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Empirical firm-size data show concavity when size is plotted against rank (largest firm = rank 1) on a log-log scale, a departure from the Pareto distribution which shows a straight line. Two reasons for this departure are proposed. One is an exponential decay in the effect of a unit increase in size upon the firm's future growth. It is shown analytically that the greater the decay rate, the greater the concavity. The other is an effect of mergers and acquisitions. Based on Federal Trade Commission data, the actual firm-size distribution in 1969 is compared with a hypothetical one where all mergers and acquisitions in 1948-69 were "undone" to demonstrate their effect.

SummaryThe authors have studied the combined data on claims in fire insurance of dwelling houses reported 1958-1969 by Swedish fire insurance companies The claims were cleared of deductibles and adjusted according to a suitable index Only losses above the largest deductible (in real value) applied during the observation period were included.The material contains four different classes according to the fire resistibillty of the building construction For international comparisons, the pure classes B1 (“stone” dwellings) and B4 (wooden houses) are of interest The distribution of the claims could be well approximated by the log-normal distribution in B1and by the Pareto distribution in B4 An equally good or better fit was obtained by assuming the original loss, reported or not, being distributed according to these distributions and applying the distributions, conditioned by the loss being larger than the deductible in both cases the distribution parameters are functions of the insurance amount in such a way, that the mean value of the loss is described as a power of this amount.The authors refrain from any theoretical arguments for the general applicability of the distributions used They observe, however, the good approximation by wellknown parametric distributions which facilitates many actuarial taks, such as the determination of first loss premiums, deductible premium factors, excess-of-loss premiums etc The agreement between model and reality make these functions fit for use in the models underlying the general risk theory and in the more comprehensive models of the non-life insurance business.

Suppose that in a random sample of size n from a population with probability density function f(x), the order statistics are X(1)

Given a random sample from a Pareto type distribution with the cdf F(x) = 1 — [1 + (x — α)/β]-y for x ≥ α, the problems of estimating (1) the scale parameter β when α and y are known, (2) the location parameter α when β and y are known, (3) α and β when y is known, are considered. Particular attention is given to the best linear unbiased estimates based on the complete sample and (for (1) and (3)) the asymptotically best linear unbiased estimates based on a few selected order statistics, but some alternative estimates are also considered. The problem of optimum spacing of the order statistics is solved analytically and numerically.

Price changes over a fixed number of transactions may have a Gaussian distribution. Price changes over a fixed time period may follow a stable Paretian distribution, whose variance is infinite. Since the number of transactions in any time period is random, the above statements are not necessarily in disagreement. A possible explanation is proposed by Taylob, and then shown by Mandelbrot to be intimately related to an earlier discussion of the specialists' function of ensuring the continuity of the market.

From a comparison between high altitude measurements of the polarization of visible radiation over various natural surfaces with those computed for molecular and turbid atmospheres, some estimates of the aerosol parameters such as the exponent of the power law distribution and the limiting radii for the particles obtained are presented.

A queuing system is described in which service times are conditioned upon a random parameter μ, such that the conditional service distribution is exponential and μ has a gamma density. It is shown that the resultant unconditional distribution of service times is Paretian. Several measures of effectiveness are discussed and the question of statistical estimation of service parameters is also explored.

Being one of the oldest branches of operations research, actuarial science has accumulated a substantial store of knowledge about the risks associated with living. The present paper will discuss one such question. Although it is relative to a specific problem of fire casualty, it illustrates more generally why the Paretian distribution of incomes and fortunes should constitute “a source of anxiety for the risk theory of insurance.” Very similar mechanisms apply in many other problems.

The density distribution and the zenith angle distribution of extensive air showers were measured at 2,760 meters elevation, geomagnetic latitude 25°N, by a cloud chamber in connection with a hodoscope consisted of Geiger counter trays shielded by lead absorbers. Results obtained were as follows: (1) The integral density spectrum of the extensive air showers followed the power law distribution and its exponent was 1.70±0.13, (2) The projected angle distribution of the extensive air showers followed the empirical law of cos n θ, where n =6.9±1.3.

