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Simulated Minimum Hellinger distance Estimation for Some Continuous Financial and Actuarial Models

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Open Journal of Statistics 2017年第4期7卷 743-759页

作者： Andrew Luong Claire Bilodeau école d’actuariat Université Laval Québec Canada

Minimum Hellinger distance (MHd) estimation is extended to a simulated version with the model density function replaced by a density estimate based on a random sample drawn from the model distribution. The method does not require a closed-form expression for the density function and appears to be suitable for models lacking a closed-form expression for the density, models for which likelihood methods might be difficult to implement. Even though only consistency is shown in this paper and the asymptotic distribution remains an open question, our simulation study suggests that the methods have the potential to generate simulated minimum Hellinger distance (SMHd) estimators with high efficiencies. The method can be used as an alternative to methods based on moments, methods based on empirical characteristic functions, or the use of an expectation-maximization (EM) algorithm.

关键词： Infinitely divisible distribution Mixture distribution Hellinger distance Robustness

Simulated Minimum Cramér-Von Mises distance Estimation for Some Actuarial and Financial Models

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Open Journal of Statistics 2017年第5期7卷 815-833页

作者： Andrew Luong Christopher Blier-Wong école d’actuariat Université Laval Québec Canada

Minimum Cramér-Von Mises distance estimation is extended to a simulated version. The simulated version consists of replacing the model distribution function with a sample distribution constructed using a simulated sample drawn from it. The method does not require an explicit form of the model density functions and can be applied to fitting many useful infinitely divisible distributions or mixture distributions without closed form density functions often encountered in actuarial science and finance. For these models likelihood estimation is difficult to implement and simulated Minimum Cramér-Von Mises (SMCVM) distance estimation can be used. Asymptotic properties of the SCVM estimators are established. The new method appears to be more robust and efficient than methods of moments (MM) for the models being considered which have more than two parameters. The method can be used as an alternative to simulated Hellinger distance (SMHd) estimation with a special feature: it can handle models with a discontinuity point at the origin with probability mass assigned to it such as in the case of the compound Poisson distribution where SMHd method might not be suitable. As the method is based on sample distributions instead of density estimates it is also easier to implement than SMHd method but it might not be as efficient as SMHd methods for continuous models.

关键词： Compound Poisson distribution double Exponential Jump diffusion distribution Mixture distribution Robustness Influence Functions

Unified Asymptotic Results for Maximum Spacing and Generalized Spacing Methods for Continuous Models

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Open Journal of Statistics 2018年第3期8卷 614-639页

作者： Andrew Luong école d’actuariat Université Laval Québec Canada

Asymptotic results are obtained using an approach based on limit theorem results obtained for α-mixing sequences for the class of general spacings (GSP) methods which include the maximum spacings (MSP) method. The MSP method has been shown to be very useful for estimating parameters for univariate continuous models with a shift at the origin which are often encountered in loss models of actuarial science and extreme models. The MSP estimators have also been shown to be as efficient as maximum likelihood estimators in general and can be used as an alternative method when ML method might have numerical difficulties for some parametric models. Asymptotic properties are presented in a unified way. Robustness results for estimation and parameter testing results which facilitate the applications of the GSP methods are also included and related to quasi-likelihood results.

关键词： Maximum Product of Spacings M-Estimators Quasi-Likelihood Ratio Test Statistic α-Mixing Sequences

Asymptotic Normality distribution of Simulated Minimum Hellinger distance Estimators for Continuous Models

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Open Journal of Statistics 2018年第5期8卷 846-860页

作者： Andrew Luong Claire Bilodeau école d’actuariat Université Laval Québec Canada

Certain distributions do not have a closed-form density, but it is simple to draw samples from them. For such distributions, simulated minimum Hellinger distance (SMHd) estimation appears to be useful. Since the method is distance-based, it happens to be naturally robust. This paper is a follow-up to a previous paper where the SMHd estimators were only shown to be consistent;this paper establishes their asymptotic normality. For any parametric family of distributions for which all positive integer moments exist, asymptotic properties for the SMHd method indicate that the variance of the SMHd estimators attains the lower bound for simulation-based estimators, which is based on the inverse of the Fisher information matrix, adjusted by a constant that reflects the loss of efficiency due to simulations. All these features suggest that the SMHd method is applicable in many fields such as finance or actuarial science where we often encounter distributions without closed-form density.

