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.     

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.     

Statistisch onderzoek van contrôle-analyse cijfers: (De verdeling van het droge stofgehalte van het brood tijdens de tweede wereldoorlog)

Summary  (Statistical investigation of the distribution of data for the solids of bread (in loaves analysed in the Food Inspection Laboratory in Amsterdam))
The distributions of the data of the solids of bread as analysed during the years of the war are investigated. The means and the standard deviations are calculated, also χ², kurtosis and skewness supposing the distributions to be normal. An example of calculation is given in table I. Actual numbers for different years are given in table II and in table III. The distributions were tested on normality because former investigations showed that the distribution of under survey prepared loaves did not deviate significantly from the normal.
It is found that generally the investigated distributions cannot be regarded as normal. Though symmetric they show leptokurtosis and the χ²-test for the goodness of fit of normal equation gives values of P 0,01 (or a little more). Similar distributions were found by Clancey¹⁾ in his investigation of numbers of chemical analyses of industrial products (about 10% of the distributions showed this shape, some 10% were truncated leptokurtic curves) and by us for the fat percentage of meals from the governmental eating-houses. The distributions are represented on probability-paper. This way of representing results gives a clear view of the variations of the mean and the standarddeviation in the course of the years (fig. 1). The deviations of the shape of the normal straight line on probability paper by special causes is investigated (fig. 3, fig. 4, fig. 5, fig. 6 to be compared with fig. 2). With this "spectrum" of possibilities of deviations from the normal distribution in mind the special cause for the leptokurtic shape in our special case has been discussed.     

De vervaardiging van een systeem functie-ponskaarten

While making a series of 10,000 "function" -punched cards with the data of the well-known Barlow's tables and with the logarithms of the integers from the tables of Schrön, we took the opportunity to examine the accuracy with which our personnel and machines are working under normal conditions. We examined for instance in how many cases the first punching operator did not notice an error she made and how many of those errors remained after control-punching. These last-mentioned errors were traced by control-calculations with the punched cards, e.g. by calculating and checking n × n²= n ³ or by the calculation of differences. In this way it appeared that the first punching operator did not notice 270 errors in the part coming from Barlow's tables, being 0.055% of the number of punched columns or 2,7% of the number of cards. Further it was found, that only one error of the first operator was not noticed (or not corrected) by the controling operator.
Besides this a number of printing-errors, or anyway errors which were the result of the bad legibility of the digits, were found in Barlow's tables, where the punching operator and the controling operator had come to the same interpretation.
The article gives in some detail what precautions were taken by wiring the machines to attain that an error would be noticed as such.     

Assortimentsbepaling van seizoenprodukten

This paper deals with the analysis of a problem often met when determining purchasing policies. From a number of different kinds of goods, which are to be bought, the total quantity is restricted by storing capacity or financial limits. The problem "how much of each kind will be bought" is solved for a discrete as well as for a continuous distribution of demand for the various goods.     

Afleiding van kwartaalcijfers uit jaartotalen

Summary  Quarterly figures may be required when only a sequence of annual totals is available. This article presents two reasonable methods for "interpolating" quarterly figures by means of appropriate minimizing procedures.
The first method relates to minimizing the sum of squared second differences of the quarterly figures, the other one is a linear programming procedure which refers to the absolute second differences. Both methods are illustrated by a numerical example, the first one giving somewhat better results.     

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.     

Optimality of block designs with variable block sizes and random block effects

We consider the simple block model with random block effects, the block effects having variance _b ² =²;, with ² the variance of the errors. It is assumed that the experimenter can vary the sizes of the blocks. The universal optimality of certain designs for all over all designs with the same number of blocks and the same number of observations is shown. It is of interest to note that if Balanced Incomplete Block Designs compete, then they perform equally well for =0 and for =, i.e. in the one way classification model and in the simple block model with fixed block effects, but they perform worse for every (0, ).The result is, however, theoretical in nature. It treats a situation which is not very likely to happen in practice. The interest lies in the fact that it provides a counterexample to a conjecture on optimality of designs in mixed models.     