It is well known that the family of Pareto distributions with densities \begin{equation*}\begin{align*}f(x; a, p) = pa^p/x^{p+1},\qquad x > a > 0, \\ = (1.1) \\ 0,\qquad x \leqq a, p > 0,\end{align*}\end{equation*} provides reasonably good fits to many empirical distributions, e.g., to distributions of income and of property values. In most of these cases, ancillary information is present, which could be utilized if an appropriate multivariate Pareto distribution were available. The objects of this note are (i) to suggest two families of bivariate Pareto distributions with the property that both marginal distributions are of univariate Pareto form; (ii) to extend these to multivariate forms; and (iii) to discuss estimation of the parameters in the bivariate distributions.

One statistical model that has been proposed for the generation of errors in telephone circuits consists of errors with successive inter-arrival times drawn independently from a Pareto distribution, resulting in errors that tend to occur in bursts. These error statistics are applied to a class of burst correcting codes due to Hagelbarger, with particular attention to codes capable of correcting very long error bursts. For such codes, asymptotic expressions are derived for the expected number of output errors per data digit error and also the expected number of false corrections per data digit error. These expressions are what one might intuitively expect, indicating that the results obtained here can perhaps be extended to other codes by simple intuitive reasoning.

The spectral distribution and its time development for X-rays below 10 A during solar flares are deduced from the measurements of integrated flux levels by satellite-borne detectors in the 0.5–3 A and 1–8 A bands. An exponential distribution and a power law distribution are considered, as well as a composite spectrum with two different electron temperatures in the bands 0.5–3 A and 3–8 A. Analysis of X-ray response for selected flare events gave spectral temperatures of about (1–3) × 107 °K for λ 3 A. The power law representation gave a distribution of λm, with m having values between 1–2 at the peak of the flare. Gradual changes in spectral temperatures are shown for some events.

In a coherent search rader, the pulse-to-pulse Doppler shift of a signal is generally not known a priori. Given the distribution of this parameter, the best test variable for detection is the average of the likelihood ratio with respect to target Doppler frequency. Most coherent search radars employ a maximum-likelihood ratio detector, that is, a bank of independent Doppler filters, for detection. The average-likelihood and maximum-likelihood tests are compared here for a target with the Rayleigh amplitude distribution. It is shown that, over a wide range of detection and false-alarm probabilities, the performances of the two tests do not differ significantly. For this target model, the likelihood ratio has the Pareto distribution, which arises in some statistical problems in economics. The new results obtained here for the distribution of the sum of two or more Pareto-distributed variables are of considerable general interest.

When firm sizes are distributed according to the Pareto law, mergers or acquisitions may either increase or decrease business concentration as measured by the slope, on a double-log scale, of the distribution. Whether mergers and acquisitions will increase or decrease concentration depends upon the size distribution of the acquired assets among the firms that survive. Some data from the U.S. economy are consistent with a set of conditions under which industrial concentration is unaffected by mergers, thus offering an explanation for the observed fact that the overall concentration of U.S. firms, as measured by the slope of the Pareto curve, has not changed substantially since the turn of the century. The findings challenge the widely held view that mergers held view that mergers "obviously" increase concentration and reduce competition.

Berger and Mandelbrot in 1963 proposed a particular renewal process as a model for the occurrence of errors in data transmitted over telephone circuits. Besides the assumed independence between the successive intervals between errors, they assume that the intervals have a Pareto distribution. Their graphical analyses of large amounts of data indicated departures from the model which Mandelbrot proposed in 1964 to account for in an extended model. Some simple formal statistical procedures are given for analyzing this sort of data, procedures that are not affected by the possibility that the population mean-interval-between-errors is infinite. The departures from independence of intervals noted by Berger and Mandelbrot are formally verified from the analysis of data from a single data transmission test. A separate analysis of another set of data is also made and the results are compared to see what features of the error patterns are general, and what features are particular to different transmission conditions. In both sets of data analyzed, the outstanding feature detected is the strong positive correlation between successive long intervals between errors. Evidence is also found which indicates that the upper tail of the marginal distribution of intervals between errors does not follow a hyperbolic law.

A sequential sampling procedure is developed to estimate the scale parameter θ of the Pareto distribution when the shape parameterθ is unknown using the cost function , where A is a positive constant. The probability distribution of the stopping time for this procedure is tabulated and the expected stopping time and cost under the sequential sampling procedure are computed and compared with the optimum sample size and minimum cost under the fixed sample size procedure (for the case when the shape parameter is known).

