Goodness of Fit for Discrete Random Variables Using the Conditional Density

In this paper we find a new test of goodness of fit in the case of discrete random variables. The main advantage of the methodology proposed in this paper relies on the fact that given the sample, we can control the probability of the type I error, that is α, and then find the exact value of the probability of the type II error, β, associated, in some cases. The results are not asymptotic, but exact. Also a conditional test for two alternatives is obtained. We also include some simulations in order to check the power of the procedures.Mathematics Subject Classification (2000) Primary 62G10 · 62B05 · Secondary 62E10     

Monotonicity of the (reversed) hazard rate of the (maximum) minimum in bivariate distributions

It is well known that in the case of independent random variables, the (reversed) hazard rate of the (maximum) minimum of two random variables is the sum of the individual (reversed) hazard rates and hence the onotonicity of the (reversed) hazard rate of the marginals is preserved by the monotonicity of the (reversed) hazard rate of the (maximum) minimum. However, for the bivariate distributions this property is not always preserved. In this paper, we study the monotonicity of the (reversed) hazard rate of the (maximum) minimum for two well known families of bivariate distributions viz the Farlie-Gumbel-Morgenstern (FGM) and Sarmanov family. In case of the FGM family, we obtain the (reversed) hazard rate of the (maximum) minimum and provide several examples in some of which the (reversed) hazard rate is monotonic and in others it is non-monotonic. In the case of Sarmanov family the (reversed) hazard rate of the (maximum) minimum may not be expressed in a compact form in general. We consider some examples to illustrate the procedureResearch of the second author is supported by a grant from Natural Sciences and Engineering Research Council and the research of the other two authors is partially supported by a travel grant from the Canadian American Center of the University of Maine     

On a family of counting distributions and recursions for related compound distributions

Abstract

This paper considers a family of counting distributions whose densities satisfy certain second order difference equations. Recursions for the evaluation of related compound distributions are developed in the case of severity distributions which are concentrated on the non-negative integers. From these a characterization of the considered counting distributions is obtained, and it is shown that most of these are compound Poisson distributions.     

Die Lebensversicherung mit exponentiell ansteigender Prämienreserve (Sparversicherung)

Abstract

Die gebräuchlichste Form der Lebensversicherung ist heute die Kapitalversicherung auf den Todes- und Erlebensfall. Bei ihr wird im Versicherungsfall ein bestimmtes Kapital fällig; als Gegenleistung sind hierfür vom Versicherungsnehmer laufend Prämien zu entrichten in Form von gleichen Jahresbeträgen oder auch gleichen unterjährigen Raten. Dieses Prinzip, einem gleichmässigen Versicherungsschutz einerseits gleichmässig fällig werdende Prämien gleicher Höhe andererseits gegenüberzustellen, wird durch die ausgleichend wirkende Funktion der Prämionreserve möglich, welche die bisher nicht benötigten Teile (Sparprämien) der eingezahlten Prämien jeweils reserviert für die sich zukünftig durch Tad oder Ablauf mehr und mehr häufenden Versicherungsfälle. Damit erfüllt die Prämionrosorve als Regulator eine wichtige versicherungstechnische Aufgabe. Dennoch bleibt sie dem Versicherungsnehmer im allgemeinen unbekannt, weil ihre jeweilige Hohe nur mit Hilfe von umfangreichen Tabellen nach mathematischen Formeln berechnet werden kann, so dass ihr verschwiegenes Dasein nicht selten zu Missverständnissen Anlass gibt.     

The Empirical Distribution of UK and US Stock Returns

There is now substantial evidence that daily equity returns are not normally distributed but instead display significant leptokurtosis and, in many cases, skewness. Considerable effort has been made in order to capture these empirical characteristics using a range of ad hoc statistical distributions. In this paper, we investigate the distribution of daily, weekly and monthly equity returns in the UK and US using two very flexible families of distributions that have been recently introduced: the exponential generalised beta (EGB) and the skewed generalised- t (SGT). These distributions permit very diverse levels of skewness and kurtosis and, between them, nest many of the distributions previously considered in the literature. Both the EGB and the SGT provide a very substantial improvement over the normal distribution in both markets. Moreover, for daily returns, we strongly reject the restrictions on the EGB and SGT implied by most of the distributions that are commonly used for modelling equity returns, including the student- t , the power exponential and the logistic distributions. Instead, our preferred distributions for daily returns are the generalised- t for the US and the skewed- t for the UK, both of which are members of the SGT family. For weekly returns, our preferred distributions are the student- t for the UK and the skewed- t for the US, while for monthly returns, our preferred distributions are the EBR12 for the UK and the logistic for the US. We consider the implications of our findings for the implementation of value-at-risk, a risk management methodology that rests heavily on the distributional characteristics of returns.     

Characterizing systems of distributions by quantile measures

Modelling an empirical distribution by means of a simple theoretical distribution is an interesting issue in applied statistics. A reasonable first step in this modelling process is to demand that measures for location, dispersion, skewness and kurtosis for the two distributions coincide. Up to now, the four measures used hereby were based on moments.
In this paper measures are considered which are based on quantiles. Of course, the four values of these quantile measures do not uniquely determine the modelling distribution. They do, however, within specific systems of distributions, like Pearson's or Johnson's; they share this property with the four moment-based measures.
This opens the possibility of modelling an empirical distribution—within a specific system—by means of quantile measures. Since moment-based measures are sensitive to outliers, this approach may lead to a better fit. Further, tests of fit—e.g. a test for normality—may be constructed based on quantile measures. In view of the robustness property, these tests may achieve higher power than the classical moment-based tests.
For both applications the limiting joint distribution of quantile measures will be needed; they are derived here as well.     

The distribution of sectoral TFP growth rates: International evidence

This paper investigates the distributional properties of TFP growth rates for countries in the G7 group. Our findings lend support to the hypothesis that multifactor productivity shocks can be plausibly fitted by a symmetric non-Gaussian stable distribution model. This leads to non-negligible implications for business cycle analysis.     

The Distributional Characteristics of a Selection of Contracts Traded on the London International Financial Futures Exchange

This study examines the distributional properties of futures prices for contracts traded on LIFFE. A filtering process is employed to remove day of the week and holiday effects, a maturity effect, moving average effects and the influence of an asset's conditional variance from the raw returns series. Alternative distributional models from the stable paretian and ARCH families are examined for their applicability to futures data using a stability under additions. The results conclusively reject the hypothesis that futures returns are normally distributed with findings in favour of two related hypotheses – the mixtures of stable distribution and the ordinary stable distribution.     

Three dual regression schemes for the Lukacs theorem

In the paper we study regressional versions of Lukacs' characterization of the gamma law. We consider constancy of regression instead of Lukacs' independence condition in three new schemes. Up to now the constancy of regressions of U=X/(X + Y) given V=X + Y for independent X and Y has been considered in the literature. Here we are concerned with constancy of regressions for X and Y while independence of U and V is assumed instead.     

Asymmetric persistence in GDP? A deeper look at depth

If an economic time series behaves asymmetrically, then an interpretation of economic fluctuations based on linear time-series models could be misleading. Beaudry and Koop (1993) recently argued that for post-war US GDP data there exists a statistically significant difference in persistence between negative and positive shocks. We demonstrate that their test has two pitfalls: First, the t-statistic for testing asymmetry in persistence does not have a conventional interpretation. Second, a highly significant t-value may come from sources different from asymmetry. Using international data, we investigate for the presence of asymmetric persistence across the G-7 countries.