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This study examines population and small sample properties of the standardized fifth and sixth moments – the "higher moments" – for assessing univariate normality against bimodal and selected unimodal alternatives. Population parameters and distributions for selected bimodal mixtures are calculated and contrasted with those for the normal distribution. Using Gram-Charlier series expansion methods, an omnibus goodness of fit test incorporating the higher moments is specified and Monte Carlo simulation used to compare test power with parametric tests based on the standardized third and fourth sample moments: the asymptotic and size corrected versions of the Jarque-Bera score test and the omnibus D'Agostino K2 test. The studentized range and directional tests using the third through sixth moments are also considered. The results demonstrate that incorporating the fifth and sixth moments can provide enhanced parametric normality test power for bimodal normal mixture alternatives but not for various unimodal alternatives. [ABSTRACT FROM AUTHOR]

Bimodal data sets are very common in different areas of knowledge. The crude birth rates data, fish length data, egg diameter data, the eruption and interruption times of the Old Faithful geyser, are examples of this type of data. In this paper, a new class of symmetric density functions for modeling bimodal data as described above are presented. From density functions with support on [ 0 , + ∞) , the symmetry is getting by reflecting the density function in the negative semi-axis with their respective normalization. In this way, if the primitive density function is unimodal, then the resulting density will be bimodal. We introduce asymmetry parameters and study their behavior, in particular the values of their modes and some other statistical values of interest. The cases for densities generated by Gamma, Weibull, Log-normal, and Birnbaum-Saunders densities, among others are studied. Statistical inference is performed from a classical perspective. A small simulation study to evaluate the benefits and limitations of the new proposal. In addition, an application to a data set related to the fetal weight in grams obtained through ultrasound in a sample of 500 units is also presented; the results show the great usefulness of the model in practical situations. [ABSTRACT FROM AUTHOR]

Modelling is challenging topic and using parametric models is important stage to reach flexible function for modelling. Weibull distribution has shape and scale parameters which play the main role for modelling. Bimodality parameter is added and so bimodal Weibull distribution can capture real data set with bimodality which can be actually combination of two populations. The properties of the proposed distribution and estimation method are examined extensively to show its usability in modelling accurately and safely for practitioners. After examination as first stage in modelling issue, it is appropriate to use bimodal Weibull for modelling bimodality in real data sets if it exists. Two estimation methods including objective functions are used to estimate the parameters of shape, scale and bimodality parameters of function. The second stage in modelling is overcome by using heuristic algorithms for optimization of function according to parameters due to the fact that converging to global point of objective function is performed by heuristic algorithms from stochastic optimization. Real data sets are provided to show the modelling competence of objective functions from bimodal forms of Weibull and Gamma distributions having well defined shape, scale and bimodality parameters and potentially less parameters when compared with the existing distributions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a bimodal gamma distribution using a quadratic transformation based on the alpha-skew-normal model. We discuss several properties of this distribution such as mean, variance, moments, hazard rate and entropy measures. Further, we propose a new regression model with censored data based on the bimodal gamma distribution. This regression model can be very useful to the analysis of real data and could give more realistic fits than other special regression models. Monte Carlo simulations were performed to check the bias in the maximum likelihood estimation. The proposed models are applied to two real data sets found in the literature. [ABSTRACT FROM AUTHOR]

A parameter estimation strategy for a conceptual rainfall–runoff (CRR) model applied to a storm sewer system in an urban catchment (Chassieu, Lyon, France) is proposed on the basis of event-by-event Bayesian local calibrations. The marginal distribution formed by locally-estimated parameters is divided into conditional functions, clustering the event-based parameters based on their transferability to similar rainfall events. The conditional functions showed to be consistent with an observed bimodality in the marginal representation, reflecting two different hydrological conditions mainly related to the magnitude of the rainfall intensities (high or low). The improvements achieved by expressing the parameter probability functions into a conditional form are shown in terms of accuracy (Nash-Sutcliffe efficiency criterion), precision (average relative interval length) and reliability (percentage of coverage) for simulating flow rate in 255 and 110 calibration/verification events. [ABSTRACT FROM AUTHOR]

Vila et al. proposed and studied a bimodal gamma (BGamma) distribution using a quadratic transformation based on the alpha-skew-normal model. The log-likelihood function and first and second-order partial derivatives are not correct for the case n>1. [ABSTRACT FROM AUTHOR]

This article proposes new symmetric and asymmetric distributions applying methods analogous as the ones in Kim (2005) and Arnold et al. (2009) to the exponentiated normal distribution studied in Durrans (1992), that we call the power-normal (PN) distribution. The proposed bimodal extension, the main focus of the paper, is called the bimodal power-normal model and is denoted byBPN(α) model, where α is the asymmetry parameter. The authors give some properties including moments and maximum likelihood estimation. Two important features of the model proposed is that its normalizing constant has closed and simple form and that the Fisher information matrix is nonsingular, guaranteeing large sample properties of the maximum likelihood estimators. Finally, simulation studies and real applications reveal that the proposed model can perform well in both situations. [ABSTRACT FROM AUTHOR]