When the cumulative frequency of the best hundred results of a number of track and field events each year is plotted on a log-log scale, a straight line is generally obtained. Though this distribution is formally similar to the Pareto distribution, which, e.g., describes the distribution of income, the coefficient b of the regression equations is much larger in the sports, from −16 to +116, than in the distribution of income, where it is approximately −1.5. The distribution of athletic ability thus comes nearer to a semilog relation. A high value of b means that the difference between the result of the best and the 100th man is small in relation to the total time used, centimeters jumped, etc., and vice versa. The highest values of b were obtained in events—like 100-meter and 800-meter runs—in which the result is evidently determined to a large extent by relatively few physiological factors, like the speed of muscle contraction, maximum oxygen uptake, or oxygen debt. The lowest values of b were a...

In Part 1 a wide-band spectrum of atmospheric pressure extending from periods of 1 week to 0.2 second is shown. The various frequency ranges are discussed with particular attention given to the midfrequency range. It is shown that convective activity and internal gravity waves can greatly modify the normal spectral distribution at midfrequencies and a specific example is discussed in detail. The amplitude spectrum of waves on the inversion layer is computed for this example from the surface pressure perturbations. In Part 2 evidence is presented indicating that the spectra of passive scalars aloft show a systematic departure from the power law distribution to be expected for a fully developed inertial subrange. The band of wave numbers analyzed in this report corresponds to scale sizes of 3 to 1000 feet. The ‘energy’ at the larger scales (small wave numbers) is quite variable and depends on altitude and atmospheric stratification. At the small scales the ‘energy’ is remarkably constant, and a source of refractive index variance at scales of 10 to 100 feet is indicated. Very thin layers of exceptionally large mean-square fluctuation are found in regions of gradient of scalars.

Order statistics drawn from a sample of independent random variables with the distribution function F are considered. Excluding lattice distribution functions F, the difference and the order statis...

It is well known that a Pareto distribution provides reasonably good fit to distributions of income and of property values. Seal [13] conceived the idea of using the discrete Pareto distribution, to represent the distribution of multiplicate life insurance policies among the policy holders of one or more offices. For detailed arguments on the existence of such distributions in economic life the reader is referred to the discussions by Davis [1], Hagstroem [2, 3], and Mandlebrot [8].

It is well known that a Pareto distribution provides reasonably good fits to distributions of income and of property values. A detailed discussion on the existence of such distributions in economic life is given by Hagstroem in [1, 2].

The method of flame scintillation spectral analysis has been adopted for the determination of the size distribution of sodium-containing particles. Near the seashore, these aerosols are identical to sea salt particles, amount to approximately 5% of the total Aitken particle count, and exhibit a power law distribution with β = 3.8. Measurements made at Mauna Loa Observatory (above the trade wind inversion) indicate a similar size distribution. However, only about 1% of the concentration is transported through the inversion, and removal efficiency tends to favor the removal of larger particles.

Showers of muons which result from interactions of primary cosmic rays high in the atmosphere appear deep underground as bundles of nearly parallel muons. For threshold energies of 1 TeV and arrival zenith angles of 45 degrees, those events having a small mean number of muons are described by a power law distribution in shower size having an exponent of ≅−3.7 and a radial density distribution that decreases with distance R from the shower axis a little less rapidly than exp(− R /5.4 m). Comparison of new data on four muon spatial correlations in terms of the same phenomenology with data on two and three muon spatial correlations from previous experiments suggests that the shower size distribution steepens and the shower radial density distribution broadens with increasing shower size (i.e., with increasing mean energy of the primary cosmic ray at a given threshold energy of the muons observed). These results might be explained by a basically geometrical effect due either to a total inelastic cross section which rises with energy or by an enhanced contribution from the interactions of secondaries. If the effect does not have a geometrical origin, then it may be necessary to reformulate the input to shower development calculations which have assumed that interaction products are described by factored, uncorrelated single particle statistical distributions. The relationships between the inclusive distributions predicted by theory and those of the empirical phenomenology are explored in some detail.

The problem of estimating the scale parameter in the Pareto distribution from grouped observations is considered. Several estimators—the maximum likelihood estimator and four variants of least squares—are evaluated. Most of these have identical BAN properties but require nonlinear computations. A linearized BAN estimator is constructed, and numerical illustrations are provided.

