B.B. van der Genugten

Testing in the restricted linear model using canonical partitions

Abstract This paper treats the normal linear model y ∼ N n ( Xβ , σ 2 I n ) under the restriction C β = 0 for arbitrary C . Considered is the identifiable testing problem H 0 : Dβ = 0 against H 1 : Dβ ≠ 0. A canonical partition of the rows of C admits a simple form of involved linear subspaces. This leads to a numerically tractable form of the usual F -statistic for this testing problem.

A Weakened Form of Fictitious Play in Two-Person Zero-Sum Games

Fictitious play can be seen as a numerical iteration procedure for determining the value of a game and corresponding optimal strategies. Although convergence is slow, it needs only a modest computer storage. Therefore it seems to be a good way for analysing large games. In this paper we introduce a weakened form of fictitious play, where players at each stage do not have to make the best choice against the total of past choices of the other player but only an increasingly better one. Theoretical bounds for convergence are derived. Furthermore, it is shown that this new form can speed up convergence considerably in practice. It is seen that weakened fictitious play can be extended to models in which the game matrix itself becomes better known as the number of stages increases.

A General Model for Repeated Audit Controls Using Monotone Subsampling

Abstract In categorical repeated audit controls, fallible auditors classify sample elements in order to estimate the population fraction of elements in certain categories. To take possible misclassifications into account, subsequent checks are performed with a decreasing number of observations. In this paper a model is presented for a general repeated audit control system, where k subsequent auditors classify elements into r categories. Two different subsampling procedures will be discussed, named “stratified” and “random” sampling. Although these two sampling methods lead to different probability distributions, it is shown that the likelihood inferences are identical. The MLE are derived and the situations with undefined MLE are examined in detail; it is shown that an unbiased MLE can be obtained by stratified sampling. Three different methods for constructing confidence upper limits are discussed; the Bayesian upper limit seems to be the most satisfactory. Our theoretical results are applied to two cases with r = 2 and k = 2 or 3, respectively.

Measuring Skill in More-Person Games with Applications to Poker

In several jurisdictions, commercially exploiting a game of chance (rather than skill) is subject to a licensing regime. It is obvious that roulette is a game of chance and chess a game of skill, but the law does not provide a precise description of where the boundary between the two classes is drawn. We build upon the framework of Borm and Van der Genugten (2001) and Dreef et al. (2004) and propose a modification of the skill concept for more-person games. We apply our new skill measure to a simplified version of poker called Straight Poker and conclude that this game should be classified as a game of skill.

Blackjack in Holland Casino's: basic, optimal and winning strategies

This paper considers the card game Blackjack according to the rules of Holland Casinos in the Netherlands. Expected gains of strategies are derived with simulation and also with analytic tools. New efficiency concepts based on the gains of the basic and the optimal strategy are introduced. A general method for approximating expected gains for strategies based on card counting systems is developed. In particular it is shown how Thorps Ten Count system and the High-Low system should be used in order to get positive expected gains. This implies that in Holland Casinos it is possible to beat the dealer in practice.

Iterated weighted least squares in heteroscedastic lineaipmod%81è

Iterated weighted least squares (IWLS) is investigated for estimating the regression coefficients in a linear model with symmetrically distributed errors. The variances of the errors are not specified; it is neither assumed that they are unknown functions of the explanatory variables nor that they are given in some parametric way IWLS is carried out in a random number of steps, of which the first one is OLS. In each step the error variance at time t is estimated with a weighted sum of m squared residuals in the neighbourhood of t and the coefficients are estimated using WLS. Furthermore an estimate of the covariance matrix is obtained. If this matrix is somehow smaller than the one before, a new step is carried out unless an upper bound has been reached. Large sample properties of IWLS are derived for fixed m. Some particular cases show that the asymptotic efficiency can be increased by allowing more than two steps. For a particular example some finite-sample properties are evaluated on the basis of simul...

A Mixed Model for Double Checking Fallible Auditors

The paper discusses the problem of a fallible auditor who assesses the values of sampled records, but may make mistakes.To detect these mistakes, a subsample of the checked elements is checked again, now by an infallible expert. We propose a model for this kind of double check, which takes into account that records are often correct; however, if they are incorrect, the errors can take on many different values - as is often the case in audit practice.The model therefore involves error probabilities as well as distributional parameters for error sizes.We derive maximum likelihood estimators for these model parameters and derive from them an estimator for the mean size of the errors in the population.A simulation study shows that the latter outperforms some other - previously introduced - estimators.

Asymptotics of Multivariate Regression with Consecutively Added Dependent Varibles

We consider multivariate regression where new dependent variables are consecutively added during the experiment (or in time).So, viewed at the end of the experiment, the number of observations decreases with each added variable. The explanatory variables are observed throughout.In a previous paper we determined the least squares and maximum likelihood estimators for the parameters in this model.In this paper we discuss the estimation technique of iterative least squares to calculate the maximum likelihood estimates and we prove the consistency of the estimators in each iteration.Moreover, we introduce a general class of estimators for the regression parameters based on arbitrary starting estimators for the covariance matrix.We prove the consistency of these new estimators and - for sake of completeness - of the previously obtained least squares and maximum likelihood estimators as well.

Density of the quotient of non-negative quadratic forms in normal variables with application to the F-statistic

The density of the quotient of two non-negative quadratic forms in normal variables is considered. The covariance matrix of these variables is arbitrary. The result is useful in the study of the robustness of theF-test with respect to errors of the first and second kind. An explicit expression for this density is given in the form of a proper Riemann-integral on a finite interval, suitable for numerical calculation.