Rechtszekerheid bij onzekerheid

Legal security under uncertainty
There are two sources of losses in decision making: collective judgment and uncertainty. The situation can be summarized as follows:
Minimization of losses can be achieved by using optimal decision rules. The effect of this method differs in the three cases treated. The analogies and differences of the three cases are discussed in this paper.
Lezing gehouden op de Statistische Dag 1969.     

Uitbijternegerende schatters bij duplobepalingen

Outlyer-ignoring estimators for measurement in duplo.
By hypothesis a measurement u is the sum of two independent random variables, the normal random variable with expectation μ, and standard error σ, and a random error φ:

Basically two independent measurements u₁ and u₂ over u are to give the estimate x=1/2(u₁+ u₂) over μ.
However, to reduce the effect of the error φ on a final estimate of μ, one adds, according to a common practice, a third or even a fourth measurement u₃, u₄, in the case that the basic pair differs by more than a number A. For this extended set of measurements two outlyer-ignoring estimator y and z of μ are defined, and investigated against three specifications fo the error φ. Also an outlyer-ignoring estimate of σ is considered, and its application is illustrated by an example.     

Winstgevendheid en winstkansen bij valuta-arbitrage

Profits and Profit Probabilities in Exchange Arbitrage.
For the case in which in each of a number of countries buying and selling prices for the currencies of the other countries are quoted by a number of banks, a method is developed by means of which the chains of transactions which yield the highest profit, if any, are traced systematically. This method is applied to constellations of buying and selling prices in five countries at three different points of time; the maximum proft of chains was0.090, 0.477 and 0.143 per thousand, respectively.
When it is assumed that the logarithms of changes of buying and selling prices are normally distributed around o with constant variance and temporally independent, the probability that there is a positive profit can be determined for each chain at each point of time as soon as an estimate of the variance is available. In the same way the mathematical expectation of profit, when possible losses are considered as o (no action being taken), can be determined. Both quantities are calculated for two points of time for a limited number of chains.     

Sequente analyse bij keuring van goederen

Sequential sampling
A simple survey of Walds theory of sequential sampling is given, followed by a modification of the rejection and acceptation lines. The rejection and acceptation lines of the Waldtest are lines of a constant ratio between the likelihoods of two hypotheses. The modified rejection and acceptation lines are curves, giving a constant ratio between the likelihood of one hypothesis and the maximum likelihood.
The modified lines closely fit those of Wald, but ensure a natural end to the sampling before a given sample number is reached. A simple manner to evaluate rejection and acceptation probabilities of a hypothesis is still lacking, the computation now being quite inconvenient.     

Statistische hulpmiddelen bij de kwaliteitsbeheersing I

An exposition is given of the methods of statistical quality control as applied to the results of measurements on a continuous scale. Examples are given of the information which may be obtained about production processes by using control charts for the mean and range of small samples taken with regular intervals out of a flow of products from a particular machine.     

Enkele opmerkingesn over de mogelijkheid van toepassing van mathematisch-statistische methoden bij tijdstudieproblemen en in het bijzonder bij het Bedaux-systeem

Summary  (Some remarks on the possibility of applying mathematical-statistical methods to time-study problems and in particular to the Bedaux system.)
Time and motion study are based on observations of the workman's movements and an estimation of his efficiency.
The time spent on a certain action in mass-production is not constant but shows a considerable variation, due to various causes. The time-study-man who investigates this action and expresses it in a rate, causes still more variations by certain inaccuracies in using the stopwatch and calculating the rate itself. Considered from a statistical point of view, these cumulating variations result in a standard-error in the calculated rate; this gives rise to some interesting phenomena.
A more statistical approach tot the whole problem is shown in this article to be possible.
In the discussion, the application of the Beaux-system is assumed, but the conclusions hold good for all time study systems.     

Schatting van een populatiegemiddelde bij ongelijke trekkingskansen

Summary This article presents a simple method of estimating the population mean of a variate and setting approximate confidence limits when probabilities of selection are proportional to some quantity of the selected elements, which is observed along with the variate concerned. The method is closely related to the method of ratio estimators.     

