Distributions Of Exponential Type Research Articles

Stochastic comparisons, differential entropy and varentropy for distributions induced by probability density functions

AbstractStimulated by the need of describing useful notions related to information measures, we introduce the ‘pdf-related distributions’. These are defined in terms of transformation of absolutely continuous random variables through their own probability density functions. We investigate their main characteristics, with reference to the general form of the distribution, the quantiles, and some related notions of reliability theory. This allows us to obtain a characterization of the pdf-related distribution being uniform for distributions of exponential and Laplace type as well. We also face the problem of stochastic comparing the pdf-related distributions by resorting to suitable stochastic orders. Finally, the given results are used to analyse properties and to compare some useful information measures, such as the differential entropy and the varentropy.

Nonlocal Probability Theory: General Fractional Calculus Approach

Nonlocal generalization of the standard (classical) probability theory of a continuous distribution on a positive semi-axis is proposed. An approach to the formulation of a nonlocal generalization of the standard probability theory based on the use of the general fractional calculus in the Luchko form is proposed. Some basic concepts of the nonlocal probability theory are proposed, including nonlocal (general fractional) generalizations of probability density, cumulative distribution functions, probability, average values, and characteristic functions. Nonlocality is described by the pairs of Sonin kernels that belong to the Luchko set. Properties of the general fractional probability density function and the general fractional cumulative distribution function are described. The truncated GF probability density function, truncated GF cumulative distribution function, and truncated GF average values are defined. Examples of the general fractional (GF) probability distributions, the corresponding probability density functions, and cumulative distribution functions are described. Nonlocal (general fractional) distributions are described, including generalizations of uniform, degenerate, and exponential type distributions; distributions with the Mittag-Leffler, power law, Prabhakar, Kilbas–Saigo functions; and distributions that are described as convolutions of the operator kernels and standard probability density.

Comparison analysis of the ETAS model with Gutenberg–Richter (GR), Tapered-GR and characteristic magnitude distributions

SUMMARY In this paper, we carry out a comparison analysis of the Epidemic Type Aftershock Sequence (ETAS) model for the earthquake process, embedded with the three main exponential-type distributions adopted in practical applications to describe the magnitudes of seismic events, that are, the Gutenberg–Richter (GR), the tapered Gutenberg–Richter (TGR) and the CHaracteristic (CH) frequency–magnitude distributions (FMDs). The first law is a pure-power decreasing function, while both the other two introduce a more rapid decay in the tail of the distribution: a soft taper in the TGR model and a sharp cut-off in the CH one. To perform the comparison, we first investigate some theoretical features of the ETAS model with CH-distributed magnitudes (ETAS-CH), which have not been deeply analysed in the literature as much as for ETAS-TGR and ETAS-GR. In particular, we explicitly compute the branching ratio, we analyse its asymptotics in relation to its parameters, and we derive the proper stability conditions. We then move to the comparison among the three ETAS-GR, ETAS-TGR and ETAS-CH processes, to highlight differences and similarities. This is done by carrying out both a theoretical analysis, mainly focused on the three models’ branching ratios and the relative sensitivity, and a simulation analysis of realistic synthetic catalogues to compare the processes’ numbers, events’ magnitude distribution and temporal evolution. The results we obtained show that the ETAS-TGR and ETAS-CH processes have very similar features. They both have also less restrictive non-explosion conditions than for ETAS-GR; in fact, differently from this latter case, their branching ratios exist for any value of the parameters and are lower than the one of ETAS-GR, to which they converge for large magnitudes.

