Parent Distribution Research Articles

Bayesian Alternatives to the F Test for Two Population Variances

Bayesian alternatives to the classical F test comparing two population variances are explored. Shoemaker (2003) suggested two adjustments to the F test due to it being very sensitive to the normal assumption of the two populations. A simulation study is performed to compare the Bayesian alternatives to the F test and Shoemaker's adjusted F tests as well as to the Levene/Brown–Forsythe and the squared rank nonparametric tests. The Bayesian alternatives assume a normal parent distribution and non informative priors and the conjugate prior for the variances; in addition, an exponential power distribution is considered as the parent distribution with a non informative prior for the variances. The latter looks to be very promising provided that a suitable value of a parameter which measures the extent of non normality is chosen.

Some characterization and ordering results based on entropies of current records

The aim of this paper is to investigate the entropy properties of current records. Several characterizations are obtained based on the entropy of these statistics. It is proved that the equality of the entropy of the endpoints of record coverage is a characteristic property of symmetric distribution. It is shown that the parent distribution can be identified uniquely by the entropy properties of the endpoints of record coverage. Considering the records coming from distribution with decreasing (increasing) density, the monotonicity of entropy of current records is discussed. We also prove that the dispersive ordering of the parent distributions implies the entropy ordering of their respective current records.

Pitman closeness of record values to population quantiles

In this paper, we examine the Pitman closeness of record statistics to the population quantiles of a location-scale family of distributions and study its monotonicity properties. Even though in general it depends on the parent distribution, exact expressions are derived for the required probabilities in the case of Uniform (−1,1) and exponential distributions. For the population median, it is shown that the first upper record is the Pitman-closest among all upper record values. Moreover, for the population median, in the case of symmetric distributions, the Pitman closeness probabilities of records are shown to be distribution-free and explicit expressions are also derived for these probabilities.

Model selection techniques for the frequency analysis of hydrological extremes

The frequency analysis of hydrological extremes requires fitting a probability distribution to the observed data to suitably represent the frequency of occurrence of rare events. The choice of the model to be used for statistical inference is often based on subjective criteria, or it is considered a matter of probabilistic hypotheses testing. In contrast, specific tools for model selection, like the well‐known Akaike information criterion (AIC) and the Bayesian information criterion (BIC), are seldom used in hydrological applications. The objective of this study is to verify whether the AIC and BIC work correctly when they are applied to identifying the probability distribution of hydrological extremes, i.e., when the available samples are small and the parent distribution is highly asymmetric. An additional model selection criterion, based on the Anderson‐Darling goodness‐of‐fit test statistic, is here proposed, and the performances of the three methods are compared through an extensive numerical analysis. The capability of the three criteria to recognize the correct parent distribution from the available data samples varies from case to case, and it is rather good in some cases (in particular when the parent is a two‐parameter distribution) and unsatisfactory in others. An application to flood peak time series from 1000 catchments located in the United Kingdom provides some further information on the qualities and drawbacks of the considered criteria. From the numerical simulations and data‐based analyses it can be concluded that the three model selection techniques considered here produce results of comparable quality.

Some notes on the location–scale Cucconi test

The best known and most used rank test for the location–scale problem is due to Lepage [Y. Lepage, A combination of Wilcoxon's and Ansari–Bradley's statistics, Biometrika 58 (1971), pp. 213–217.], but this paper is focused on the location–scale rank test of Cucconi [O. Cucconi, Un nuovo test non parametrico per il confronto tra due gruppi campionari, Giorn. Econom. XXVII (1968), pp. 225–248.], proposed earlier but not nearly as well-known. The test is of interest because, contrary to the other location–scale tests, it is not a quadratic form combining a test for location and a test for scale differences, and it is based on squared ranks and squared contrary-ranks. Moreover, it is easier to compute the test of Cucconi than those of Lepage, Manly–Francis, Büning–Thadewald, Neuhäuser, Büning and Murakami. Exact critical values for the test have been computed for the very first time. The power of the Cucconi test has been studied for the very first time and compared with that of the Lepage and other tests that include several Podgor–Gastwirth efficiency robust tests. Simulations show that the test of Cucconi maintains a size very close to α and is more powerful than the Lepage test, and therefore should be taken into account as a better alternative when it is not possible to develop an efficiency robust procedure for the problem at hand. The simulation study considers also the case of different shapes for the parent distributions, and the case of tied observations which is generally not considered in power studies. The presence of ties does not lower the performance of the Cucconi test, the contrary happens for the Lepage test. The tests are applied to real and fictitious biomedical data.

