Andrew Luong

Efficient estimators for the good family

We consider the problem of estimating the two parameters of the discrete Good distribution. We first show that the sufficient statistics for the parameters are the arithmetic and the geometric means. The maximum likelihood estimators (MLEs) of the parameters are obtained by solving numerically a system of equations involving the Lerch zeta function and the sufficient statistics. We find an expression for the asymptotic variance-covariance matrix of the MLEs, which can be evaluated numerically. We show that the probability mass function satisfies a simple recurrence equation linear in the two parameters, and propose the quadratic distance estimator (QDE) which can be computed with an ineratively reweighted least-squares algorithm. the QDE is easy to calculate and admits a simple expression for its asymptotic variance-covariance matrix. We compute this matrix for the MLEs and the QDE for various values of the parameters and see that the QDE has very high asymptotic efficiency. Finally, we present a numer...

Quadratic distance estimators for the zeta family

The zeta distribution is a discrete distribution which has been relatively little used in actuarial science and statistics, a reason being that most estimators proposed in the literature for the parameter of this distribution require iterative methods or the extensive use of tables for its calculation, due to the complicated form of its probability mass function. In this paper, we propose a new estimator, based on quadratic distance, asymptotically fully efficient for parameter values greater than 2 and highly efficient for smaller values, but computationally more appealing than the maximum likelihood estimator; we also compare its asymptotic variance with that of the moment estimator and the estimator based on the ratio of the observed frequencies of the first two classes.

Generalized least squares estimators for covariance parameters for credibility regression models with moving average errors

Abstract Weighted least squares methods are developed for the estimation of variance–covariance parameters of credibility regression models with moving average dependent errors. The estimators proposed are shown to be useful for constructing empirical Bayes estimators and credibility type estimators. Numerical examples are included to illustrate the proposed methods.

Minimum cramér-von mises distance methods for complete and grouped data

In this paper, we consider the minimum Cramer-von Mises distance estimator for parametric families of distributions and we derive its asymptotic properties. We treat the situation where complete data are available as well as the one where data are grouped into intervals. The influence function for the estimator is derived and used to show that the estimator is asymptotically normal. We are able to compute the estimators approximate variance for a few selected models, but more generally, we show how to consistently estimate both the influence function and the variance of the estimator. Moreover, we see that in many cases, the influence function of the estimator is bounded, a robustness property which is desirable in the context of estimation of insurance loss distributions.

Some Simple Method of Estimation for the Parameters of the Discrete Stable Distribution with the Probability Generating Function

In this article, we develop a method to estimate the two parameters of the discrete stable distribution. By minimizing the quadratic distance between transforms of the empirical and theoretical probability generating functions, we obtain estimators simple to calculate, asymptotically unbiased, and normally distributed. We also derive the expression for their variance–covariance matrix. We simulate several samples of discrete stable distributed datasets with different parameters, to analyze the effect of tuncation on the right tail of the distribution.

Goodness of fit test statistics for the zeta family

Abstract Goodness of fit test procedures for the zeta parametric family based on quadratic distances and the Box-Cox transform are developed. Test statistics based on quadratic distances are shown to follow a chi-square distribution asymptotically. Test procedures based on the Box-Cox transform make use of the estimator of the parameter introduced by the Box-Cox transform, and numerical computations are based on the nonlinear weighted least squares algorithms.

Cramér–Von Mises distance estimation for some positive infinitely divisible parametric families with actuarial applications

Cramér–Von Mises (CVM) inference techniques are developed for some positive flexible infinitely divisible parametric families generalizing the compound Poisson family. These larger families appear to be useful for parametric inference for positive data. The methods are based on inverting the characteristic functions. They are numerically implementable whenever the characteristic function has a closed form. In general, likelihood methods based on density functions are more difficult to implement. CVM methods also lead to model testing, with test statistics asymptotically following a chi-square distribution. The methods are for continuous models, but they can also handle models with a discontinuity point at the origin such as the case of compound Poisson models. Simulation studies seem to suggest that CVM estimators are more efficient than moment estimators for the common range of the compound Poisson gamma family. Actuarial applications include estimation of the stop loss premium, and estimation of the present value of cash flows when interest rates are assumed to be driven by a corresponding Lévy process.

General quadratic distance methods for discrete distributions definable recursively

Abstract Quadratic distance (QD) methods for inference and hypothesis testing are developed for discrete distributions definable recursively. The methods are general and applicable to many families of discrete distributions including those with complicated probability mass functions (pmfs). Even if no explicit expression for the pmf of some distributions exists, QD methods are relatively simple to implement: the QD estimator can be computed numerically using a non-linear least-squares method. The asymptotic properties of the QD estimator are studied. Test statistics for goodness-of-fit are formulated and shown to follow asymptotically a chi-square distribution under the null hypothesis. Estimation and model testing are treated in a unified way. Simulation results presented indicate that the QDE protects against a certain form of mis-specification of the distribution, which makes the maximum likelihood estimator (MLE) biased, while keeping the QDE unbiased.

Minimum quadratic distances methods for censored data

Quadratic distances methods are developed under the randomly censored model.Asymptotic properties of quadratic estimators are given. Test statistics based on quadratic distances are proposed for testing the simple null hypothesis and composite hypothesis. They are shown to follow a chi– squared distribution under the null hypothesis.

Minimum Quadratic Distance Methods Using Grouped Data for Parametric Families of Copulas

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.