A New Alpha Power Weibull Model for Analyzing Time-to-Event Data: A Case Study from Football

Statistical methodologies have wider applications in exercise science, sports medicine, sports management, sports marketing, sports science, and other related sciences. These methods can be used to predict the winning probability of a team or individual in a match, the number of minutes that an individual player will spend on the ground, the number of goals to be scored by an individual player, the number of red/yellow cards that will be issued to an individual player or a team, etc. Keeping in view the importance and applicability of the statistical methodologies in sport sciences, healthcare, and other related sectors, this paper introduces a novel family of statistical models called new alpha power family of distributions. It is shown that numerous properties of the suggested method are similar to those of the new Weibull-X and exponential type distributions. Based on the novel method, a special model, namely, a new alpha power Weibull distribution, is studied. The new model is very flexible because the shape of its probability density function can either be right-skewed, decreasing, left-skewed, or increasing. Furthermore, the new distribution is also able to model real phenomena with bathtub-shaped failure rates. Finally, the usefulness/applicability of the proposed distribution is shown by analyzing the time-to-event datasets selected from different football matches during 1964–2018.

Randomly stopped minima and maxima with exponential-type distributions

Let {ξ1, ξ2,...} be a sequence of independent real-valued and possibly nonidentically distributed random variables. Suppose that η is a nonnegative, nondegenerate at 0 and integer-valued random variable, which is independent of {ξ1, ξ2,...}. In this paper, we consider conditions for {ξ1, ξ2,...} and η under which the distributions of the randomly stopped maxima and minima as well as randomly stopped maxima of sums and randomly stopped minima of sums belong to the class of exponential distributions.

Randomly stopped sums with exponential-type distributions

Assume that {ξ1, ξ2, …} are independent and possibly nonidentically distributed random variables. Suppose that η is a nonnegative, nondegenerate at zero and integer-valued random variable, which is independent of {ξ1, ξ2, …}. In this paper, we consider conditions for η and {ξ1, ξ2, …} under which the distribution of the random sum {ξ1 + ξ2 + … + ξη} belongs to the class of exponential distributions.

Bayesian Prediction Bounds for the Exponential-Type Distribution Based on Ordered Ranked Set Sampling

Abstract We suggest a ranked set sample method to improve Bayesian prediction intervals. The paper deals with the Bayesian prediction intervals in the context of an ordered ranked set sample from a certain class of exponential-type distributions. A proper general prior density function is used and the predictive cumulative function is obtained in the two-sample case. The special case of linear exponential distributed observations is considered and completed with numerical results.

Gabor frames and asymptotic behavior of Schwartz distributions

We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor frames. Our characterizations are indeed Tauberian theorems for shift asymptotics (S-asymptotics) in terms of short-time Fourier transforms with respect to windows generating Gabor frames. For it, we show that the Gabor coefficient operator provides (topological) isomorphisms of the spaces of tempered distributions S'(Rd) and distributions of exponential type K'1(Rd) onto their images.

The short-time Fourier transform of distributions of exponential type and tauberian theorems for shift-asymptotics

We study the short-time Fourier transform on the space K01(Rn) of distributions of exponential type. We give characterizations of K01(Rn) and some of its subspaces in terms of modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform.

Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology

Abstract. The need of understanding and modelling the space–time variability of natural processes in hydrological sciences produced a large body of literature over the last thirty years. In this context, a multifractal framework provides parsimonious models which can be applied to a wide-scale range of hydrological processes, and are based on the empirical detection of some patterns in observational data, i.e. a scale invariant mechanism repeating scale after scale. Hence, multifractal analyses heavily rely on available data series and their statistical processing. In such analyses, high order moments are often estimated and used in model identification and fitting as if they were reliable. This paper warns practitioners against the blind use in geophysical time series analyses of classical statistics, which is based upon independent samples typically following distributions of exponential type. Indeed, the study of natural processes reveals scaling behaviours in state (departure from exponential distribution tails) and in time (departure from independence), thus implying dramatic increase of bias and uncertainty in statistical estimation. Surprisingly, all these differences are commonly unaccounted for in most multifractal analyses of hydrological processes, which may result in inappropriate modelling, wrong inferences and false claims about the properties of the processes studied. Using theoretical reasoning and Monte Carlo simulations, we find that the reliability of multifractal methods that use high order moments (>3) is questionable. In particular, we suggest that, because of estimation problems, the use of moments of order higher than two should be avoided, either in justifying or fitting models. Nonetheless, in most problems the first two moments provide enough information for the most important characteristics of the distribution.