LetP(x) be a polynomial monotone increasing on ( − ∞, +∞). The probability distribution possessing the distribution functionis called the polynomial logistic distribution with associated polynomialP. This has recently been introduced by Koutras et al., who have also demonstrated its importance for modeling financial data. In this article, the properties of the polynomial logistic distribution with an associated polynomial of degree 3 have been investigated in detail. An example of polynomial logistic distribution describing daily exchange rate fluctuations for the US dollar versus the Turkish lira is provided. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new extension of the Birnbaum–Saunders (BS) distribution as a follow-up to the family of skew-flexible-normal distributions. This extension produces a family of BS distributions including densities that can be unimodal as well as bimodal. This flexibility is important in dealing with positive bimodal data, given the difficulties experienced by the use of mixtures of distributions. Some basic properties of the new distribution are studied including moments. Parameter estimation is approached by the method of moments and also by maximum likelihood, including a derivation of the Fisher information matrix. Three real data illustrations indicate satisfactory performance of the proposed model. [ABSTRACT FROM AUTHOR]

This paper formulates a comprehensive methodology for analyzing, quantifying and identifying congestion characteristics based on speed distribution. Utilizing vehicle speed data, a mathematical approach is applied, in order to characterize roadway segments, in terms of travel reliability, congestion severity and duration. We argue that the Gaussian mixture model (GMM) and its parameter combination is the appropriate tool if we are to obtain quantitative congestion measures and rank roadway performance. A significant contribution of our approach is that it is based on assumptions regarding mixed components as well as speed distribution and can be applied to large databases. We test our framework on the greater Toronto and Hamilton area in Ontario, Canada, and conclude that congestion quantification through the application of the GMM can be successfully accomplished. Results indicate that speed patterns differ significantly between counties as well as days of the week. [ABSTRACT FROM AUTHOR]

The influence of the number of detectors is observed by comparing a given bimodal particle size distribution (PSD) and a calculated bimodal PSD by computer simulation and experimental method. A nonlinear iteration based on the Twomey method is used for calculating PSD. An observed range of scattering angles is 5.0 × 10−4to 2.5 rad, and the Mie light scattering theory is used. The number of detectors was changed from 6 to 81 elements by the computer simulations. The algorithm of the inversion problem is applied with mean diameters of lognormal distribution in a range from 0.546 to 422 µm at a standard deviation of 0.01 and 0.27. Experimental bimodal samples are prepared by certified monodisperse polystyrene latex standards, and results are obtained by three different kinds of optics, which has 21, 41, and 81 elements detector, respectively. The polydisperse bimodal PSD converges from 42 elements of the number of detectors. The bimodal PSD of monodisperses or size range of PSD under 50 µm converges from 21 elements of the number of detectors. The accuracy of calculated PSD is effected not only by the number of detectors but also the shape of the scattering function. [ABSTRACT FROM PUBLISHER]

In order to increase the present understanding of bimodal emulsion drop size distributions, systematic series of experiments have been carried out to investigate the effects of formulation variables on bimodal drop size distributions, and probability distribution functions were proposed to analyze the distribution. The results show that, the span of the drop size and Sauter mean diameter become larger when the dispersed phase volume fraction becomes higher and rotor speed becomes lower; the Frechet function represents the experimental data satisfactorily. The prediction model of Sauter mean diameter established by combining the prediction theory of the maximum stable drop diameter and experimental analysis results can fit the experimental data well. [ABSTRACT FROM AUTHOR]

In this paper, we shall develop a novel family of bimodal univariate distributions (also allowing for unimodal shapes) and demonstrate its use utilizing the well-known and almost classical data set involving durations and waiting times of eruptions of the Old-Faithful geyser in Yellowstone park. Specifically, we shall analyze the Old-Faithful data set with 272 data points provided in Dekkinget al.[3]. In the process, we develop a bivariate distribution using a copula technique and compare its fit to a mixture of bivariate normal distributions also fitted to the same bivariate data set. We believe the fit-analysis and comparison is primarily illustrative from an educational perspective for distribution theory modelers, since in the process a variety of statistical techniques are demonstrated. We do not claim one model as preferred over the other. [ABSTRACT FROM PUBLISHER]

Mechanical damage in cotton fiber processing both shifts the fiber length distribution and alters its shape. This leads to length distributions with complex shapes, often bimodal, that cannot be classified based on simple summary statistics. In this research, we quantified the distribution departure from unimodality and its alteration from the seed to the card sliver in order to determine where in the cotton process the complex bimodal shape appears, and to identify its potential sources of variability observed at the bale stage. Results presented in this paper indicate that bimodality is a transient state associated with intermediate fiber damage levels starting from the gin stand. The distribution is unimodal or shows a non-significant departure from unimodality at both the low and high damage levels, i.e. seed and card sliver, respectively. Fiber maturity plays a major role in determining the variability of the distribution modality at any processing stages. A mature cotton typically shows an extended intermediate damage state associated with a bimodal distribution, while an immature cotton will reach an advanced damage level from the early stages of mechanical processing, and thus will not exhibit the bimodality that is characteristic of an early/intermediate stage of the fiber breakage process. [ABSTRACT FROM PUBLISHER]

