Afschatting van verdelingen van kwadratische vormen met toepassingen op aanpassingstoetsen voor normaliteit en exponentialiteit

Summary  In this paper a simple approximation is given for the distribution of the quadratic form

being a weighted sum of squares of independent, identically distributed standardized normal variates. Using the formulae, the Monte Carlo results concerning some goodness of fit tests for normality and exponentiality are verified and extended.     

Maxima voor de eenzijdige overschrijdingskans en voor de grootte van de gemiddelde overschrijding bij discontinue en continue, eentoppige verdelingen

The well-known inequality of Bienaymé-Tschébyschef (for short B-T), generalized by Camp and Meidell (for short C-M) for continuous, unimodal distributions gives specific limits for total probabilities outside the ± to limits.
In many cases however, especially in the field of industrial applications we are interested only in the probability of one tail of the distribution, which of course must be smaller than the limits given by the B-T and C-M formula.
For these cases the maximum probability of surpassing the to limit on one side equals under B-T conditions and under C-M conditions instead of the two-sided values of 1/t² and 47/9 · 1/t² respectively (cf e.g. Uspensky: "Introduction to mathematical probability", 1937, p. 198) These results set upper limits for the value of

Alternatively we may also set an upper limit for the integral

which measures in terms of σ the average amount by which the limit + tσ is exceeded. This problem is also discussed and under C-M conditions an upper limit

is derived.
Some practical applications of these results are considered.     

Tabel en nomogram voor het aflezen van jaarlijkse groeipercentages van grafieken op semi-logaritmisch papier

Summary  A table and a nomogram are given to find the growth rate of a regression line drawn on so-called semi-logarithmic graph paper. The growth rate is determined by: a) the slope of the regression line, b) the length ratio between the logarithmic and the metric unit, in other words, the number of years on the time axis (if a year is the time unit) corresponding in length to one cycle on the logarithmic axis (say, 1 to 10, or 100 to 1000). With an example, the use of the table and nomogram is explained.     

Tabel en nomogram voor het aflezen van jaarlijkse groeipercentages van grafieken op semi-logaritmisch papier

Summary A table and a nomogram are given to find the growth rate of a regression line drawn on so-called semi-logarithmic graph paper. The growth rate is determined by: a) the slope of the regression line, b) the length ratio between the logarithmic and the metric unit, in other words, the number of years on the time axis (if a year is the time unit) corresponding in length to one cycle on the logarithmic axis (say, 1 to 10, or 100 to 1000). With an example, the use of the table and nomogram is explained.     

Transformatie van niet-normale verdelingen

In statistical analysis and operational research many techniques are being applied, which are based upon the fact that the phenomenon under consideration has a distribution, which is approximately normal. This distribution-type has got a number of very nice properties and a constant pattern, so that "normal" phenomenona can be predicted. In case of non-normal, but deviated or skew distributions, analysis and forecasting become much more complicated. In this article a method has been described in which some non-normal variables can be transformed into normal ones, so that techniques based on normal distributions may be used.     

Statistische eigenschappen van een vlambeveiliging met behulp van een gasontladingsbuis welke gevoelig is voor ultraviolet licht

Summary The output voltage of a detector circuit containing a gas-filled tube sensitive to ultraviolet radiation can be used to indicate whether a flame is still burning or not. As long as the flame is burning the random nature of the discharges of the tube leads to random fluctuations in the output voltage, which can be considered as a Markov-process with dependent increments. The problem is to select an alarm level for the output voltage with a suitable chosen probability of exceedance while the flame is burning, in order to limit the probability of false alarm. In this article a method is given for determining upper and lower bounds for the distribution function of the output voltage at any moment, answering the question of the alarm level to be chosen. Results are given for a special circuit.     

Statistische eigenschappen van een vlambeveiliging met behulp van een gasontladingsbuis welke gevoelig is voor ultraviolet licht

Summary  The output voltage of a detector circuit containing a gas-filled tube sensitive to ultraviolet radiation can be used to indicate whether a flame is still burning or not. As long as the flame is burning the random nature of the discharges of the tube leads to random fluctuations in the output voltage, which can be considered as a Markov-process with dependent increments. The problem is to select an alarm level for the output voltage with a suitable chosen probability of exceedance while the flame is burning, in order to limit the probability of false alarm. In this article a method is given for determining upper and lower bounds for the distribution function of the output voltage at any moment, answering the question of the alarm level to be chosen. Results are given for a special circuit.     