Random Translation of Records from Geometric Distributions

It is well known that the distributions of two records from a sequence of iid geometric random variables are related by a random translation. The distribution of the random translation is a negative binomial distribution. In this article, we investigate if this property characterizes the geometric parent distribution. We show that the answer is positive when weak records are considered. On the other hand, when ordinary records are considered, it is possible to characterize tail-geometric distributions if the records are consecutive, but in the case of non consecutive ordinary records it is possible to find out distributions that do not seem to be tail-geometric satisfying the property of random translation.

Questions about floral (dis)integration

Questions about floral (dis)integration

Phase equilibrium of colloidal suspensions with particle size dispersity: A Monte Carlo study

We have studied the crystalline-amorphous coexistence for systems of polydisperse soft spheres that interact via a purely repulsive power law potential. Potential softness quantified by the exponent of the potential was a primary input in our simulations. Simulations were performed in the isobaric semigrand statistical ensemble, i.e., the composition of the parent distribution was not fixed in our systems. Gibbs-Duhem integration was used to trace the coexistence pressure as a function of potential softness for monodisperse systems. A second Gibbs-Duhem integration, initiated from the monodisperse coexistence curve, was employed to determine coexistence pressure versus imposed variance of the activity distribution. Amorphous-crystalline coexistence densities and volume fractions were determined to be monotonically increasing functions of the breadth of particle size dispersity. Semigrand ensemble simulations testified to the existence of a terminal diameter dispersity, i.e., a dispersity above which no amorphous-crystalline phase coexistence was observed. At the terminus size dispersity increases from 5.8% to 6.1% to 6.4% and to 6.7% and 6.5% for the crystalline phase as the steepness parameter n, takes on smaller values: from 100 to 50 to 12 to 10 and 8, respectively. In sharp contrast to the crystalline phases' enhanced, by potential softness, allowable size dispersity the amorphous phase exhibits an opposite trend, as potential interactions soften. Furthermore, amorphous phases accommodate, on average, smaller particles than those of the ordered (fcc) phase. Contrary to widely accepted intuition crystalline phases composed of size-disperse particulates exhibit a higher degree of local order than their monodisperse counterparts, admittedly at differing thermodynamic conditions.

Regional daily maximum rainfall estimation for Cekerek Watershed by L‐moments

AbstractThe estimation of maximum daily rainfall (PDmax) is usually required for the estimation of design flood (the maximum flood that any hydraulic structure can safely pass). However, PDmax estimation is usually required for watersheds where rainfall data are either not available or only available in short periods from various sites and so are unsuitable for maximum daily rainfall estimation. In this study, the regional PDmax of the Cekerek watershed in Turkey is estimated using the method of l‐moments using 17 rainfall stations in the region. The discordant test for outlier stations showed no discordant station in the region. Applying the homogeneity measure, Hi, the homogeneous region was identified. To find the best regional distribution, the ZDIST goodness‐of‐fit test was applied. This test introduced two distributions as the candidates for regional parent distributions; Generalized Extreme Values (GEV) and 3‐parameter Log Normal (LOGN3) distributions. The LOGN3 distribution was selected as the best regional distribution as it has the smaller absolute value of the statistics (ZDIST) based on the goodness‐of‐fit‐test. Copyright © 2009 Royal Meteorological Society

RADIATION PRESSURE FORCE AND BLACK HOLE MASS DETERMINATION IN LOW-REDSHIFT TYPE-I AND TYPE-II ACTIVE GALACTIC NUCLEI

The distributions of L(oiii 5007), black hole (BH) mass and L/L(Edd) in two large samples of type-I and type-II active galactic nuclei (AGNs) are compared in order to test the suggestion that radiation pressure force is affecting the gas velocity in the broad line region and hence the BH mass determination. The samples are drawn from the SDSS archive and are modified to represent the same parent distribution at 0.1 < z < 0.2. BH masses in type-I sources are calculated in two different ways, one using a simple virial mass assumption and the other by taking into account the effect of radiation pressure force on the gas. The simple virial mass estimate results in good agreement with the sigma*-based BH mass and L/L(Edd) estimates in type-II sources. In contrast, there is a clear disagreement in the L/L(Edd) distributions when radiation pressure-based estimates are used. This indicates that radiation pressure force is not important in 0.1 < z < 0.2 AGNs with L(5100)=10^{42.8-44.8} erg/s. This has important implications to the physics of the gas producing the broad emission lines in AGNs, in particular the existence of extremely large (about 10^24 cm^(-2)) column density clouds.