Het verdelen van steekproeven over subpopulaties bij accountantscontroles

Summary  "Stratificationprocedures for a typical auditing problem".
During the past ten years, much experience was gained in The Netherlands in using random sampling methods for typical auditing problems. Especially, a method suggested by VAN. HEERDEN [2] turned out to be very fruitful. In this method a register of entries is considered to be a population of T guilders, if all entries total up to T guilders. The sample size n ₀ is determined in such a way that the probability β not to find any mistake in the sample, if a fraction p ₀ or more of T is incorrect, is smaller than a preassigned value β₀. So n ₀ should satisfy (l- p )ⁿ⁰≤β₀ for p ≥ p ₀. A complication arises if it is not possible to postpone sampling until the whole population T is available. One then wants to take samples from a population which is growing up to T . Suppose one is going to take samples n _i from e.g. r subpopulations

Using the minimax procedure, it is shown, that in this case one should choose the sizes n _i equal to ( T _i/ T ) n ₀. The minimax-value of the probability not to find any incorrect guilder in the r samples, taken together is equal to β₀.     

Het verdelen van steekproeven over subpopulaties bij accountantscontroles

Summary “Stratificationprocedures for a typical auditing problem”. During the past ten years, much experience was gained in The Netherlands in using random sampling methods for typical auditing problems. Especially, a method suggested by VAN. HEERDEN [2] turned out to be very fruitful. In this method a register of entries is considered to be a population of T guilders, if all entries total up to T guilders. The sample size n₀ is determined in such a way that the probability β not to find any mistake in the sample, if a fraction p₀ or more of T is incorrect, is smaller than a preassigned value β₀. So n₀ should satisfy (l-p)ⁿ⁰≤β₀ for p≥p₀. A complication arises if it is not possible to postpone sampling until the whole population T is available. One then wants to take samples from a population which is growing up to T. Suppose one is going to take samples n_i from e.g. r subpopulations Using the minimax procedure, it is shown, that in this case one should choose the sizes n_i equal to (T_i/T)n₀. The minimax-value of the probability not to find any incorrect guilder in the r samples, taken together is equal to β₀.     

Over de spreiding bij het droge-stofonderzoek van brood

The accuracy of the determination of the amount of solids in bread.
Of a loaf with a nominal weight of 800 grams only about 480 grams remains after complete drying. This so-called 'amount of solids in bread' is regularly checked by samples. From a batch of loaves a group of at least 5. but usually more, are taken at random and weighed. From this group one is selected with a weight as nearly as possible equalling the average weight of the whole group; for two slices of this sample loaf the decrease in weight upon drying is determined in the laboratory, and is considered as representative for the entire batch.
The accuracy of these determinations is analysed on the basis of 35 samples which were investigated in 3 different laboratories. The standard error is estimated at 5 grams per loaf. The influence of various factors, such as the method of sampling which may be altered in many ways, is briefly considered, while in a final section the results obtained are discussed from a practical point of view.     

Produktiviteitsverhoging bij accountantscontrole door toepassing van gelaagde steekproeven

The auditor's productivity: application of stratified sampling to an auditing problem.
In this paper it is shown that the use of stratified sampling can lead to a considerable saving of costs in auditing operations and thus increases the auditor's productivity.
The sampling method to be described will render it possible to say that the total value of the transactions will not differ more than a fraction φ, given in advance, from the correct total value, apart from a probability not surpassing a preassigned level e, on condition that a certain acceptance criterion is fulfilled. Taking into account the auditing costs, the author gives an optimal sampling design and calculates the savings obtained.
Finally the theoretical results are applied to a population of clerical transactions, with an exponential and a logarithmico-normal distribution respectively.     

Mogelijkheden en moeilijkheden bij het toepassen van de sequentietest

Summary  (Possibilities and Difficulties in Applying Sequential Sampling)
The application of sequential sampling schemes may be much simplified by chasing H and b ( in Barnard's notation ) in such a way that H/(b+ 1) = integer and (b + 1) = integer. A decision to accept can now be taken only after each (b + 1) items and samples of (b + 1) items may therefore be chosen from the batch.
A handicap of H/(b + 1) points is now allowed to the batch. One point is added to the score whenever no defectives are found in the sample; 0, 1, 2, points are subtracted whenever respectively 1, 2, 3. … defectives are found in a sample. The acceptance boundary is 2H/(b + 1) points; the rejection boundary is 0 points.
For given 1 in 20 producer's and consumer's risk points ( p ₁% and p ₂%), values of H and b are given in table 1 and fig. 3.     

Het quotiënt van Von Neumann bij regressie-analyse

Some remarks are made concerning the application of the VON NEUMANN ratio in testing the residuals of a regression against serial correlation.
Special attention is paid to the fact that in general the VON NEUMANN ratio increases when first differences of the original variables are used.