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The usual mathematical formulation of availability assumes an exponential distribution for failure and repair times. While such an assumption is sometimes correct for reliability, it is not valid for maintainability. This study was conducted primarily in order to verify that the lognormal distribution is suitable descriptor for corrective maintenance repair times, and to estimate the error caused in assuming an exponential distribution for availability and maintainability calculations when in fact the distribution is lognormal. Approximately 20 sets of existing maintainability demonstration repair time data, of essentially electronic systems, were analyzed using the methods of probability plotting and statistical testing for distributional assumption. The results show that the lognormal distribution assumption cannot be rejected in most of the cases, while the exponential distribution is rejected. However, the error caused when assuming an exponential distribution for MTTR is found to be negligible.

The determination of the luminosity function (LF) in gamma ray bursts (GRBs) depends on the adopted cosmology, each one characterized by its corresponding luminosity distance. Here we analyse three cosmologies: the standard cosmology, the plasma cosmology, and the pseudo-Euclidean universe. The LF of the GRBs is firstly modeled by the lognormal distribution and the four broken power law, and secondly by a truncated lognormal distribution. The truncated lognormal distribution fits acceptably the range in luminosity of GRBs as a function of the redshift.

Quantum Monte Carlo data are often afflicted with distributions that resemble lognormal probability distributions and consequently their statistical analysis can not be based on simple Gaussian assumptions. To this extent a method is introduced to estimate these distributions and thus give better estimates to errors associated with them. This method is applied to a simple quantum model utilizing the single-thread Monte Carlo algorithm to estimate ground state energies.

Quantum Monte Carlo data are often afflicted with distributions that resemble lognormal probability distributions and consequently their statistical analysis can not be based on simple Gaussian assumptions. To this extent a method is introduced to estimate these distributions and thus give better estimates to errors associated with them. This method is applied to a simple quantum model utilizing the single-thread Monte Carlo algorithm to estimate ground state energies.

I propose a method to fit the probability distribution function (hereafter PDF) of the large scale density field rho, motivated by a Lagrangian version of the continuity equation. It consists in applying the Edgeworth expansion to the quantity Phi=log rho - < log rho >. The method is tested on the matter particle distribution in two cold dark matter N-body simulations of different physical sizes to cover a large dynamic range. It is seen to be very efficient, even in the non-linear regime, and may thus be used as an analytical tool to study the effect on the PDF of the transition between the weakly non-linear regime and the highly non-linear regime.

Small sample size tables are presented which facilitate reliability-- mission life calculations at a given confidence level. The tables are calculated using the exact sampling distribution of the normal reliability estimator. A computer program is included which will enable one to calculate more extensive tables for different sample sizes and confidence levels.

How to quantify the impact of a researcher's or an institution's body of work is a matter of increasing importance to scientists, funding agencies, and hiring committees. The use of bibliometric indicators, such as the h-index or the Journal Impact Factor, have become widespread despite their known limitations. We argue that most existing bibliometric indicators are inconsistent, biased, and, worst of all, susceptible to manipulation. Here, we pursue a principled approach to the development of an indicator to quantify the scientific impact of both individual researchers and research institutions grounded on the functional form of the distribution of the asymptotic number of citations. We validate our approach using the publication records of 1,283 researchers from seven scientific and engineering disciplines and the chemistry departments at the 106 U.S. research institutions classified as "very high research activity". Our approach has three distinct advantages. First, it accurately captures the overall scientific impact of researchers at all career stages, as measured by asymptotic citation counts. Second, unlike other measures, our indicator is resistant to manipulation and rewards publication quality over quantity. Third, our approach captures the time-evolution of the scientific impact of research institutions.

This work explores the mathematical properties of a distribution introduced by Basu & Jones (2004), and applies it to model the stellar initial mass function (IMF). The distribution arises simply from an initial lognormal distribution, requiring that each object in it subsequently undergoes exponential growth but with an exponential distribution of growth lifetimes. This leads to a modified lognormal with a power-law tail (MLP) distribution, which can in fact be applied to a wide range of fields where distributions are observed to have a lognormal-like body and a power-law tail. We derive important properties of the MLP distribution, like the cumulative distribution, the mean, variance, arbitrary raw moments, and a random number generator. These analytic properties of the distribution can be used to facilitate application to modeling the IMF. We demonstrate how the MLP function provides an excellent fit to the IMF compiled by Chabrier (2005) and how this fit can be used to quickly identify quantities like the mean, median, and mode, as well as number and mass fractions in different mass intervals.

