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.     

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 beschouwingen in verband met textiel

Owing to the variability of the raw material and the large number of production units in textile industry the application of statistical methods is indispensable to obtain a high quality product.
The author discusses examples of the statistical problems cropping up in various phases of the production process.     

Een opmerking over het aanpassen van functies aan een groot aantal waarnemingsuitkomsten

Note on the fitting of a function to a large number of observations.
The paper deals with various ways in which a set of functions may be chosen to describe a given body of observations, the parameters determining the function being determined by the data.
The special case when the functions of the set are given in the form of a table or a graph is emphasized.     

Suggestie betreffende het combineren van overschrijdingskansen

Summary In this paper a new method for combining P -values is suggested. This method takes into account that the P-values to be combined may be based upon different numbers of degrees of freedom.     

Een meer-loketten probleem met overwerk

In an office a number of employees do the same kind of work. The jobs arrive at random and the holding time is exponential; the queuediscipline is first in, first served. The mean queuelength is shortened by work done in overtime according to the rule that on every day on which the total number of jobs exceeds a certain number N, one or more extra hours of work will be done. The distribution of the number of jobs in the office, the mean value of this number and the probability of overtime are given. The model is illustrated with a numerical example.     

Een mathematisch model voor een reserveringsprobleem

A mathematical model for a booking problem
Air companies have several problems in connection with booking. Clients may book for a certain flight some time in advance and are also allowed to cancel their bookings.
As a result of this it is possible that an airplane leaves with some places empty or that the company, expecting some clients to cancel their booking, has sold to many places.
In both cases losses are the result. In this paper a booking policy that minimizes the expected losses is developed.     

Regen, een toepassing van dynamische programering

Veel mensen zal het wel eens overkomen zijn dat ze tijdens een wandeling overvallen werdcn door een regenbui. De eerste reactie in zo'n situatie is in de regel, hard te gaan lopen teneinde zo droog mogelijk thuis te komen. Bij sommigen echter zal ongetwijfeld wel eens de vraag zijn opgekomen hoeveel effect dat harde lopen nu heeft; als je hard loopt ben je weliswaar eerder thuis maar je vangt ook meer regen per tijdseenheid, en stel je voor dat het al gauw ophoudt met regenen. In dit verhaaltje wordt het probleem van de optimale strategie aan de orde gesteld, wat is de optimale snelheid in verschillende omstandigheden. Als criterium gebruiken we de (verwachte) totale hoeveelheid opgevangen regen. Daarbij gaan we er van uit dat de regen recht naar beneden valt en homogeen verdeeld is over de ruimte. In paragraaf 1 wordt aangenomen dat het blijft regenen. In paragraaf 2 wordt rekening gehouden met de mogelijkheid dat het ophoudt met regenen en daarna droog blijft. In paragraaf 3 nemen we aan dat het steeds opnieuw kan gaan regenen.     

Constructie van kansellipsen in een correlatiediagram

Summary  (Construction of chance ellipses in a correlation diagram).
Chance ellipses in a correlation diagram are calculated from the formula of the normal correlation surface (the chance ellips P contains the area of the correlation diagram, where P% of the observations are found). The construction of these ellipses is explicated. A table with factors to construct the ellipses for different values of P is added.     

Vergelijking van een tweetal toetsen tegen verloop voor een aantal kansen

Two test statistics t and b , testing equality of probabilities p_i of success in k different series against the hypothesis of trend with given numbers g_i (weights) of the series, are compared. The first teststatistic, due to C. van Ee den en J, Hemelrijk [1] is

with n_i1 equal to the number of successes in n_i trials and

Defining trend by

it appears that the teststatistic t gives rise to a test consistent for the complete set of alternatives τ≠ 0.
The other teststatistic is

b gives rise to a test which is not consistent for the general set of alternatives τ≠ 0 but for a rather important subset of these alternatives, i.e. those alternatives which show a lineair trend.
Neither of the tests in necessarily unbiased. The asymptotic relative efficiency of test t with respect to b is equal to or lower than unity. (Equal in case
n₁= n₂=…= n_k with b = t;
in this case b is also consistent for the set of alternatives τ≠ 0). The variances of the teststatistics can be estimated with

Both tests are based on the approximately normal distribution of the teststatistics. To judge this approximation the 3rd and 4th cumulants of the distributions of the statistics are evaluated in terms of number of elements of a binomial distribution.
It is concluded that in case of possible non-lineair relationships the teststatistic t is preferable as it gives rise to a consistent, designfree test. In case a lineair relationship has to be tested against the hypothesis of no trend the teststatistic b has to be prefierred, especially if the number of trials in the series are very different. An example is discussed.     

Een model voor discreet verdeelde grootheden met negatieve correlatie

Summary
A model for negatively correlated variables.
A mathematical model is described with which a bivariate distribution of two negatively correlated binomially distributed variables can be constructed. Let be given a vase with W white, Z black and R red balls; N = W + Z + R; P1= W: N;P2 = Z: N.
Let be drawn a random sample with replacement of n balls. Let be w = number of white en z = number of black balls; w + z < n. Then and are binomially distributed variates, correlated according to (1).
At the end of the paper the author formulates the following statistical problem: let be drawn a finite number of times () such random samples of n balls. The pairs w, z furnish a correlation coefficient r, which itself is a stochastical variable. How is the distribution of? In particular what are the expectation (, N) and the standard deviation (k, n, N)     

