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Volume 38 - Issue 2 - December 2013
It is known that a large part of data in the real world has the property that the first digit 1 appears about 30% of the time, the first digit 2 appears about 17% of the time, and so on, with the first digit 9 appearing about 5% of the time. This phenomenon is known as Benford's law. In this paper we give an account of the various explanations suggested for its occurrence, and the subsequent rejection of them by others in the field. We also give an account of two recent explanations, one when the data have been growing exponentially over time, and the other where we have the situation of selecting house numbers at random from streets, with the number of houses in the street itself a random number.
Thomas Koshy and Zhenguang Gao
We investigate some divisibility properties of Catalan numbers with Mersenne numbers Mk as their subscripts, and then extend them for k = Fn, Ln, Pn, Qn, and Cn, where Fn, Ln, Pn, Qn, and Cn denote the nth Fibonacci, Lucas, Pell, Pell–Lucas, and Catalan numbers, respectively. We also compute CM_k moduli 4, 8, and 64; and CM_k+1 modulo 128.
Thomas Koshy, Zhenguang Gao and Angelo Di Domenico
We employ a Vandermonde-like technique to develop a combinatorial identity and then use it to establish a new result. We then deduce three Catalan formulas from the latter identity.
Leslie M. Golden
The technique of the linear programming model, which determines the optimal means of allocating finite resources among competing entities, has been applied to a wide variety of business, communication, engineering, and scientific problems. Although its applicability to problems of public policy is recognized, it has been underutilized despite its great promise in aiding the setting of budgets, decision making, and promoting intergovernmental cooperation. Here we show how it can be used not only to allocate finite resources but also to increase government efficiency, provide a documented, transparent objectivity, and improve government accountability while retaining the ability to incorporate different philosophies of government. We present a formalism in which the objective function incorporates the priorities set for each proposed project and the decision variables are the number of units of each project for which resources are requested. An index is calculated which quantifies the level of optimization.
The odds-theorem and the corresponding solution algorithm (odds-algorithm) are tools to solve a wide range of optimal stopping problems. Its generality and tractability have aroused much attention. (Google for instance `Bruss odds' to obtain a quick overview.) Many extensions and modifications of this result have appeared since its publication in 2000 (see Bruss (2000)). This article reviews the important new developments and applications in this field. The spectrum of application comprises fields as different as secretary problems, more general stopping problems, robotic maintenance problems, and compassionate use clinical trials, amongst others. This review also includes a new contribution of our own.
M. Shelton Peiris
This paper considers an application of estimating functions (EF) in the estimation of regression models with heteroscedastic errors. It is shown that the class of estimators using the estimating function approach (on the choice of the estimating function) has smaller variance than that of the usual least squares estimators. An example is added to illustrate this theory.
L. Bryan and W. F. Scott
We present extensive tables for the Kolmogorov–Smirnov and the Cramér–von Mises tests for exponentiality, with unknown population mean. The tables were constructed using Monte Carlo methods and are in good agreement with those already published (see Lilliefors (1969), Durbin (1975), Stephens (1986), and Weibull.com (2009)). We also mention a method of testing the fit of a Weibull distribution (with known shape parameter, but unknown scale parameter) and of a Pareto distribution. A comparison of the power of the two tests is given, and there is a practical example.
Jonathan R. Bradley and David L. Farnsworth
Ignoring possible misclassifications when testing for a proportion can lead to erroneous decisions. A statistical test is described that incorporates misclassification rates into the analysis. Easily checked safeguards that ensure that the test is appropriate are given. Additionally, the test provides a procedure when the hypothesis stipulates that the proportion is zero. Applications of the test are illustrated with examples which show that it is practical. Comprehensive guidance is supplied for the practitioner.