몬테카를로 시뮬레이션에 의한 미소 비커스 경도의 Weibull 통계 해석

In the present study, the Weibull statistical analysis using the Monte-Carlo simulation has been performed to investigate the micro-Vickers hardness measurement reliability considering the variability. Experimental indentation test were performed with a micro-Vickers hardness tester for as-received and quenching and tempering specimens in SCM440 steels. The distribution of micro-Vickers hardness is found to be 2-parameter Weibull distribution function. The mean values and coefficients of variation (COV) for both data set are compared with results based on Weibull statistical analysis. Finally, Monte-Carlo simulation was performed in order to evaluate the effect of sample size on the micro-Vickers hardness measurement reliability. For the parent distribution with shape parameter 30.0 and scale parameter 200.0 (COV=0.040), the number of sample data required to obtain the true Weibull parameters was founded by 20. For the parent distribution with shape parameter 10.0 and scale parameter 200.0 (COV=0.1240), the number of sample data required to obtain the true Weibull parameters was founded by 30.

Tight evaluations for expectations of small order statistics from symmetric and symmetric unimodal populations

We present sharp non-positive upper bounds on the deviations of the expectations of small order statistics from the population mean, measured in the scale units generated by the population absolute central moments, when the parent distribution is either symmetric or symmetric unimodal.

Extreme limit distribution of truncated models for daily rainfall

AbstractWe propose truncated and power‐transformed (TPT) models for daily rainfall and we derive the generalized extreme value (GEV) limit distributions for these models. We find that these limit distributions belong to the domain of attraction of the Fréchet family when the parent distribution of the daily values is a TPT t‐Student model. In this case, the shape parameter of the limiting GEV model depends on the degrees of freedom and the power transformation parameter. When the parent distribution of the daily values is a TPT Normal model, the limiting GEV model is independent of the parameters of the parent model. We perform a detailed inference and predictive analysis to verify these theoretical results using a Bayesian approach. Markov Chain Monte Carlo methods (MCMC) were used to estimate the posterior distribution of the parameters of the t‐Student model for daily rainfall on one hand, and to estimate the posterior distribution of the parameters of the GEV model for the annual maxima on the other hand. Numerical results are presented for two locations: Maiquetía (Vargas State), and La Mariposa (Miranda State), Venezuela. Simulations from the predictive distribution of the daily values suggest a good approximation between the extreme distribution of the TPT t‐Student model and the Fréchet model found by standard extreme value limit theory. Copyright © 2009 John Wiley &amp; Sons, Ltd.

The statistical characters of PM 10 in Belgrade area

The concentrations of air pollutants depend on meteorological conditions and pollutant emission level. From the statistical properties of air pollutants the number of times the daily average concentrations exceed the assigned air quality standard (AQS) can be estimated, as well as the level of reduction of particle matter emission sources required to meet the AQS. In this paper three statistical distributions (lognormal, Weibull and type V Pearson distribution) were used to fit the complete set of PM 10 data for the Belgrade urban area during a three-year period (2003–2005). The method of moments and the method of least squares were both used to estimate the parameters of the three theoretical distributions. The type V Pearson distribution represented the PM 10 daily average concentration most closely. However, the parent distributions sometimes diverged in predicting a high PM 10 concentration and therefore asymptotic distributions of extreme values were used to fit the high PM 10 concentration distribution more correctly. This method can successfully predict the return period and exceedances over a critical concentration in succeeding years. The estimated emission source reduction of PM 10 to meet the assigned standard varied from 53% to 63% in the Belgrade urban area. The results provide useful information for air quality management and could be used to examine the similarities and differences among air pollution types in diverse areas.

A New Criterion for Goodness-of-Fit in Extreme Statistics and Its Applicability

A weighted least squares method for extreme statistics has been developed to estimate the parameters of candidate distribution functions. The method employs variances of the order statistics of extreme data, which are analytically evaluated by virtue of asymptotic distributions of the order statistics. A new criterion for goodness-of-fit in extreme statistics has been proposed to select a distribution function best fit to the parent distribution. Monte Carlo simulations with 5,000 and 10,000 samples have been carried out to verify an applicability of the new criterion for goodness-of-fit. The simulations show that the present criterion can select a correct distribution function for the case of FT-I type and generalized extreme value distributions, and Weibull distributions with the shape parameters of 0.75 and 1.0.