This paper studies a Stieltjes-type moment problem defined by the generalized lognormal distribution, a heavy-tailed distribution with applications in economics, finance and related fields. It arises as the distribution of the exponential of a random variable following a generalized error distribution, and hence figures prominently in the EGARCH model of asset price volatility. Compared to the classical lognormal distribution it has an additional shape parameter. It emerges that moment (in)determinacy depends on the value of this parameter: for some values, the distribution does not have finite moments of all orders, hence the moment problem is not of interest in these cases. For other values, the distribution has moments of all orders, yet it is moment-indeterminate. Finally, a limiting case is supported on a bounded interval, and hence determined by its moments. For those generalized lognormal distributions that are moment-indeterminate Stieltjes classes of moment-equivalent distributions are presented.

We present a Bayesian reconstruction algorithm that infers the three-dimensional large-scale matter distribution from the weak gravitational lensing effects measured in the image shapes of galaxies. The algorithm assumes that the prior probability distribution of the matter density is lognormal, in contrast to many existing methods that assume normal (Gaussian) distributed density fields. We compare the reconstruction results for both priors in a suite of increasingly realistic tests on mock data. We find that in cases of high noise levels (i.e. for low source galaxy densities and/or high shape measurement uncertainties), both normal and lognormal priors lead to reconstructions of comparable quality. In the low-noise regime, however, the lognormal model produces significantly better reconstructions than the normal model: The lognormal model 1) enforces non-negative densities, while negative densities are present when a normal prior is employed, 2) better traces the extremal values and the skewness of the true underlying distribution, and 3) yields a higher correlation between the reconstruction and the true density distribution. Hence, the lognormal model is to be preferred over the normal model, in particular since these differences become relevant for data from current and futures surveys.

The statistical distribution of biological phenomena is generally assumed to be normal or Gaussian. In some instances, however, the distribution is lognormal, that is, the logarithm of the variable is normally distributed. This report presents a technique for determining the mean and standard deviation of these distributions from the cumulative distribution function. The actual distribution is compared against a standardized cumulative distribution function of mean, 5, and standard deviation, 1, (probit transformation). The relationship between the random variable and the probit is found by linear regression and the corresponding mean and standard deviation determined. Two programs are presented for solving this problem, one written for a Hewlett-Packard 9820A programmable calculator with plotter, and the other written for a digital computer in Fortran IV. As an example of the use of these programs, the platelet size distribution in a fresh blood sample, obtained from rats subjected to hyperbaric exposures and subsequently decompressed, is solved. (Author)

Bibliography: p. 18-19

Bibliography: p. 18-19

ADA072320

This thesis developed modified goodness-of-fit tests for the three parameter lognormal distribution when the location and scale parameters must be estimated from the sample. The critical values were generated for the Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises goodness-of-fit tests, using the Monte Carlo methods of 5000 repetitions, to simulate samples of size 5,10,...,30 and the second part of the research also involved a Monte Carlo simulation of 5000 repetitions for sample sizes of 5,15, and 25. From these observations, the power of the test was determined by counting the number of times the modified goodness-of-fit tests incorrectly accepted null hypothesis that the distribution was lognormally distributed. The data used in this power comparison came from the lognormal distribution (shape = 1.0 and 3.0), Weibull, gamma, beta, exponential, and normal distributions. The third and and final phase of research was to determine the functional relationship, in any, between the known shape parameter and the new modified critical values. This was completed by using SAS. Keywords: Maximum likelihood estimation, Computer programs. (Author)

The usual mathematical formulation of availability assumes an exponential distribution for failure and repair times. While such an assumption is sometimes correct for reliability, it is not valid for maintainability. This study was conducted primarily in order to verify that the lognormal distribution is a suitable descriptor for corrective maintenance repair times, and to estimate the error caused in assuming an exponential. distribution for availability and maintainability calculations when in fact the distribution is lognormal. Approximately 20 sets of existing maintainability demonstration repair time data, of essentially electronic systems, were analyzed using the methods of probability plotting and statistical testing for distributional assumption. The results show that the lognormal distribution assumption cannot be rejected in most of the cases, while the exponential distribution is rejected. However, the error caused when assuming an exponential distribution for MTTR is found to be negligible.

