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Open Journal of Statistics 2020年第3期10卷 516-539页

作者： andrew luong école d’actuariat Université Laval Ste Foy Québec Canada

Generalized method of moments based on probability generating function is considered. Estimation and model testing are unified using this approach which also leads to distribution free chi-square tests. The estimation methods developed are also related to estimation methods based on generalized estimating equations but with the advantage of having statistics for model testing. The methods proposed overcome numerical problems often encountered when the probability mass functions have no closed forms which prevent the use of maximum likelihood (ML) procedures and in general, ML procedures do not lead to distribution free model testing statistics.

关键词： Mixture Distributions Consistent Chi-Square Tests Infinitely Divisible Distributions Mixture Distributions Distribution Free Test Statistics Model Testing

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

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

Maximum Entropy Empirical Likelihood Methods Based on Laplace Transforms for Nonnegative Continuous Distribution with Actuarial Applications

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Open Journal of Statistics 2017年第3期7卷 459-482页

作者： andrew luong É cole d’actuariat Université Laval Ste Foy Québec Canada

Maximum entropy likelihood (MEEL) methods also known as exponential tilted empirical likelihood methods using constraints from model Laplace transforms (LT) are introduced in this paper. An estimate of overall loss of efficiency based on Fourier cosine series expansion of the density function is proposed to quantify the loss of efficiency when using MEEL methods. Penalty function methods are suggested for numerical implementation of the MEEL methods. The methods can easily be adapted to estimate continuous distribution with support on the real line encountered in finance by using constraints based on the model generating function instead of LT.

关键词： Quasi-Likelihood Projection Power Mixture Operator Quadratic Distance Methods Insurance Premium Stop-Loss Premium

Pseudodistance Methods Using Simultaneously Sample Observations and Nearest Neighbour Distance Observations for Continuous Multivariate Models

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Open Journal of Statistics 2019年第4期9卷 445-457页

作者： andrew luong école d’actuariat Université Laval Ste Foy Québec Canada

Using the fact that a multivariate random sample of n observations also generates n nearest neighbour distance (NND) univariate observations and from these NND observations, a set of n auxiliary observations can be obtained and with these auxiliary observations when combined with the original multivariate observations of the random sample, a class of pseudodistance?Dh?is allowed to be used and inference methods can be developed using this class of pseudodistances. The Dh?estimators obtained from this class can achieve high efficiencies and have robustness properties. Model testing also can be handled in a unified way by means of goodness-of-fit tests statistics derived from this class which have an asymptotic normal distribution. These properties make the developed inference methods relatively simple to implement and appear to be suitable for analyzing multivariate data which are often encountered in applications.

关键词： Goodness-of-Fit Statistics Robust Estimators Multivariate Density Estimate Information Matrix Model Testing

Likelihood and Quadratic Distance Methods for the Generalized Asymmetric Laplace Distribution for Financial Data

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Open Journal of Statistics 2017年第2期7卷 347-368页

作者： andrew luong É cole d’actuariat Université Laval Canada

Maximum likelihood (ML) estimation for the generalized asymmetric Laplace (GAL) distribution also known as Variance gamma using simplex direct search algorithms is investigated. In this paper, we use numerical direct search techniques for maximizing the log-likelihood to obtain ML estimators instead of using the traditional EM algorithm. The density function of the GAL is only continuous but not differentiable with respect to the parameters and the appearance of the Bessel function in the density make it difficult to obtain the asymptotic covariance matrix for the entire GAL family. Using M-estimation theory, the properties of the ML estimators are investigated in this paper. The ML estimators are shown to be consistent for the GAL family and their asymptotic normality can only be guaranteed for the asymmetric Laplace (AL) family. The asymptotic covariance matrix is obtained for the AL family and it completes the results obtained previously in the literature. For the general GAL model, alternative methods of inferences based on quadratic distances (QD) are proposed. The QD methods appear to be overall more efficient than likelihood methods infinite samples using sample sizes n ≤5000 and the range of parameters often encountered for financial data. The proposed methods only require that the moment generating function of the parametric model exists and has a closed form expression and can be used for other models.

关键词： M-Estimators Cumulant Generating Function Chi-Square Tests Generalized Hyperbolic Distribution Simplex Pattern Search Variance Gamma Minimum Distance Value at Risk Entropic Value at Risk European Call Option

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

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 Results for Goodness-of-Fit Tests Using a Class of Generalized Spacing Methods with Estimated Parameters

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Open Journal of Statistics 2018年第4期8卷 731-746页

作者： andrew luong école d’actuariat Université Laval Ste Foy Québec Canada

A class of pseudo distances is used to derive test statistics using transformed data or spacings for testing goodness-of-fit for parametric models. These statistics can be considered as density based statistics and expressible as simple functions of spacings. It is known that when the null hypothesis is simple, the statistics follow asymptotic normal distributions without unknown parameters. In this paper we emphasize results for the null composite hypothesis: the parameters can be estimated by a generalized spacing method (GSP) first which is equivalent to minimize a pseudo distance from the class which is considered;subsequently the estimated parameters are used to replace the parameters in the pseudo distance used for estimation;goodness-of-fit statistics for the composite hypothesis can be constructed and shown to have again an asymptotic normal distribution without unknown parameters. Since these statistics are related to a discrepancy measure, these tests can be shown to be consistent in general. Furthermore, due to the simplicity of these statistics and they come a no extra cost after fitting the model, they can be considered as alternative statistics to chi-square statistics which require a choice of intervals and statistics based on empirical distribution (EDF) using the original data with a complicated null distribution which might depend on the parametric family being considered and also might depend on the vector of true parameters but EDF tests might be more powerful against some specific models which are specified by the alternative hypothesis.

关键词： Density Based Tests EDF Tests Anderson-Darling Statistic Hellinger Distance Statistic Pseudo-Distance Maximum Spacing Method

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