On monotone optimal decision rules and the stay-on-a-winner rule for the two-armed bandit

Summary Consider the following optimization problem: Find a decision rule such thatw(x, (x))=max _a w(x, a) for allx under the constraint (x)D (x). We give conditions for the existence of monotone optimal decision rules . The term monotone is used in a general sense. The well-known stay-on-a-winner rules for the two-armed bandit can be characterized as monotone decision rules by including the stage number intox and using a special ordering onx. This enables us to give simple conditions for the existence of optimal rules that are stay-on-a-winner rules. We extend results ofBerry andKalin/Theodorescu to the case of dependent arms.     

RAM: A Range Adjusted Measure of Inefficiency for Use with Additive Models,and Relations to Other Models and Measures in DEA

Generalized Efficiency Measures (GEMS) for use in DEA are developed and analyzed in a context of differing models where they might be employed. The additive model of DEA is accorded a central role and developed in association with a new measure of efficiency referred to as RAM (Range Adjusted Measure). The need for separately treating input oriented and output oriented approaches to efficient measurement is eliminated because additive models effect their evaluations by maximizing distance from the efficient frontier (in ₁, or weighted ₁, measure) and thereby simultaneously maximize outputs and minimize inputs. Contacts with other models and approaches are maintained with theorems and accompanying proofs to ensure the validity of the thus identified relations. New criteria are supplied, both managerial and mathematical, for evaluating proposed measures. The concept of approximating models is used to further extend these possibilities. The focus of the paper is on the physical aspects of performance involved in technical and mix inefficiencies. However, an Appendix shows how overall, allocative and technical inefficiencies may be incorporated in additive models.     

Some brief notes on the “unfinished business” of qualitative inquiry

The paper argues that there are a variety of implicit issues in qualitative inquiry that need to be addressed if the area is to develop in some normal science sense. This unfinished business is concerned with a deeper investigation of basic terms that are now simply taken for granted, such as theme and pattern. It also includes the need to develop rules which will assist in making and justifying how qualitative interpretations are made from the implicit processes of inference. Specific suggestions are made for accomplishing these issues.     

Sexual harassment in organizations

Sexual harassment is now recognized as a potential problem for most organizations. The purpose of this paper is to provide an introduction to the various aspects of sexual harassment as a problem: its definition (including both the economic injury and hostile environment types of harassment), its characteristics, and a discussion of how to deal with such problems in the workplace. Organizations are encouraged to act affirmatively by (1) establishing a policy prohibiting sexual harassment in the organization; (2) establishing and making employees aware of a grievance procedure for sexual harassment complaints; and (3) educating employees about the problem of sexual harassment through a combination of awareness and experiential training techniques.     

The large-sample power of the 195-1195-1195-1test for multidimensional contingency tables

Summary The asymptotic power of the frequency ² test depends on a noncentrality parameter, ,Mitra [1958] offered a general expression for , which is rather difficult to apply. This work provides simplified formulae for in various models associated with multidimensional contingency tables.     

Realization of closed,convex, and symmetric subsets of the unit square as regions of risk for testing simple hypotheses

It is well-known that the region of risk for testing simple hypotheses is some closed, convex, and (1/2, 1/2)-symmetric subset of the unit square, which contains the points (0, 0) and (1, 1). It is shown that for any such subsetR of the unit square and any atomless probability measureP on some -algebra there exists some probability measureQ on the same -algebra such thatR is the corresponding region of risk for testingP againstQ. This generalizes a result of [4] and is as a first step derived here for the special case, whereP is equal to the uniform distribution on the unit interval. The corresponding distributionQ is given explicitly in this case and the general case is treated by some well-known measure-isomorphism. This method of proof shows thatQ might be chosen to be of typeQ=Q ₁+(1–)Q ₂ for some satisfying 01, whereQ ₁ is a probability measure, which is absolutely continuous with respect toP andQ ₂ is a one-point mass.     

