Common Probability Distribution Research Articles

Statistical analysis of the radar cross section of two small fixed‐wing drones using typical flights

AbstractThe presence and importance of small drones (unmanned aerial vehicles) are increasing. With traditional radar detection the target radar cross section and its statistical properties are needed in order to design the detector and to assess detection performance. The authors make a statistical analysis of the radar cross section of two small fixed‐wing drones using measured data from typical flights. Using data from a typical flight instead of measurements or electromagnetic calculations as a function of aspect angle is a convenient method. Work results of the authors in mean values, medians and standard deviations, in analyses of common target probability distributions and in decorrelation times for the radar cross section, are necessary for design and assessment of the detection performance. The authors suggest a new target probability distribution which agrees better with the measured data than the others tested. The authors also note significant differences between the drones.

Predicting the neutrinoless double- β -decay matrix element of Xe136 using a statistical approach

Calculation of the nuclear matrix elements (NMEs) for double-beta decay is of paramount importance for guiding experiments and for analyzing and interpreting the experimental data, especially for the search of the neutrinoless double beta decay mode ($0\nu\beta\beta$). However, there are currently still large differences between the NME values calculated by different methods, hence a quantification of their uncertainties is very much required. In this paper we propose a statistical analysis of $0\nu\beta\beta$ NME for the $^{136}Xe$ isotope, based on the interacting shell model, but using three independent effective Hamiltonians, emphasizing the range of the NMEs' most probable values and its correlations with observables that can be obtained from the existing nuclear data. Consequently, we propose a common probability distribution function for the $0\nu\beta\beta$ NME, which has a range of (1.55 - 2.65) at 90\% confidence level, with a mean value of 1.99 and a standard deviation of 0.37.

Robustness of iterated function systems of Lipschitz maps

AbstractLet $\{X_n\}_{n\in{\mathbb{N}}}$ be an ${\mathbb{X}}$ -valued iterated function system (IFS) of Lipschitz maps defined as $X_0 \in {\mathbb{X}}$ and for $n\geq 1$ , $X_n\;:\!=\;F(X_{n-1},\vartheta_n)$ , where $\{\vartheta_n\}_{n \ge 1}$ are independent and identically distributed random variables with common probability distribution $\mathfrak{p}$ , $F(\cdot,\cdot)$ is Lipschitz continuous in the first variable, and $X_0$ is independent of $\{\vartheta_n\}_{n \ge 1}$ . Under parametric perturbation of both F and $\mathfrak{p}$ , we are interested in the robustness of the V-geometrical ergodicity property of $\{X_n\}_{n\in{\mathbb{N}}}$ , of its invariant probability measure, and finally of the probability distribution of $X_n$ . Specifically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with autoregressive conditional heteroscedastic errors, and for IFS under roundoff error or under thresholding/truncation. Moreover, we provide a general set of assumptions covering the classical Feller-type hypotheses for an IFS to be a V-geometrical ergodic process. An accurate bound for the rate of convergence is also provided.

Statistical analysis for the neutrinoless double- β -decay matrix element of Ca48

Neutrinoless double beta decay ($0\nu\beta\beta$) nuclear matrix elements (NME) are the object of many theoretical calculation methods, and are very important for analysis and guidance of a large number of experimental efforts. However, there are large discrepancies between the NME values provided by different methods. In this paper we propose a statistical analysis of the $^{48}$Ca $0\nu\beta\beta$ NME using the interacting shell model, emphasizing the range of the NME probable values and its correlations with observables that can be obtained from the existing nuclear data. Based on this statistical analysis with three independent effective Hamiltonians we propose a common probability distribution function for the $0\nu\beta\beta$ NME, which has a range of (0.45 - 0.95) at 90\% confidence level of, and a mean value of 0.68.

Investigating Hill’s question for some probability distributions

Benford’s law asserts that the lower first significant digit (FSD) occurs more frequently than the higher FSD in naturally produced datasets. The applications of the law vary from detecting election, tax, and Covid-19 data fraud to checking abnormalities in the stock market. Hence, it is vital to know which common probability distributions satisfy Benford’s law, which is called Hill’s question. Many research studies have been performed to answer this question by using various methods. The purpose of the work is to give a more simple and intuitive method to address the question for some common probability distributions. Moreover, statistical simulation is adopted to test their conformity to Benford’s law.

