A. Hald

The Compound Hypergeometric Distribution and a System of Single Sampling Inspection Plans Based on Prior Distributions and Costs

The main results of this paper, apart from the limit theorems, were given in two lectures at the University of London, and in a seminar at Imperial College of Science and Technology, London, January 1959. The complete paper was presented for discussion in a meeting at Imperial College in February 1960. The paper reviews present sampling inspection plans for attributes placing particular emphasis on their underlying assumptions. A model is then proposed based upon prior distributions and costs, and optimum sampling plans are derived which minimize the average costs for any prior distribution. Tables and examples are provided.

The Determination of Single Sampling Attribute Plans with Given Producer's and Consumer's Risk

Abstract : The purpose of the paper is (1) to give a survey of exact and approximate solutions already known, (2) to discuss the exact solution for the binomial case by means of a new table, and (3) to present some new approximate solutions and discuss their accuracy.

Bayesian Single Sampling Attribute Plans for Continuous Prior Distributions

The purpose of the paper is to derive and discuss the properties of a system of single sampling attribute plans obtained by miniiizing the average costs under the assumptions that sampling and decision costs are linear in lot size, sample size, and the number of defectives in the lot and the sample, that sampling is without replacement, and that the distribution of lot quality is a mixed binomial distribution, i.e., each lot is produced by a process in binomial control but the process average varies from lot to lot according to a frequency distribution which is assumed to be differentiable in the neighbourhood of the break-even value. It is shown that the optimum sample size is approximately a linear function of the square root of the lot size, and that the optimum acceptance number is approximately a linear function of the sample size. The accuracy of the approximation has been evaluated numerically for the beta distribution as prior. Some auxiliary tables are given with examples of applications. A short...

A Note on the Determination of Attribute Sampling Plans of Given Strength

Simple and accurate approximation formulas for the determination of single sampling attribute plans in the binomial and hypergeometric case are given based on the Poisson solution. The solution is generalized to multiple sampling.

On the early history of life insurance mathematics

Abstract An outline is given of the history of life insurance mathematics in the latter half of the 17th century, stressing basic ideas and results and adding a few new interpretations.

On the History of the Correction for Grouping, 1873–1922

The correction for grouping is a sum of two terms, the first depending on the length of the grouping interval, the second being a periodic function of the position. Thiele (1873) studied the second term, but missed the first. Sheppard (1898) studied the first term, but missed the second. Bruns (1906) derived the first term as the aperiodic term of a Fourier series and the second as the sum of the periodic terms. He found the correction to the coefficients of the Gram-Charlier series and proved that the second term is negligible for a grouped normal distribution with at least eight groups. Independently, Fisher (1922) used the same method to derive the correction to the moments. For the normal distribution with a grouping interval less than the standard deviation Fisher proved that the second term is negligible compared with the first and with the standard error of the first four moments. Moreover, he proved that the estimates of the mean and the standard deviation obtained by the method of moments for a grouped sample with Sheppards corrections have nearly the same variances as the maximum likelihood estimates, thus providing a new and compelling reason for using Sheppards corrections.

Galileo's Statistical Analysis of Astronomical Observations

Summary It is shown that Galileos statistical analysis of astronomical observations in 1632 contains the rudiments of a theory for comparing hypotheses by means of the sums of the absolute deviations of the observations from the hypothetical values. Galileo did not state the method unambiguously and he did not carry out all the necessary calculations.

Some Limit Theorems for the Dodge-Romig LTPD Single Sampling Inspection Plans.

Explicit asymptotic formulas for sample size as function of lot size and for accepe ante number as function of sample size are derived for the Dodge-Romig LTPD single sampling inspection plans. Numerical investigations show that a simple finite population correction of the asymptotic formulas leads to a good approximation to the Dodge-Romig solution. Tables and graphs are provided for the asymptotic solution. Asymptotic formulas are also given for the producers risk and for the minimum amount of inspection for lots of process average quality. The main results are that sample size asymptotically is proportional to the logarithm of lot size and that the highest allowable fraction defective in the sample converges to the tolerance fraction defective, the difference being of order 1/√n.

Computerized hot-wire anemometry--principles of calculation.

Principles of calculation of respiratory parameters based on a hot-wire anemometer with special reference to computer monitoring were evaluated. Flow-rate, gas-pressure, and flow-direction signals were recorded simultaneously on magnetic tape. Subsequent quantitative analyses were performed on a general purpose digital minicomputer. An analysis epoch of 256 s was selected from the 3 channels. After identification of one cycle baseline values of flow-rate and pressure were determined. Different time-lags in one respiratory cycle (inspiratory time, pause time and expiratory time) could be determined. Inspiratory and expiratory volumes were obtained by integration. Peak of the thoracic cage and the lungs were calculated using the above mentioned parameters. Finally, the respiratory frequency was calculated.

A History of Mathematical Statistics from 1750 to 1930.

DIRECT PROBABILITY 17501805. Some Results and Tools in Probability Theory (By Bernoulli, de Moivre, and Laplace). The Distribution of the Arithmetic Mean, 17561781. Chance or Design: Tests of Significance. Theory of Errors and Methods of Estimation. Fitting of Equations to Data, 17501805. INVERSE PROBABILITY BY BAYES AND LAPLACE, WITH COMMENTS ON LATER DEVELOPMENTS. Induction and Probability: The Philosophical Background. Bayes, Price, and the Essay, 17641765. Equiprobability, Equipossibility, and Inverse Probability. Laplaces Applications of the Principle of Inverse Probability in 1774. Laplaces General Theory of Inverse Probability. The Equiprobability Model and the Inverse Probability Model for Games of Chance. Laplaces Methods of Asymptotic Expansion, 1781 and 1785. Laplaces Analysis of Binomially Distributed Observations. Laplaces Theory of Statistical Prediction. Laplaces Sample Survey of the Population of France and the Distribution of the Ratio Estimator. THE NORMAL DISTRIBUTION, THE METHOD OF LEAST SQUARES, AND THE CENTRAL LIMIT THEOREM. GAUSS AND LAPLACE, 18091828. The Early History of the Central Limit Theorem, 18101853. Derivations of the Normal Distribution as a Law of Error. Gausss Linear Normal Model and the Method of Least Squares, 1809 and 1811. Laplaces Large-Sample Theory of Linear Estimation, 18111827. Gausss Theory of Linear Unbiased Minimum Variance Estimation, 18231828. SELECTED TOPICS IN ESTIMATION THEORY 18301930. On Error and Estimation Theory, 18301890. Bienaym?s Proof of the Multivariate Central Limit Theorem and His Defense of Laplaces Theory of Linear Estimation, 1852 and 1853. Cauchys Method for Determining the Number of Terms to be Included in the Linear Model and for Estimating the Parameters, 18351853. Orthogonalization and Polynomial Regression. Statistical Laws in the Social and Biological Sciences, Poisson, Quetelet, and Galton, 18301890. Sampling Distributions under Normality. Fishers Theory of Estimation, 19121935, and His Immediate Precursors. References. Index.