Information geometry of target tracking sensor networks

In this paper, the connections between information geometry and performance of sensor networks for target tracking are explored to pursue a better understanding of placement, planning and scheduling issues. Firstly, the integrated Fisher information distance (IFID) between the states of two targets is analyzed by solving the geodesic equations and is adopted as a measure of target resolvability by the sensor. The differences between the IFID and the well known Kullback–Leibler divergence (KLD) are highlighted. We also explain how the energy functional, which is the “integrated, differential” KLD, relates to the other distance measures. Secondly, the structures of statistical manifolds are elucidated by computing the canonical Levi–Civita affine connection as well as Riemannian and scalar curvatures. We show the relationship between the Ricci curvature tensor field and the amount of information that can be obtained by the network sensors. Finally, an analytical presentation of statistical manifolds as an immersion in the Euclidean space for distributions of exponential type is given. The significance and potential to address system definition and planning issues using information geometry, such as the sensing capability to distinguish closely spaced targets, calculation of the amount of information collected by sensors and the problem of optimal scheduling of network sensor and resources, etc., are demonstrated. The proposed analysis techniques are presented via three basic sensor network scenarios: a simple range-bearing radar, two bearings-only passive sonars, and three ranges-only detectors, respectively.

Two sample Bayesian prediction intervals for order statistics based on the inverse exponential-type distributions using right censored sample

Abstract In this paper, two sample Bayesian prediction intervals for order statistics (OS) are obtained. This prediction is based on a certain class of the inverse exponential-type distributions using a right censored sample. A general class of prior density functions is used and the predictive cumulative function is obtained in the two samples case. The class of the inverse exponential-type distributions includes several important distributions such the inverse Weibull distribution, the inverse Burr distribution, the loglogistic distribution, the inverse Pareto distribution and the inverse paralogistic distribution. Special cases of the inverse Weibull model such as the inverse exponential model and the inverse Rayleigh model are considered.

Expansions for quantiles and moments of extremes for distributions of exponential power type

Let M nr be the rth largest of a random sample of size n from a distribution F of exponential power type on R. That is, 1-F(z) = O(x d exp(−x)) as x = (z/σ) α → ∞. For example, the exponential, gamma, chi-square, Laplace and normal distributions are of this type. We obtain an asymptotic expansion in powers of u 1 = −log(1 − u) and u 2 = log u 1, for the quantile F −1(u) near u = 1. From this, we obtain a double expansion in inverse powers of (log n, n) for the moments of M nr /n 1/n 1/α , with the coefficient a polynomial in log log n. We also discuss a possible application to an optimal stopping problem.

Bayesian prediction based on generalized order statistics using multiply type-II censoring

The paper deals with the Bayesian prediction intervals for generalized order statistics (GOS) based on a certain class of exponential-type distributions. This class of distributions includes several important distributions such as Weibull, Burr-XII and Pareto distributions. A general class of prior density functions is used and the predictive reliability function is obtained in the one sample case. The investigation of multiply type-II censored GOS samples generalizes results of ordinary order statistics (OS), sequential order statistics (SOS) and censored GOS. The special case of Pareto distributed observations is considered and completed with numerical results.

Records in a changing world

In the context of this paper, a record is an entry in a sequence of random variables (RVs)that is larger or smaller than all previous entries. After a brief review of the classictheory of records, which is largely restricted to sequences of independent andidentically distributed (i.i.d.) RVs, new results for sequences of independent RVs withdistributions that broaden or sharpen with time are presented. In particular,we show that when the width of the distribution grows as a power law in timen, the mean number of records is asymptotically of orderlnn for distributions with a power law tail (the Fréchet class of extreme value statistics), of order(ln n)2 for distributions of exponential type (Gumbel class), and of ordern1/(ν+1) for distributions of bounded support (Weibull class), where the exponentν describes the behaviour of the distribution at the upper (or lower) boundary. Simulationsare presented which indicate that, in contrast to the i.i.d. case, the sequence of recordbreaking events is correlated in such a way that the variance of the number of records isasymptotically smaller than the mean.

