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Simulated Minimum Hellinger Distance Inference Methods for Count Data 被引量:4
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作者 Andrew Luong Claire Bilodeau Christopher Blier-Wong 《Open Journal of Statistics》 2018年第1期187-219,共33页
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 deter... 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. 展开更多
关键词 BREAK Down POINTS Robustness Power MIXTURE Esscher Transform MIXTURE Discrete DISTRIBUTIONS CHI-SQUARE Tests STATISTICS
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Asymptotic Normality Distribution of Simulated Minimum Hellinger Distance Estimators for Continuous Models 被引量:1
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作者 Andrew Luong Claire Bilodeau 《Open Journal of Statistics》 2018年第5期846-860,共15页
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 metho... 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. 展开更多
关键词 Continuous DISTRIBUTION KERNEL Density ESTIMATE CONTINUITY in PROBABILITY DIFFERENTIABILITY in PROBABILITY Hellinger Distance
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Unified Asymptotic Results for Maximum Spacing and Generalized Spacing Methods for Continuous Models 被引量:1
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作者 Andrew Luong 《Open Journal of Statistics》 2018年第3期614-639,共26页
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 MS... 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. 展开更多
关键词 MAXIMUM Product of SPACINGS M-ESTIMATORS QUASI-LIKELIHOOD Ratio Test Statistic Α-MIXING Sequences
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Simulated Minimum Hellinger Distance Estimation for Some Continuous Financial and Actuarial Models 被引量:1
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作者 Andrew Luong Claire Bilodeau 《Open Journal of Statistics》 2017年第4期743-759,共17页
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... 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. 展开更多
关键词 INFINITELY Divisible DISTRIBUTION MIXTURE DISTRIBUTION Hellinger Distance ROBUSTNESS
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Maximum Entropy Empirical Likelihood Methods Based on Bivariate Laplace Transforms and Moment Generating Functions
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作者 Andrew Luong 《Open Journal of Statistics》 2018年第2期264-283,共20页
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 close... 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. 展开更多
关键词 BIVARIATE Power Mixture Operator Laplace Transform COPULAS CHI-SQUARE Test Statistics BIVARIATE NORMALITY Infinite DIVISIBILITY Empirical Likelihood
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Simulated Minimum Quadratic Distance Methods Using Grouped Data for Some Bivariate Continuous Models
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作者 Andrew Luong 《Open Journal of Statistics》 2018年第2期362-389,共28页
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 vers... 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. 展开更多
关键词 Projected SCORE Functions Adaptive BASIS Complete BASIS CHI-SQUARE Tests Statistics Random Cells CONTINGENCY Table
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Minimum Quadratic Distance Methods Using Grouped Data for Parametric Families of Copulas
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作者 Andrew Luong 《Open Journal of Statistics》 2018年第3期427-456,共30页
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 contingenc... 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. 展开更多
关键词 Influence Functions Quasi-Monte Carlo Numbers CHI-SQUARE Tests Statistics Random Cells CONTINGENCY TABLES
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Simulated Minimum Cramér-Von Mises Distance Estimation for Some Actuarial and Financial Models
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作者 Andrew Luong Christopher Blier-Wong 《Open Journal of Statistics》 2017年第5期815-833,共19页
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 simulate... 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. 展开更多
关键词 Compound POISSON DISTRIBUTION Double EXPONENTIAL JUMP Diffusion DISTRIBUTION MIXTURE DISTRIBUTION Robustness Influence Functions
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Robust Continuous Quadratic Distance Estimation Using Quantiles for Fitting Continuous Distributions
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作者 Andrew Luong 《Open Journal of Statistics》 2019年第4期421-435,共15页
Quadratic distance estimation making use of the sample quantile function over a continuous range is introduced. It extends previous methods which are based only on a few sample quantiles and it parallels the continuou... Quadratic distance estimation making use of the sample quantile function over a continuous range is introduced. It extends previous methods which are based only on a few sample quantiles and it parallels the continuous GMM method. Asymptotic properties are established for the continuous quadratic distance estimators (CQDE) and the implementation of the methods are discussed. The methods appear to be useful for balancing robustness and efficiency and useful for fitting distribution with model quantile function being simpler than its density function or distribution function. 展开更多
关键词 COVARIANCE Kernel Influence Function HILBERT Space Linear Operator GMM Estimation Spectral Decomposition
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Asymptotic Results for Goodness-of-Fit Tests Using a Class of Generalized Spacing Methods with Estimated Parameters
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作者 Andrew Luong 《Open Journal of Statistics》 2018年第4期731-746,共16页
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 ex... 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. 展开更多
关键词 Density Based TESTS EDF TESTS Anderson-Darling Statistic Hellinger Distance Statistic PSEUDO-DISTANCE Maximum SPACING Method
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Pseudodistance Methods Using Simultaneously Sample Observations and Nearest Neighbour Distance Observations for Continuous Multivariate Models
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作者 Andrew Luong 《Open Journal of Statistics》 2019年第4期445-457,共13页
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 ob... 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. 展开更多
关键词 GOODNESS-OF-FIT STATISTICS Robust ESTIMATORS MULTIVARIATE Density ESTIMATE Information Matrix Model Testing
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Generalized Method of Moments and Generalized Estimating Functions Using Characteristic Function
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作者 Andrew Luong 《Open Journal of Statistics》 2020年第3期581-599,共19页
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 characte... 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. 展开更多
关键词 Generalized Normal Laplace Distribution Generalized Asymmetric Laplace Distribution Optimum Estimating Functions Infinitely Divisible Distribution Simulated Estimation Method
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Generalized Method of Moments and Generalized Estimating Functions Based on Probability Generating Function for Count Models
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作者 Andrew Luong 《Open Journal of Statistics》 2020年第3期516-539,共24页
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... 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. 展开更多
关键词 Mixture Distributions Consistent Chi-Square Tests Infinitely Divisible Distributions Mixture Distributions Distribution Free Test Statistics Model Testing
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Efficiency of Some Estimators for a Generalized Poisson Autoregressive Process of Order 1
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作者 Louis G. Doray Andrew Luong El-Halla Najem 《Open Journal of Statistics》 2016年第4期637-650,共14页
Various models have been proposed in the literature to study non-negative integer-valued time series. In this paper, we study estimators for the generalized Poisson autoregressive process of order 1, a model developed... Various models have been proposed in the literature to study non-negative integer-valued time series. In this paper, we study estimators for the generalized Poisson autoregressive process of order 1, a model developed by Alzaid and Al-Osh [1]. We compare three estimation methods, the methods of moments, quasi-likelihood and conditional maximum likelihood and study their asymptotic properties. To compare the bias of the estimators in small samples, we perform a simulation study for various parameter values. Using the theory of estimating equations, we obtain expressions for the variance-covariance matrices of those three estimators, and we compare their asymptotic efficiency. Finally, we apply the methods derived in the paper to a real time series. 展开更多
关键词 Discrete Time Series Autoregressive Process Moment Estimator QUASI-LIKELIHOOD EFFICIENCY Generalized Poisson Quasi Binomial Distribution
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