Generalized piecewise linear histograms

We extend the concept of piecewise linear histogram introduced recently by Beirlant, Berlinet and Györfi. The disadvantage of that histogram is that in many models it takes on negative values with probability close to 1. We show that for a wide set of models, the extended class of estimates contains a bona fide density with probability tending to 1 as the sample size n increases to infinity. The mean integrated absolute error in the extended class of estimators decreases with the same rate n^–2/5 as in the original narrower class.     

Calculation of policy-values for different rates of interest

Abstract

If we calculate the Policy-Values of a Whole Life Assurance for a given Age at Entry and a given Duration, but for several equidistant rates of interest, we find that the proportion of two successive values is very nearly constant. This fact is illustrated by Tables 1 a and 1 b, giving certain Policy Values on the basis H^M and A 24–29 respectively, for the rates of 3, 4, 5 and 6 % of interest. Tables 2 a and 2 b show, that the same applies, if to a smaller degree, to Endowment Assurances.     

Die Wahrscheinlichkeit einer bestimmten Auszahlungssumme

I.

Die Prämien der Lebensversicherung werden unter der Voraussetzung berechnet, dass der Verlauf der Sterblichkeit und etwaiger anderer Ausscheideursachen genau vorgegebenen Annahmen entspricht, die in der Sterbetafel bzw. in anderen Ausscheideordnungen enthalten sind. Unter dieser Voraussetzung kann der Barwert aller Yerbindlichkeiten des Versicherers eindeutig festgesetzt und hieraus die Nettoprämie, d. h. die Prämie ohne Berücksichtigung der Kosten und eines eventuellen Gewinnzuschlages, berechnet werden.     

Der derzeitige Stand der Bausparmathematik

Die aussergewöhnlichen Verhältnisse auf dem Kapitalmarkt und die grosse Wohnungsnot haben in Deutschland im Jahre 1924, unmittelbar nach der Überwindung der Inflation, welche alle Kapitalien zerstört hatte, eiii schon älteres Projekt reifen lassen und es kam durch G. Kropp in einem kleinen Städtchen im Bundesstaat Thüringen, in Wüstenrot, zur Gründung der ersten deutschen “Bausparkasse”, der sogenannten “Gesellschaft der Freunde”. Es handelt sich hier um eine Institution, die, ähnlich den englischen Building Societies und den amerikanischen Building and Loan Associations, doch nach einem abweicheriden System, Gelder von Sparern dazu verwendet, um Bauvorhaben dieser selben Sparer zu finanzieren. Dies geschieht nach folgender Methode: eine Anzahl von Sparern, die die Bausparkasse bilden, zahlen in regelmässigen Zeitabständen Sparbeiträge, welche einen festen Prozentsatz der Summe ausmachen, die für den Zweck des Hausbaues vorgesehen ist und zwischen dem Bausparer und der Bausparkasse vereinbart wurde. Wenn in der Kasse genügend viel Geld vorhanden ist, um einem Oder mehreren Bausparern die vereinbarte Summe auszuzahlen, so geschieht dies and die (z. B. durchs Los bestimmten) so beteilten Bausparer haben nun die Verpflichtung, das erhaltene Darlehen, nälich die Ergänzung ihres bereits eingezahlten Guthabens (das mit oder ohne Zinsen berechnet wird) auf die voile Bausparsumme, zu tilgen. Dieser Tilgung wird wieder ein Zinsfuss zu Grunde gelegt, der auch Null sein kann. Die so einlaufenden Gelder — Spar- und Tilgungsbeiträge — werden firmer wieder zu Zuteilungen verwendet. Verwaltungskosten werden aus gesondert zu zahlenden Beiträgen gedeckt. Auf diese Weise ist es den meisten Bausparern möglich, früher zu einem Hans zu gelangen, als wenn sie ohne Hilfe der Gesamtheit die ganze Summe zum Zins der Kasse selbst hätten ansparen müssen.     

On efficiency of estimation and testing with data quantized to fixed number of cells

Metrika - In continuous parametrized models with i.i.d. observations we consider finite quantizations. We study asymptotic properties of the estimators minimizing disparity between the observed and...     

Minimum kolmogorov distance estimates of parameters and parametrized distributions

Minimum Kolmogorov distance estimates of arbitrary parameters are considered. They are shown to be strongly consistent if the parameter space metric is topologically weaker than the metric induced by the Kolmogorov distance of distributions from the statistical model. If the parameter space metric can be locally uniformly upper-bounded by the induced metric then these estimates are shown to be consistent of ordern ^−1/2. Similar results are proved for minimum Kolmogorov distance estimates of densities from parametrized families where the consistency is considered in theL ₁-norm. The presented conditions for the existence, consistency, and consistency of ordern ^−1/2 are much weaker than those established in the literature for estimates with similar properties. It is shown that these assumptions are satisfied e.g. by all location and scale models with parent distributions different from Dirac, and by all standard exponential models. Supported by the scientific exchange program between the Hungarian Academy of Sciences and the Royal Belgian Academy of Sciences, and by GACR grant 201/93/0232.     

Analytical studies in stop-loss reinsurance

Abstract

Introduction.

Consider a unit of risk, say the whole portfolio of an office, or a comprehensive contract of a branch of casualty insurance, which can give rise to a variety of total amounts of claims during a chosen period, say one year. The total claims of the years i =- 1, 2, ... will be denoted by x ₁. They follow some frequency distribution and we assume that during the years considered they are independent from year to year and subject to the same parent distribution. This means, implicitly, that the volume of business and the value of money have remained unaltered and this assumption will be made, since the adjustments otherwise needed are technically trivial and we are not dealing here with the commercial aspect (dif. ficult though it may be of solution) arising out of changes in monetary value. The frequency distribution mentioned can then be regarded as given by a sample from a population whose probability distribution is given by p (x), say, so that      

Asymptotic distributions of φ-divergences of hypothetical and observed frequencies on refined partitions

For a wide class of goodness-of-fit statistics based on φ-divergences between hypothetical cell probabilities and observed relative frequencies, the asymptotic normality is established under the assumption n / m _n →γ∈(0,∞), where n denotes sample size and m _n the number of cells. Related problems of asymptotic distributions of φ-divergence errors, and of φ-divergence deviations of histogram estimators from their expected values, are considered too.     

Necessary and sufficient conditions for consistency of generalizedM-estimates

GeneralizedM-estimates (minimum contrast estimates) and their asymptotically equivalent approximate versions are considered. A relatively simple condition is found which is equivalent with consistency of all approximateM-estimates under wide assumptions about the model. This condition is applied in several directions. (i) A more easily verifiable condition equivalent with consistency of all approximateM-estimates is derived and illustrated on models with stationary and ergodic observations. (ii) A condition sufficient for inconsistency of all approximateM-estimates is obtained and illustrated on models with i.i.d. observations. (iii) A simple necessary and sufficient condition for consistency of all approximateM-estimates in linear regression with i.i.d. errors is found. This condition is weaker than sufficient conditions for consistency ofM-estimators known from the literature. A linear regression example is presented where theM-estimate is consistent and an approximateM-estimate is incosistent.Supported by CSAS grant N. 17503.