The probabilistic termination tool amber

We describe the Amber tool for proving and refuting the termination of a class of probabilistic while-programs with polynomial arithmetic, in a fully automated manner. Amber combines martingale theory with properties of asymptotic bounding functions and implements relaxed versions of existing probabilistic termination proof rules to prove/disprove (positive) almost sure termination of probabilistic loops. Amber supports programs parametrized by symbolic constants and drawing from common probability distributions. Our experimental comparisons give practical evidence of Amber outperforming existing state-of-the-art tools.

Goodness of fit tests for random multigraph models

Goodness of fit tests for two probabilistic multigraph models are presented. The first model is random stub matching given fixed degrees (RSM) so that edge assignments to vertex pair sites are dependent, and the second is independent edge assignments (IEA) according to a common probability distribution. Tests are performed using goodness of fit measures between the edge multiplicity sequence of an observed multigraph, and the expected one according to a simple or composite hypothesis. Test statistics of Pearson type and of likelihood ratio type are used, and the expected values of the Pearson statistic under the different models are derived. Test performances based on simulations indicate that even for small number of edges, the null distributions of both statistics are well approximated by their asymptotic -distribution. The non-null distributions of the test statistics can be well approximated by proposed adjusted -distributions used for power approximations. The influence of RSM on both test statistics is substantial for small number of edges and implies a shift of their distributions towards smaller values compared to what holds true for the null distributions under IEA. Two applications on social networks are included to illustrate how the tests can guide in the analysis of social structure.

Fragment size distributions in brittle deformed rocks

The production of breccias and cataclasites is commonly proposed to result in power-law or log-normal probability distributions for fragment (grain) size. We show that in both natural and experimental examples, the common best fit probability distributions for the complete distributions are members of the Generalised Gamma (GG), Extreme Value (GEV) and Pareto (GP) families; power-law and log-normal distributions are commonly, but not always, poor fits to the data. An hierarchical sequence, GG → GEV → GP, emerges as the sample mean of the fragment size decreases. The physical foundations (self-similar fragmentation, collisional fragmentation, shattering) for these distributions are discussed. Particularly important is the shattering continuous phase transition that results in the simultaneous development of both coarse fragments and ultra-fine particles (dust). This phase transition leads to Generalised Pareto fragment size distributions for the coarse fragments. Also included is a discussion of the relations between fragment size distribution, processes and deformation history in the context of monomineralic rocks. The overall reported size distributions are compatible with theoretical developments but the topic would benefit from observations and experiments conducted with the theories in mind.

Aperture and permeability of aged concrete joints

Often concrete structures cannot be regarded as monolithic since discontinuities like cracks and joints develop inevitably during construction and service life. Joints are not only introduced coincidentally at interfaces of different concreting steps, they are also induced deliberately to release stresses due to thermal actions or shrinkage in a controlled manner. This results in a change of material properties at the joint and is often investigated from a mechanical point of view. However, joints are also changing locally the moisture transport properties of a structure and thus considerably affecting the durability of those. Given the lack of data in the literature, in the presented work the distribution of the joint aperture and hydraulic conductivity is measured for a set of aged concrete expansion and cold joints. It is shown that the distribution of the apertures can be modeled with common probability distribution functions, which allow a relation between saturation and capillary potential. The parallel plate model is found to be a good estimate for the hydraulic conductivity of the joints when the available void areas of the joint are considered.

Intensity and frequency of extreme novel epidemics

Observational knowledge of the epidemic intensity, defined as the number of deaths divided by global population and epidemic duration, and of the rate of emergence of infectious disease outbreaks is necessary to test theory and models and to inform public health risk assessment by quantifying the probability of extreme pandemics such as COVID-19. Despite its significance, assembling and analyzing a comprehensive global historical record spanning a variety of diseases remains an unexplored task. A global dataset of historical epidemics from 1600 to present is here compiled and examined using novel statistical methods to estimate the yearly probability of occurrence of extreme epidemics. Historical observations covering four orders of magnitude of epidemic intensity follow a common probability distribution with a slowly decaying power-law tail (generalized Pareto distribution, asymptotic exponent = -0.71). The yearly number of epidemics varies ninefold and shows systematic trends. Yearly occurrence probabilities of extreme epidemics, Py, vary widely: Py of an event with the intensity of the "Spanish influenza" (1918 to 1920) varies between 0.27 and 1.9% from 1600 to present, while its mean recurrence time today is 400 y (95% CI: 332 to 489 y). The slow decay of probability with epidemic intensity implies that extreme epidemics are relatively likely, a property previously undetected due to short observational records and stationary analysis methods. Using recent estimates of the rate of increase in disease emergence from zoonotic reservoirs associated with environmental change, we estimate that the yearly probability of occurrence of extreme epidemics can increase up to threefold in the coming decades.