Poisson-proces met interrupties, een waarschijnlijkheids-model voor het machinaal dunnen van bieten.

Starting from the assumption that {even in case of dense precision sowing), the occurence of beet plants in a row can be taken as a realisation of a Poisson proces, the influence of mechanical thinning is studied.
The distribution of the distance between two plants, originally exponential, is changed by mechanical thinning in a way, dependent on the lengths of the intervals a (the potential survivor-zone) and b (the length of the thinningblade).
The distribution of the distance between plants after mechanical thinning, is examined, and the "best" way of thinning, leading to the distribution of the distance between plants with the smallest variance, is derived.     

Tabellen en nomogrammen voor het onderscheidingsver-mogen van de 5% en 10%-F-toets voor het gebruik bij de gewarde blokkenproef

Tables and graphs of the power of 5% and 10%-F-tests for use in randomized block experiments.
As randomized block experiments occur so often in agricultural and other research, it was thought worth while to adapt existing tables of the power of 5% F-tests for this special case: The author hopes that in this way the design of experiments in terms of power will be facilitated. Moreover new tables were computed for the power of 10%-F-tests.
The tables have been summarized in graphs that greatly facilitate their use. Illustrations are given in practical problems.     

Een nomogram voor betrouwbaarheidsgrenzen van quantielen en overschrijdingskansen bij normaal verdeelde grootheden

A nomogram for confidence intervals and exceedance probabilities.
In this paper two problems are considered regarding the probability β that an observation on a normally (μ, σ²)-distributed random variable exceeds a given value W:

If μ and σ² are unknown, the two problems are as follows:
1)if Wis given, to determine a confidence interval for β and
2)if β is given, to determine a confidence interval for W.
For these two essentially equivalent problems graphs are given from which the confidence intervals can be determined. The graphs are given in terms of:

and are based on an approximation for the distribution of x¯ +k s .     

Identificeren op grond van waarnemingen

This paper is an exposition of some elements of Wald's decision theory. Concepts like a priori distribution, decision function, loss function, risk, Bayes procedures, admissible procedures, minimax procedures, least favourable distribution are introduced all in connection with the problem of classification of observations into two given populations. The exact treatment of the procedures and their mutual relations are illustrated by numerical examples concerning univariate and multivariate normal populations. The extension towards the classification into more than two given populations concludes the paper.     

Een criterium voor het kiezen van een proefopzet

A characterization of D-optimality is given together with several examples where D-optimal designs are computed.     

Enige ervaringen met kursussen in de OR met behulp van management games

Samenvatting Binnen Akzo is veel ervaring opgedaan met speciaal voor dit doel ontwikkelde managementgames in OR-cursussen. Het blijkt dat deze van grote waarde zijn zowel om de motivatie te verhogen als om het inzicht te verdiepen in de toepassingsmogelijkheden van OR in de praktijk.
In dit artikel worden een tweetal cursussen beschreven en de plaats hierin van de games. Een aantal praktische ervaringen worden genoemd.
De cursussen die hier beschreven zijn bereiken door de grote motivatie van de cursisten en de vrijere wijze van doceren veel beter hun doel dan meer technische cursussen. Het instrument van het management game draagt hier zeer wezenlijk toe bij.     

Bepaling van de personeelssterkte op kostenbasis*

Determining the optimal number of staff in a department using a method based on costs. How many men do you appoint in a department of a firm? This number can be determined in an optimal way, if the following data are known: - The number of men required per unit of time (may be given as a stochastic quantity). - The mean probability of absence per man in the department (due to illness, holidays etc.). - Wages per man. - Costs (proceeds), if in one unit of time there are less (more) men available than required. Generally the solution will be obtained by an iterative method. However, when the parameters show a certain behaviour it is possible to find the optimum solution by analytical and graphical means.