borne entirely by the provider. The mortality-indexed annuity on the other hand, transfers systematic longevity risk completely back to the annuitants by fully adjusting benefits to mortality experience. In this paper, we propose the partial mortality-indexed annuity (PMIA), which aims to seek a balance between the two ends of the risk-sharing spectrum. Through a simulation study, we show that the PMIA achieves risk sharing and benefits both the provider and the annuitant.

scholarly journals Optimal heavy tail estimation, Part I: Order selection

2017 ◽  
Author(s):  
Manfred Mudelsee ◽  
Miguel A. Bermejo
Keyword(s):  
Time Series ◽  
Heavy Tails ◽  
Characteristic Exponent ◽  
Tail Probability ◽  
Heavy Tail ◽  
Infinite Variance ◽  
Accurate Estimation ◽  
Monte Carlo Experiment ◽  
River Elbe ◽  
Tail Estimation

Abstract. The tail probability, P, of the distribution of a variable is important for risk analysis of extremes. Many variables in complex geophysical systems show heavy tails, where P decreases with the value, x, of a variable as a power law with characteristic exponent, α. Accurate estimation of α on the basis of data is currently hindered by the problem of the selection of the order, that is, the number of largest x-values to utilize for the estimation. This paper presents a new, widely applicable, data-adaptive order selector, which is based on computer simulations and brute force search. It is the first in a set of papers on optimal heavy tail estimation. The new selector outperforms competitors in a Monte Carlo experiment, where simulated data are generated from stable distributions and AR(1) serial dependence. We calculate error bars for the estimated α by means of simulations. We illustrate the method on an artificial time series. We apply it to an observed, hydrological time series from the river Elbe and find an estimated characteristic exponent of 1.48 ± 0.13. This result indicates finite mean but infinite variance of the statistical distribution of river runoff.

scholarly journals POSTERIOR CONSISTENCY IN CONDITIONAL DENSITY ESTIMATION BY COVARIATE DEPENDENT MIXTURES

2013 ◽  
Vol 30 (3) ◽  
pp. 606-646 ◽  
Author(s):  
Andriy Norets ◽  
Justinas Pelenis
Keyword(s):  
Large Class ◽  
Mixture Models ◽  
Density Estimation ◽  
Nonparametric Estimation ◽  
Simulation Study ◽  
Conditional Density ◽  
Small Samples ◽  
Posterior Consistency ◽  
Bayesian Nonparametric Estimation ◽  
Strongly Consistent

This paper considers Bayesian nonparametric estimation of conditional densities by countable mixtures of location-scale densities with covariate dependent mixing probabilities. The mixing probabilities are modeled in two ways. First, we consider finite covariate dependent mixture models, in which the mixing probabilities are proportional to a product of a constant and a kernel and a prior on the number of mixture components is specified. Second, we consider kernel stick-breaking processes for modeling the mixing probabilities. We show that the posterior in these two models is weakly and strongly consistent for a large class of data-generating processes. A simulation study conducted in the paper demonstrates that the models can perform well in small samples.

scholarly journals Functional nonparametric estimation of conditional extreme quantiles

2010 ◽  
Vol 101 (2) ◽  
pp. 419-433 ◽  
Author(s):  
Laurent Gardes ◽  
Stéphane Girard ◽  
Alexandre Lekina
Keyword(s):  
Nonparametric Estimation ◽  
Extreme Quantiles
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