OPTIMAL SEMIPARAMETRIC INFERENCE FOR THE TAIL INDEX BASED ON RATIOS OF THE LARGEST EXTREMES

SummaryThe k largest order statistics in a random sample from a common heavy‐tailed parent distribution with a regularly varying tail can be characterized as Fréchet extremes. This paper establishes that consecutive ratios of such Fréchet extremes are mutually independent and distributed as functions of beta random variables. The maximum likelihood estimator of the tail index based on these ratios is derived, and the exact distribution of the maximum likelihood estimator is determined for fixed k, and the asymptotic distribution as k →∞. Inferential procedures based upon the maximum likelihood estimator are shown to be optimal. The Fréchet extremes are not directly observable, but a feasible version of the maximum likelihood estimator is equivalent to Hill's statistic. A simple diagnostic is presented that can be used to decide on the largest value of k for which an assumption of Fréchet extremes is sustainable. The results are illustrated using data on commercial insurance claims arising from fires and explosions, and from hurricanes.

Comparing Fisher information in record values and their concomitants with random observations

In a sequence of bivariate random variables {(X i , Y i ), i≥1} from a continuous distribution with a real parameter θ, general comparison results between the amount of Fisher information about θ contained in the sequence of the first n records and their concomitants, and the desired information in an i.i.d. sample of size n from the parent distribution are established. Some relationships between reliability properties and the proposed criteria are obtained in situations in which the univariate counterpart of the underlying bivariate family belongs to location, scale or shape families. It is also shown that in some classes of bivariate families, the concerned information property is equivalent to that of its univariate counterpart. The proposed procedure is illustrated by considering several examples.

Prevalence of High X-Ray Obscuring Columns among AGNs that Host H 2 O Masers

Of 104 AGNs known to exhibit H2O emission, X-ray data that enable estimation of column densities, or lower limits, are available for 42. Contributing to this, we report analysis of new and archival X-ray data for eight galaxies and collation of values for three more. Maser emission is indicative of large columns of cold gas, and in five of the eight new cases, spectra point toward origins in accretion disks viewed close to edge-on (a.k.a. maser systems). In these, we detect hard continuum and Fe Kα emission with equivalent widths on the order of 1 keV, which is consistent with Compton reflection, fluorescence by cold material, and obscuring columns 1024 cm−2. Reviewing the full sample of 42, 95% exhibit NH > 1023 cm−2 and 60% exhibit NH > 1024 cm−2. Half of these are now recognized to be disk masers (up from 13); in this subsample, which is likely to be more homogeneous vis-a-vis the origin of emission, 76% exhibit NH > 1024 cm−2. The probability of a common parent distribution of columns for disk masers and other AGN masers is 3%. Because ground-based surveys of AGNs to detect new disk masers are relatively unbiased with respect to X-ray brightness and comparatively inexpensive, they may also be efficient guides for the sensitive pointed X-ray observations required to identify Compton-thick objects outside of shallow surveys.

Extreme value analysis applied to pitting corrosion experiments in low carbon steel: Comparison of block maxima and peak over threshold approaches

The block maxima and peak over threshold approaches to extremes have been applied to pitting corrosion data from laboratory-simulated buried line pipe steel. Both approaches have been useful for gaining a better understanding of pitting in low carbon steel. We show that the parent distribution of the extreme depths is the right tail of the observed pit depth distribution. The threshold approach proved more robust to maximum depth reduction, which results from the dependency between pit depths. A simplified sampling method is proposed for applying the threshold approach without the need for measuring the entire pit depth distribution.

Globular Clusters in the Milky Way and Dwarf Galaxies: A Distribution‐Free Statistical Comparison

It has been found that globular clusters (GCs) in dwarf galaxies and those in the Milky Way (MW) outer halo mostly have the same parent distributions, while GCs in the MW disk and inner halo have a different origin from those in dwarf galaxies. Thus, these dwarf galaxies did not play a crucial role in the formation of the Galactic disk or inner halo. In order to investigate this phenomenon in a more objective manner, a statistical comparison of the GCs of our Galaxy and those of neighboring dwarf galaxies has been carried out by a multivariate nonparametric method. For the various parameters of GCs in the MW and in dwarf galaxies, the multivariate Gaussian assumption fails, so a nonparametric method of comparison (instead of multivariate analysis of variance [MANOVA]) has been chosen. The test is performed on GCs of the MW disk, inner halo, and outer halo separately, with GCs from neighboring dwarf galaxies Canis Major, Fornax, and Sculptor, and the LMC dwarf irregular galaxy. The test is also performed for GCs from dwarf spheroidal galaxies in the neighborhood of M31: M33, NGC 147, NGC 185, and NGC 205.