Distortion inequality for a Markov operator generated by a randomly perturbed family of Markov Maps in ℝ d

Abstract Asymptotic properties of the sequences (a) { P j } j = 1 ∞ $\{P^{j}\}_{j=1}^{\infty}$ and (b) { j − 1 ∑ i = 0 j − 1 P i } j = 1 ∞ $\{ j^{-1} \sum _{i=0}^{j-1} P^{i}\}_{j=1}^{\infty}$ are studied for g ∈ G = {f ∈ L 1(I) : f ≥ 0 and ‖f ‖ = 1}, where P : L 1(I) → L 1(I) is a Markov operator defined by P f := ∫ P y f d p ( y ) $Pf:= \int P_{y}f\, dp(y) $ for f ∈ L 1; {Py } y∈Y is the family of the Frobenius-Perron operators associated with a family {φy } y∈Y of nonsingular Markov maps defined on a subset I ⊆ ℝ d ; and the index y runs over a probability space (Y, Σ(Y), p). Asymptotic properties of the sequences (a) and (b), of the Markov operator P, are closely connected with the asymptotic properties of the sequence of random vectors x j = φ ξ j ( x j − 1 ) $x_{j}=\varphi_{\xi_{j}}(x_{j-1})$ for j = 1,2, . . .,where { ξ j } j = 1 ∞ $\{\xi_{j}\}_{j=1}^{\infty}$ is a sequence of Y-valued independent random elements with common probability distribution p. An operator-theoretic analogue of Rényi’s Condition is introduced for the family {Py } y∈Y of the Frobenius-Perron operators. It is proved that under some additional assumptions this condition implies the L 1- convergence of the sequences (a) and (b) to a unique g 0 ∈ G. The general result is applied to some families {φy } y∈Y of smooth Markov maps in ℝ d .

Retailer’s Emergency Ordering Policy When Facing an Impending Supply Disruption

Supply disruption is a common phenomenon in business activities. For the case where the supply disruption is predictable, the retailer should make an emergency procurement beforehand to decrease the inventory cost. For the scenario such that the happening time of the supply disruption obeys a certain common probability distribution but the ending time of the supply disruption is deterministic, based on minimizing the inventory cost and under two possible procurement strategies, we establish an emergency procurement optimization model. By considering the model solution in all cases, we establish a closed-form solution to the optimization model and provide an optimal emergency procurement policy to the retailer. Some numerical experiments are made to test the validity of the model and the effect of the involved parameters on the emergency procurement policy.

Assessing Flood Risk of the Chao Phraya River Basin Based on Statistical Rainfall Analysis

The Chao Phraya River Basin is one of the largest in Asia and is highly vulnerable to water-related disasters. Based on rainfall gauge data over 36 years (1981–2016), a frequency analysis was performed for this basin to understand and evaluate its overall flood risk; daily rainfall measurements of 119 rain gauge stations within the basin were considered. Four common probability distributions, i.e., Log-Normal (LOG), Gumbel type-I (GUM), Pearson type-III (PE3), and Log-Pearson type-III (LP3) distributions, were used to calculate the return period of rainfall at each station and at the basin-scale level. Results of each distribution were compared with the graphical Gringorten method to analyze their performance; GUM was found to be the best-fitted distribution among the four. Thereafter, design hyetographs were developed by integrating the return period of rainfall based on three adopted methods at basin and subbasin scales; each method had its pros and cons for hydrological applications. Finally, utilizing a Rainfall-Runoff-Inundation (RRI) model, we estimated the possible flood inundation extent and depth, which was outlined over the Chao Phraya River Basin using the design hyetographs with different return periods. This study can help enhance disaster resilience at industrial complexes in Thailand for sustainable growth.