Over 50 years ago, in a 1955 issue of JASA, a paper on a bounded continuous distribution by Topp and Leone [C.W. Topp and F.C. Leone, A family of J-shaped frequency functions, J. Am. Stat. Assoc. 50(269) (1955), pp. 209-219] appeared (the subject was dormant for over 40 years but recently the family was resurrected). Here, we shall investigate the so-called Two-Sided Generalized Topp and Leone (TS-GTL) distributions. This family of distributions is constructed by extending the Generalized Two-Sided Power (GTSP) family to a new two-sided framework of distributions, where the first (second) branch arises from the distribution of the largest (smallest) order statistic. The TS-GTL distribution is generated from this framework by sampling from a slope (reflected slope) distribution for the first (second) branch. The resulting five-parameter TS-GTL family of distributions turns out to be flexible, encompassing the uniform, triangular, GTSP and two-sided slope distributions into a single family. In addition, the probability density functions may have bimodal shapes or admitting shapes with a jump discontinuity at the 'threshold' parameter. We will discuss some properties of the TS-GTL family and describe a maximum likelihood estimation (MLE) procedure. A numerical example of the MLE procedure is provided by means of a bimodal Galaxy M87 data set concerning V-I color indices of 80 globular clusters. A comparison with a Gaussian mixture fit is presented. [ABSTRACT FROM AUTHOR]

Photon correlation spectroscopy (PCS) is widely used for nanoparticle characterization, being particularly suited for the analysis of narrow distributions. However, because of its simplicity, it would be important to extend it to other situations, namely to wide particle size distributions and to bimodal samples. In this work an evaluation of the discriminating capability of the most common algorithms used in PCS to deconvolute the measured auto-correlation function, when analyzing bimodal samples, was performed. The results show that the CONTIN algorithm enables a better description of bimodal mixtures, though its performance is dependent on the characteristics of the sample and on operating conditions such as the sampling time. Additionally, a close analysis of the auto-correlation function revealed the existence of systematic deviations in the slope of the first section of that function when a bimodal sample is compared with the equivalent monomodal one (same average diameter). The results obtained indicate that the inspection of the first part of the auto-correlation function produces important information that can reveal the presence of more than one peak when an unknown sample is analyzed. This feature is rather important since the inversion process may mask one of the peaks due to the ill-conditioned nature of the problem. This prior information can be useful to alert the user to the need for an iterative adjustment of the parameters of the inversion process in order to better reveal all the peaks in the sample. [ABSTRACT FROM AUTHOR]

For over 30 years, observers of financial markets have been puzzled by the behaviour of the market model residuals. This research offers one answer to two of the questions raised by the anomalous behaviour of the residuals. The first is the failure of cumulative abnormal residuals (CARs) to be centred at zero and normally or t distributed. The second is the failure of CARs to be homoscedastic. The authors argue that, in some cases, a bimodal distribution fits the data better. During the period 1985–1993 there were 47 Helsinki Exchanges listed firms having at least one bimodal return distribution after the publication of an interim report. This frequency represents 76% of the total number of firms observed. Bimodality occurs most prominently during the first few days after the event. Internal factors, contained in the interim reports, and external factors, available exogenously, help explain the uncertainty that gives rise to expost bimodality. Sometimes bimodality disappears during the investigation period, while at other times bimodality remains. These findings should promote the use of more sophisticated methods for non-normal return distributions and longer examination periods after the event. On the practical side, these results should help managers refine their communication practices. [ABSTRACT FROM AUTHOR]

α-Al 2 O 3 prepared by combustion technique was ball-billed in a planetary mill in toluene medium at 300 rpm in ZrO 2 pot with a ball to powder ratio 10:1 for 1, 5, 10, 15, and 20 hours. X-ray diffraction pattern from the milled materials showed super-Lorentzian peak shapes for α-Al 2 O 3 peak profiles. The super-Lorentzian peak shape has been attributed to the bimodal size distribution of the α-Al 2 O 3 particles. Rietveld analysis using two different phase fractions of α-Al 2 O 3 with different microstructural features yielded a low goodness-of-fit of the x-ray data indicating the suitability of the assumed model. The phase fraction of α-Al 2 O 3 particles with smaller size increases with the milling time. Further the particles shapes were observed to be cylindrical in this case with the cylinder axis along the crystallographic c -axis. The cylinder diameter and the length were obtained to be 86 Å and 140 Å respectively after 20 hrs of milling. The α-Al 2 O 3 particles of larger size are isotropic. It is, thus, proposed that milling induces bimodal size distribution in the initial hours of milling. [ABSTRACT FROM AUTHOR]