De oplossing van een kwaliteitsprobleem door het gebruik van proefschema's

Samenvatting  Het artikel geeft een beschrijving van de strategie en de methoden, die worden gebruikt voor het zoeken naar de oorzaak van een kwaliteitsvermindering bij dubbelspiralen voor gloeilampen. Er wordt niet alleen aandacht besteed aan de beschrijving en motivering van de gekozen proefschema's maar ook aan de praktische uitvoering van de experimenten. Het fabricage-proces is nl. dermate gecompliceerd dat niet kon worden volstaan met de instructie de experimenten in aselecte volgorde uit te voeren.
Het gehele onderzoek wordt uitqevoerd aan de hand van een drietal proefschema's. De keuze van het tweede en derde proefschema volgt logisch uit de uitkomsten van de voorafgaande proef. Na vol-tooiing van de derde proef is ondubbelzinnig aangetoond dat de belangrijkste oorzaak van de kwaliteitsvermindering is gevonden.     

Een toepassing van "importance sampling"

Summary  This paper describes an experiment with "importance sampling", to show how much reduction of the computation time and sample size can be achieved in comparison with the usual Monte Carlo method. A comparison is made between each of the three methods of "importance sampling" and the usual Monte Carlo method by the determination of the expression

Of the three methods A, B and C the first one uses the shifted exponential distribution, the second one uses the gamma distribution, and the third one uses the exponential distribution with modified parameter. These three methods have all smaller variances, ranges and sample sizes than the usual Monte Carlo method. Their order of preference is A, B, C. With respect to computing time only the method A is significantly better. So only the method A is an improvement in respect of both the sample size and the computing time.     

Aanvoer en opslag van dunne melk in een melkproduktenfabriek

Summary
In a dairy products plant the problem arose that the milk-carrying tankcars had considerable waiting times at the receiving platform. This was supposedly caused by too small a capacity of the raw milk storage room.
In order to study this problem the stream of milk through the storage room was followed for three weeks continuously. The storage room proved to operate mainly as a daily buffer. This function could be expressed in a linear equation. Its discussion led to some simple measures resulting in a considerable increase in efficiency of the storage room.
The study further proved that the waiting times of the tankcars were caused to only a small extent by the storage room capacity and that this effect could be eliminated by the above mentioned measures. It was concluded, therefore, that the storage room need not be enlarged.
The principal cause of the waiting times proved to be the limited unloading capacity of the receiving platform. And this in its turn was due to a strong periodicity in arrival frequencies of the cars. So, the solution of the problem had to be found in better car schedules, capable of breaking the periodicity.
Apart from this periodicity the capacity of the existing receiving platform was ample, as was shown by means of queuing theory.
Further, a special application of queuing theory was used to calculate the optimal design of a receiving platform. It was shown that few high capacity receiving installations are preferable to a larger number of smaller installations. The practical optimum is a platform of three installations.     

Enkele beschouwingen betreffende de bijdrage van een component tot een samengestelde variabele.

Several definitions of the effective contribution of a component to a composite are reviewed. The diversity of the corresponding values is stressed. An attempt has been made to look at the different definitions, one of which is novel, as giving complementary information about the same situation.     

De fout bij het gebruik van een tweetal benaderingsformules

Summary For the sake of efficiency, the use of simplified formulae in item analysis is recommended. For adequate use of such formulae the systematic error must be known. For two important formulae this error is estimated: the KUDER-RICHARD-SON-21 (instead of 20) and a formula for the correction of the spurious part-whole correlation. As for the KR, if itemscores are 1 andO, the KR20 is given in (I) and the KR21 in (2). The difference dr between them, given in (5), is directly proportional to the variance of the itemdifficulties. This variance turns out to be rarely greater than 0,05. If the distribution of itemdifficulties is unimodal, a value greater than 0,05 is indeed improbable in four-choice tests. The KR21, which does not need an item analysis, can be used as a substitute of the KR20. The KR21 coefficient itself is an lower limit, the KR21 plus dr - estimated by (6) - a probable upper limit of the KR20 coefficient. The part-whole correlation of tests is spurious. If one wants to correct for this spuriousness it is theoretically preferable to estimate the part-parallel test correlation instead ofthe part rest correlation. This is done here with the assumptions of classical test theory. The part-parallel test correlation is given in (12), the easily computed first approximation, called A, in (13) and the second approximation in (75), in which last formula the corrected correlation is expressed as a function of A, the standard deviations of part and whole, and the reliability of the test. One can compute with this formula a maximum A, below which the first approximation is sufficiently precise.     

De invloed van de keuze der gewichten op de hoogte van een gewogen indexcijfer

The influence of the choice of the weights on the value of an indexnumber.
Price and quantity indexnumbers are weighted averages of groups of price and quantity ratios and they are convenient instruments to indicate the general tendency of such groups, especially if the number of basic ratios is considerable. The frequent use of indexnumbers is due to the fact that they can often be applied to problems for which, strictly speaking, an indexnumber had to be used derived from the same group of ratios but based on a different set of weights.
Two typical examples of such problems are given.
The use of a set of weights differing from the appropriate one is only justified, however, when the indexnumber is rather insensitive to changes in the set of weights. A simple formula is derived showing that the relative change of an index-number due to a change in the set of weights is equal to the product of the (weighted) coefficient of variation of the basic ratios, the (weighted) standard deviation of the relative changes of the weights and the (weighted) coefficient of correlation of the ratios and of the relative changes. The system of weights used in the calculation of these three factors is the same and is equal to the set of true weights belonging to the problem under consideration.
The practical use of the formula is demonstrated at the problem of index-numbers of costs frequently encountered in the practice of cost accounting.