A statistical model for the occurrence of errors in data transmission over telephone circuits was recently proposed by Berger and Mandelbrot. In their model the intervals between successive errors are distributed independently according to the Pareto distribution t^{-\alpha} . This paper presents an analysis based on the Pareto model and compares the results with published measurements on telephone networks. Statistics, such as the probability of zero, one or two errors in a block of digits, and the probability of error-free transmission runs for no-error correction or single-error correction codes are derived. Comparison with experiments indicates that the Pareto distribution provides an excellent representation of the error statistics.

The evaluation of multiple integrals which occur in order statistics distribution theory is involved due to the fact that the integration is to be carried on over an ordered range of variables of integration. This difficulty is sometimes completely obviated by transforming the ordered variates to the unordered ones. Several such transformations are available in the Theory of Multiple Integrals. In previous papers [2, 3] the author used one such transformation, and gave alternative simplified proofs of several known results in the distribution theory of order statistics from the exponential and the power function distributions. In this paper we use such a known transformation to derive moments (and distributions if necessary) of order statistics from the Pareto distribution. Malik [4] has derived moments of order statistics from this distribution without the transformation of the ordered variates to the unordered ones. The process of direct integration used by Malik becomes complicated for dealing...

Use of the functional relationship between the exponential and the Pareto and limited distributions enables one to obtain conditional maximum-likelihood (ML) estimators, from singly censored samples, of the shape parameters of the Pareto distribution F1(y,?,K) = 1 - (y - ?)?K and the limited distribution F2(x,?,K) = 1 - (? - x)K by a simple transformation of the corresponding estimator of the scale parameter of the exponential distribution ??mn, based on the first m order statistics of a sample of size n. Use is made of the fact that K?mn|? = 1/??mn and K?mn|? = 1/??mn, where 2m?mn/? has the x2 distribution with 2m degrees of freedom, to set confidence bounds on the shape parameter K of the Pareto and limited distributions. The probability densities of K?mn|? and K?mn|?, which for a given m are the same for any n ? m, are obtained by a simple transformation of that of ??mn. The expected values of K?mn|? and K?mn|? are determined and from them the unbiasing factors by which the ML estimators must be multiplied to obtain unbiased estimators K?mn|? and K?mn|?. Expressions for the variances of the estimators and for the Cramer-Rao lower bound are found. A section on numerical examples is included.

This article owes its inspiration to one of Tinbergen’s most original contributions to economic theory [9], in which he endeavours to derive the distribution of labour incomes from the choice of occupation by individuals. This is also the aim of the present paper, which, however, uses somewhat different techniques of analysis and has a more limited scope. Whereas Tinbergen relied heavily of the differential calculus, I use mostly linear inequalities and integration in order to take account of the fact that most individuals have only one job at a time and to permit analysis of what has become known as a ‘continuum of traders’. Unlike Tinbergen, moreover, I shall not consider individuals’ non-monetary preferences concerning different types of work.

A positive random variable $X$ whose mean exists, has a Pareto distribution if, and only if, $E(X \mid X > x) = h + gx$ for $g > 1$. This characterization was motivated by the fact that the Pareto distribution has been widely used to model income. Now, suppose that individuals under-report their income for income tax purposes. If one assumes that for a given income, the average under-reporting is a constant fraction of the amount by which the income exceeds the tax exempt level, then the average under-reporting error for a given reported income is a linear function of the reported income if, and only if, incomes follow a Pareto distribution.

The research topics discussed in the paper include: Variational methods in statistics, inference in Markov dependent firing distribution; paired comparisons; sampling theory and its applications to the development of new sampling strategy for monitoring air pollutants; and, Pareto distribution.

In this paper we characterise the Pareto distribution using the properties of the order statistics of a random sample of size N. A similar characterisation theorem has been recently proved by Malik [2].