A conventional statistical problem with irregular minimax behavior

Summary The problem of testing the mean vector of the two dimensional circularly symmetrical normal distribution with unit variances, where the data consists of just one sample point inR ², is examined for stability of -maximin criteria. If the null hypothesisH ₀ is the one point set containing the origin and the alternative set equal to the whole ofR ²–H ₀, then the -maximin is not unique. If a zone of indifference _I containingH ₀ is introduced, then the problem of testingH ₀ againstR ² – _I can turn out to have a unique -maximin test. In the present paper we show a class of such _I for which this is the case. We show further that, given any -maximin test for testingH ₀ againstR ²–H ₀, there is a decreasing sequence of _I , with intersection equal toH ₀, for which the corresponding sequence of -maximin tests forH ₀ againstR ² – _I approaches a limit (in the usual weak star topology) which is not equivalent to .     

A note on randomized response models for quantitative data

Standard randomized response (RR) models deal primarily with surveys which usually require a yes or a no response to a sensitive question, or a choice for responses from a set of nominal categories. As opposed to that, Eichhorn and Hayre (1983) have considered survey models involving a quantitative response variable and proposed an RR technique for it. Such models are very useful in studies involving a measured response variable which is highly sensitive in its nature. Eichhorn and Hayre obtained an unbiased estimate for the expectation of the quantitative response variable of interest. In this note we propose a procedure which uses a design parameter (controlled by the experimenter) that generalizes Eichhorn and Hayres results. Such a procedure yields an estimate for the desired expectation which has a uniformly smaller variance.Acknowledgements We are grateful to two referees for their valuable and constructive comments.     

Charakterisierung der zweiseitigen Exponentialverteilung

LetF () be a family of distribution functions with a translation parameter such thatF (0) has a densityf. It is well known that each sample median is a maximum likelihood estimate of , iff belongs to the classE of all bilateral exponential densities which are symmetric about 0. Here it is shown that, conversely,fE holds, either if there is an evenm such that for every sample of sizem each median is an MLE of , or if there is an infinite setM such that for every sample of any sizemM at least one median is an MLE of .     

The transition from central planning to a market economy: A computable general equilibrium model

This paper develops a computable general equilibrium (CGE) model of the transition from a central planned economy to a market economy. The model is an extension of Wellisz and Findlay's (1986) model of the Soviet second economy. By distinguishing alternative assumptions about the disposition of the government budget, two model variants — the activist and non-activist — are analyzed. Equilibria of these model variants are computed for various parameter specifications of the Kantorovich ray, which represents the stringency of central planners' direction of the economy. The paper shows that increasing efficiency of the private sector, while it reduces the size of government subsidies to the state sector, does not necessarily increase the net government budget.     

Über eine Variante der empirischen Verteilungsfunktion

Summary If ₁, ₂,..., _n and ₁, ₂,..., _–1 are two ordered samples from a population with continuous distribution functionF(x), then the points ( _r ,r/n),r=1, 2,..., n–1 provide a better approximation ofF(x) than the points ( _r ,r/n),r=1, 2,..., n, in the following sense:A maximal upper deviation and a maximal lower deviation of more theny have — contrary to the points ( _r ,r/n) — equal probability for anyy0, if we deal with the points ( _r ,r/n). This probability is at least for ally in the interval , 1 less than the probability for a maximal upper deviation of more thany in the case of the points ( _r ,r/n). This is shown by a comparison of the Smirnow-Birnbaum-Tingey — formula with an analogous formula for the maximal one-sided deviations of the points( _r ,r/n).     

On estimating the variance of a finite population

Summary Observing that the estimator for a finite population variance as recommended byLiu [1974a, b] can sometimes become negative, we suggest a few non-negative alternative estimators and note some of their properties. UnlikeLiu we follow the conventional Bayesian approach to get another estimator with an optimal property of uniform admissibility.This paper, however, was prepared when the author worked in the Department of Economic Statistics, University of Sydney.