This thesis develops unbiased single order statistic and m-order- statistic estimators of the scale parameter of a truncated lognormal probability density function. In this development it is assumed that the location parameter is known and that if it is not zero, a transformed variable with a location parameter approximately zero can be obtained. The method of development of the singleorder-statistic estimator was to consider the expected value of the i-th order statistic. The development of the m-order-statistic estimator utilizes the development of the single-order-statistic estimator and then considers the variance of the m-order-statistic estimator as a Lagrangian function which is minimized to obtain the necessary weighting factors. These weighting factors are then combined with the other coefficients to obtain the desired multipliers. All the multipliers used to obtain the estimators, the variance of the estimators and their relative efficiencies are presented in tabular form as appendices.

This thesis develops unbiased single order statistic and m-order- statistic estimators of the scale parameter of a truncated lognormal probability density function. In this development it is assumed that the location parameter is known and that if it is not zero, a transformed variable with a location parameter approximately zero can be obtained. The method of development of the singleorder-statistic estimator was to consider the expected value of the i-th order statistic. The development of the m-order-statistic estimator utilizes the development of the single-order-statistic estimator and then considers the variance of the m-order-statistic estimator as a Lagrangian function which is minimized to obtain the necessary weighting factors. These weighting factors are then combined with the other coefficients to obtain the desired multipliers. All the multipliers used to obtain the estimators, the variance of the estimators and their relative efficiencies are presented in tabular form as appendices.

This thesis compares two modified maximum likelihood (ML) estimation techniques against three minimum distance (MD) estimation techniques in application to the three parameter lognormal distribution. The three parameter lognormal distribution has a location parameter (xi) and two other parameters associated with the mean (micron) and standard deviation (delta) of its parent normal population. The first modified ML technique uses linear interpolation on order statistics to estimate location while the second ML technique uses the first order statistic as the location estimate. The remaining two parameters are calculated by using the location estimate in their respective censored or uncensored ML equations and solving for the parameters. Three MD techniques are used: Kolmogrov Distance, Cramer-von Mises Statistic, and the Anderson-Darling Statistic. The MD techniques refine the location estimates which are then used in the ML equations of the other two parameters to obtain their refined estimates. Monte Carlo analysis is used to accomplish the comparison of estimation techniques. Three measures of effectiveness are used to facilitate comparisons: mean square error, relative efficiency, and the Cramer-von Mises Statistic. Comparisons of these effectiveness measures across all cases reveal a clear superiority of the MD techniques over the modified ML techniques. (Author)

This thesis compares two modified maximum likelihood (ML) estimation techniques against three minimum distance (MD) estimation techniques in application to the three parameter lognormal distribution. The three parameter lognormal distribution has a location parameter (xi) and two other parameters associated with the mean (micron) and standard deviation (delta) of its parent normal population. The first modified ML technique uses linear interpolation on order statistics to estimate location while the second ML technique uses the first order statistic as the location estimate. The remaining two parameters are calculated by using the location estimate in their respective censored or uncensored ML equations and solving for the parameters. Three MD techniques are used: Kolmogrov Distance, Cramer-von Mises Statistic, and the Anderson-Darling Statistic. The MD techniques refine the location estimates which are then used in the ML equations of the other two parameters to obtain their refined estimates. Monte Carlo analysis is used to accomplish the comparison of estimation techniques. Three measures of effectiveness are used to facilitate comparisons: mean square error, relative efficiency, and the Cramer-von Mises Statistic. Comparisons of these effectiveness measures across all cases reveal a clear superiority of the MD techniques over the modified ML techniques. (Author)

The shape of the probability distribution function (PDF) of molecular clouds is an important ingredient for modern theories of star formation and turbulence. Recently, several studies have pointed out observational difficulties with constraining the low column density (i.e. Av