T HE most frequently used estimating technique for applied economic research has been ordinary least squares (OLS). There are two theoretical justifications for its use. First, the Gauss-Markov theorem suggests that OLS estimators will outperform all other techniques in the class of linear unbiased estimators.1 Secondly, under the assumption that the error terms are normally distributed, OLS estimators can be derived as the maximum likelihood estimates. Frequently when OLS is introduced in elementary texts, the method of minimizing the sum of absolute deviations (MAD) is presented for comparison purposes, but rarely is it given serious consideration for applied uses.) Certainly one reason for such behavior stems from previous evaluations of OLS versus MAD estimators. Asher and Wallace (1963) found that the use of MAD meant " one should be prepared to give up considerable efficiency . " 3 More recently Glahe and Hunt (1970), suggested several estimators derived under the general minimization criterion. In contrast to the Asher-Wallace study, the Glahe-Hunt model was a two-equation linear simultaneous system. Their results show that neither form of absolute deviation estimator outperformed either OLS or two-stage least squares.4 The purpose of this paper is to suggest that in at least one aspect these comparisons have given OLS a differential advantage. That is, both of these studies employed errors for their hypothesized models which were drawn from normal distributions. Consequently the OLS estimators are both maximum likelihood and best linear unbiased estimators (BLUE) under such circumstances. Since recently published works by Zeckhauser and Thompson (1970), Fama (1965) and others have called into question the assumption of normally distributed error terms, attention has begun to shift to other alternatives.5 Blattberg and Sargent (1971) have examined three techniques including both OLS and MAD when the errors are drawn from a stable Paretian distribution. Their findings indicate that the MAD estimator ". . . performs sufficiently well that it deserves further study and elaboration." G Accordingly we have chosen to explore the relative merits of OLS and MAD for a single equation model whose errors are drawn from a double exponential parent distribution. This distribution was chosen because both estimators will exhibit theoretically desirable properties. OLS remains the BLUE estimator, while MAD is the maximum likelihood estimator. Furthermore, this distribution is one member of the power distribution suggested by Zeckhauser and Thompson as an alternative to the normal. This paper is divided into, three sections. The first describes the design of the experiments. Section II presents the empirical results and the last summarizes the primary findings of the paper.

A FUNCTION representing the distribution of income in a society can serve a number of purposes. It may be used to smooth out irregularities in the observed income distribution caused by the misreporting of income. In this role it is similar to the graduation formulae used in demographic work to correct an age distribution distorted by mistatements of age. An income function may also form the basis of a model explaining how an income distribution is generated. Interest here lies in the success with which the model generates a distribution close to that of the observed values and in the meaning that can be ascribed to the parameters in the model. In addition, an income function can assist in the analysis of income distributions by highlighting the more important characteristics of such distributions and providing measures, which can be compared spatially or temporally, of those characteristics. Other uses can no doubt be suggested. For a function to serve these purposes adequately, it is desirable that it should approximate observed distributions of income closely when particular values, usually estimated from the observed data, are given to the parameters. This criterion is the least satisfied by the formulae that have been suggested to date except, perhaps, over limited segments of the income range. The Pareto curve fits income distributions at the extremities of the income range but provides a poor fit over the whole income range. The log normal (or Gibrat) distribution fits reasonably well over a large part of the income range but diverges markedly at the extremities. A function suggested

ONE OF THE common problems facing an economist dealing with production functions is the problem of aggregation of factors. In a rather neglected paper, Houthakker advances an ingenious approach for explaining the possibility of finding a neoclassical production function for an industry even when production within each of the firms (or "cells")3 is done according to a fixed coefficients production function. These fixed proportions vary in a regular way from one cell to another so that the overall input-output relationship takes the form of a regular neoclassical production function. As Solow notices in a survey article on production functions4 this paper has been forgotten and not followed in any direction. In this note we try to reverse Houthakker's procedure and to show how each neoclassical production function implies some density function or distribution function over the cells. We here do it for CES production functions, but it will be obvious that the same method applies to any production function. Following Houthakker we normalize the cells so that each of them is capable of producing one unit of output. Each cell has a requirement, say t, of the variable factor and this requirement varies from one cell to another. If the wage rate in terms of output produced is p then all the cells with tp < 1 will produce a unit of output, all others will be idle. Assume that we are given a density function of the various cells by g(t). Output produced will then be Q = f Pg(t)dt and the input used A f f'IP tg(t)dt. By eliminating i/p one gets a relationship between Q and A. In this way Houthakker has shown that a Pareto distribution implies a CobbDouglas production function. Notice that the relationship between Q and A the cumulated product and factor used-is the familiar Lorenz curve. Assume that the overall relationship between output and the variable factor follows a CES production function with elasticity of substitution (a) smaller than 1