Common knowledge equilibrium of Boolean securities in distributed information market

We investigate common knowledge equilibrium of separable (or parity) and totally symmetric Boolean securities in distributed information market. We theoretically show that clearing price converges to the true value when a common prior probability distribution of information of each player satisfies some conditions.

Illumination Depth

The concept of illumination bodies studied in convex geometry is used to amend the halfspace depth for multivariate data. The proposed notion of illumination enables finer resolution of the sample points, naturally breaks ties in the associated depth-based ordering, and introduces a depth-like function for points outside the convex hull of the support of the probability measure. The illumination is, in a certain sense, dual to the halfspace depth mapping, and shares the majority of its beneficial properties. It is affine invariant, robust, uniformly consistent, and aligns well with common probability distributions. Supplementary materials for this article are available online.

Multiparticle Localization at Low Energy for Multidimensional Continuous Anderson Models

We study the multiparticle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multiparticle lower spectral edges are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multiparticle Anderson model at low energy. The proof based on the multiparticle multiscale analysis bounds needs the values of the external random potential to be independent and identically distributed, whose common probability distribution is at least Log-Hölder continuous.

Category forms of Local-Causality and Non-Signalling and their duals

Two fundamental (meta)physical principles – NS (Non-Signalling condition which states the impossibility to communicate by means of physical correlations) and LC (the principle of Local Causality which isolates classical correlations from those responsible for non-locality) are considered in the framework of category theory. The original form of these principles operates with properties of common probability distributions for outcomes of measurements implemented in two space-time regions. The suggested category form consists of some assertions about special commutative diagrams. To any common probability distribution in the matter of discourse, an arrow (morphism) in these diagrams is associated. In fact, LC turns into the condition of the arrow being able to factor through a definite standard arrow. NS looks like uniqueness of an arrow which makes commutative a special diagram incorporating the considered arrow associated to the distribution. By means of these diagrams dual forms of NS and LC are suggested.

The geometric mean?

The sample geometric mean (SGM) introduced by Cauchy in 1821, is a measure of central tendency with many applications in the natural and social sciences including environmental monitoring, scientometrics, nuclear medicine, infometrics, economics, finance, ecology, surface and groundwater hydrology, geoscience, geomechanics, machine learning, chemical engineering, poverty and human development, to name a few. Remarkably, it was not until 2013 that a theoretical definition of the population geometric mean (GM) was introduced. Analytic expressions for the GM are derived for many common probability distributions, including: lognormal, Gamma, exponential, uniform, Chi-square, F, Beta, Weibull, Power law, Pareto, generalized Pareto and Rayleigh. Many previous applications of SGM assumed lognormal data, though investigators were unaware that for that case, the GM is the median and SGM is a maximum likelihood estimator of the median. Unlike other measures of central tendency such as the mean, median, and mode, the GM lacks a clear physical interpretation and its estimator SGM exhibits considerable bias and mean square error, which depends significantly on sample size, pd, and skewness. A review of the literature reveals that there is little justification for use of the GM in many applications. Recommendations for future research and application of the GM are provided.

ForestFit: An R package for modeling plant size distributions

Modeling the diameter distribution of trees in forest stands is a frequent application in ecology that supports key biologically and economically relevant management decisions. The choice of model used to represent the diameter distribution and how to estimate its parameters has received much attention in the forestry literature; however, accessible software that facilitates comprehensive comparison of the myriad modeling approaches is not available. To this end, we developed an R package called ForestFit that simplifies estimation of common probability distributions used to model tree diameter distributions, including the two- and three-parameter Weibull distributions, Johnson’s SB distribution, Birnbaum–Saunders distribution, and finite mixture distributions. Frequentist and Bayesian techniques are provided for individual tree diameter data, as well as grouped data. Additional functionality facilitates fitting curves to height-diameter data. The package also provides a set of functions for computing probability distributions and simulating random realizations from common finite mixture models.

Probability distributions unimodality of finite sample extremes of independent Erlang random variables

Samples of independent identically distributed random non-negative values with a finite size are studied. It is posed the problem to find the sufficient conditions for their common probability distribution which guarantee the unimodality of the probability distributions and which correspond to the maximum and to the minimum of the sample, respectively. It is proved that if the distribution Q is determined by a continuously differentiable Erlang probability density q of an arbitrary order then distributions are unimodal.