关键词： Continuous distribution Kernel density Estimate Continuity in Probability differentiability in Probability Hellinger distance

Maximum Entropy Empirical Likelihood Methods Based on Bivariate Laplace Transforms and Moment Generating Functions

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Open Journal of Statistics 2018年第2期8卷 264-283页

作者： Andrew Luong école d’actuariat Université Laval Québec Canada

Maximum Entropy Empirical Likelihood (MEEL) methods are extended to bivariate distributions with closed form expressions for their bivariate Laplace transforms (BLT) or moment generating functions (BMGF) without closed form expressions for their bivariate density functions which make the implementation of the likelihood methods difficult. These distributions are often encountered in joint modeling in actuarial science and finance. Moment conditions to implement MEEL methods are given and a bivariate Laplace transform power mixture (BLTPM) is also introduced, the new operator generalizes the existing univariate one in the literature. Many new bivariate distributions including infinitely divisible(Id) distributions with closed form expressions for their BLT can be created using this operator and MEEL methods can also be applied to these bivariate distributions.

关键词： Bivariate Power Mixture Operator Laplace Transform Copulas Chi-Square Test Statistics Bivariate Normality Infinite divisibility Empirical Likelihood

Simulated Minimum Hellinger distance Inference Methods for Count data

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Open Journal of Statistics 2018年第1期8卷 187-219页

作者： Andrew Luong Claire Bilodeau Christopher Blier-Wong école d’actuariat Université Laval Québec Canada

In this paper, we consider simulated minimum Hellinger distance (SMHd) inferences for count data. We consider grouped and ungrouped data and emphasize SMHd methods. The approaches extend the methods based on the deterministic version of Hellinger distance for count data. The methods are general, it only requires that random samples from the discrete parametric family can be drawn and can be used as alternative methods to estimation using probability generating function (pgf) or methods based matching moments. Whereas this paper focuses on count data, goodness of fit tests based on simulated Hellinger distance can also be applied for testing goodness of fit for continuous distributions when continuous observations are grouped into intervals like in the case of the traditional Pearson’s statistics. Asymptotic properties of the SMHd methods are studied and the methods appear to preserve the properties of having good efficiency and robustness of the deterministic version.

关键词： Break down Points Robustness Power Mixture Esscher Transform Mixture discrete distributions Chi-Square Tests Statistics

Simulated Minimum Quadratic distance Methods Using Grouped data for Some Bivariate Continuous Models

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Open Journal of Statistics 2018年第2期8卷 362-389页

作者： Andrew Luong école d’actuariat Université Laval Québec Canada

Quadratic distance methods based on a special distance which make use of survival functions are developed for inferences for bivariate continuous models using selected points on the nonegative quadrant. A related version which can be viewed as a simulated version is also developed and appears to be suitable for bivariate distributions with no closed form expressions and numerically not tractable but it is easy to simulate from these distributions. The notion of an adaptive basis is introduced and the estimators can be viewed as quasilikelihood estimators using the projected score functions on an adaptive basis and they are closely related to minimum chi-square estimators with random cells which can also be viewed as quasilikeliood estimators using a projected score functions on a special adaptive basis but the elements of such a basis were linearly dependent. A rule for selecting points on the nonnegative quadrant which make use of quasi Monte Carlo (QMC) numbers and two sample quantiles of the two marginal distributions is proposed if complete data is available and like minimum chi-square methods;the quadratic distance methods also offer chi-square statistics which appear to be useful in practice for model testing.

关键词： Projected Score Functions Adaptive Basis Complete Basis Chi-Square Tests Statistics Random Cells Contingency Table

Minimum Quadratic distance Methods Using Grouped data for Parametric Families of Copulas

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Open Journal of Statistics 2018年第3期8卷 427-456页

作者： Andrew Luong école d’actuariat Université Laval Sainte-Foy Canada

Minimum quadratic distance (MQd) methods are used to construct chi-square test statistics for simple and composite hypothesis for parametric families of copulas. The methods aim at grouped data which form a contingency table but by defining a rule to group the data using Quasi-Monte Carlo numbers and two marginal empirical quantiles, the methods can be extended to handle complete data. The rule implicitly defines points on the nonnegative quadrant to form quadratic distances and the similarities of the rule with the use of random cells for classical minimum chi-square methods are indicated. The methods are relatively simple to implement and might be useful for applied works in various fields such as actuarial science.

关键词： Influence Functions Quasi-Monte Carlo Numbers Chi-Square Tests Statistics Random Cells Contingency Tables

Generalized Method of Moments and Generalized Estimating Functions Using Characteristic Function

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Open Journal of Statistics 2020年第3期10卷 581-599页

作者： Andrew Luong école d’actuariat Université Laval Ste Foy Québec Canada

GMM inference procedures based on the square of the modulus of the model characteristic function are developed using sample moments selected using estimating function theory and bypassing the use of empirical characteristic function of other GMM procedures in the literature. The procedures are relatively simple to implement and are less simulation-oriented than simulated methods of inferences yet have the potential of good efficiencies for models with densities without closed form. The procedures also yield better estimators than method of moment estimators for models with more than three parameters as higher order sample moments tend to be unstable.

关键词： Generalized Normal Laplace distribution Generalized Asymmetric Laplace distribution Optimum Estimating Functions Infinitely divisible distribution Simulated Estimation Method