A Class of Two-Stage Selection Procedures Using L-Statistics

A class of selection procedures for selecting the least dispersive distribution from k available distributions has been proposed. This problem finds applications in reliability and engineering. In engineering, for example, the goal of the experimenter is to select a firm whose components have least dispersive distribution from the available set of competing firms manufacturing the components of the desired specifications meant for the same purpose. The proposed procedures can be used even when the underlying distributions belong to different families. Applications of the proposed selection procedures are discussed by taking exponential, gamma and Lehmann type distributions. Performance of the proposed selection procedures is assessed through simulation study. Implementation of the proposed selection procedure is illustrated through an example.

Strong, Weak and False Inverse Power Laws

Pareto, Zipf and numerous subsequent investigators of inverse power distributions have often represented their findings as though their data conformed to a power law form for all ranges of the variable of interest. I refer to this ideal case as a strong inverse power law (SIPL). However, many of the examples used by Pareto and Zipf, as well as others who have followed them, have been truncated data sets, and if one looks more carefully in the lower range of values that was originally excluded, the power law behavior usually breaks down at some point. This breakdown seems to fall into two broad cases, called here (1) weak and (2) false inverse power laws (WIPL and FIPL, resp.). Case 1 refers to the situation where the sample data fit a distribution that has an approximate inverse power form only in some upper range of values. Case 2 refers to the situation where a highly truncated sample from certain exponential-type (and in particular, “lognormal-like”) distributions can convincingly mimic a power law. The main objectives of this paper are (a) to show how the discovery of Pareto–Zipf-type laws is closely associated with truncated data sets; (b) to elaborate on the categories of strong, weak and false inverse power laws; and (c) to analyze FIPLs in some detail. I conclude that many, but not all, Pareto–Zipf examples are likely to be FIPL finite mixture distributions and that there are few genuine instances of SIPLs.

Convolution of Lorentz Invariant Ultradistributions and Field Theory

In this work, a general definition of convolution between two arbitrary four dimensional Lorentz invariant (fdLi) Tempered Ultradistributions is given, in both: Minkowskian and Euclidean Space (Spherically symmetric tempered ultradistributions). The product of two arbitrary fdLi distributions of exponential type is defined via the convolution of its corresponding Fourier Transforms. Several examples of convolution of two fdLi Tempered Ultradistributions are given. In particular we calculate exactly the convolution of two Feynman's massless propagators. An expression for the Fourier Transform of a Lorentz invariant Tempered Ultradistribution in terms of modified Bessel distributions is obtained in this work (Generalization of Bochner's formula to Minkowskian space). At the same time, and in a previous step used for the deduction of the convolution formula, we obtain the generalization to the Minkowskian space, of the dimensional regularization of the perturbation theory of Green Functions in the Euclidean configuration space given in ref.[12]. As an example we evaluate the convolution of two n-dimensional complex-mass Wheeler's propagators.

Convolution of n-Dimensional Tempered Ultradistributions and Field Theory

In this work, a general definition of convolution between two arbitrary Tempered Ultradistributions is given. When one of the Tempered Ultradistributions is rapidly decreasing this definition coincides with the definition of J. Sebastiao e Silva. In the four-dimensional case, when the Tempered Ultradistributions are even in the variables $k^0$ and $\rho$ (see Section 5) we obtain an expression for the convolution, which is more suitable for practical applications. The product of two arbitrary even (in the variables $x^0$ and $r$) four dimensional distributions of exponential type is defined via the convolution of its corresponding Fourier Transforms. With this definition of convolution, we treat the problem of singular products of Green Functions in Quantum Field Theory. (For Renormalizable as well as for Nonrenormalizable Theories). Several examples of convolution of two Tempered Ultradistributions are given. In particular we calculate the convolution of two massless Wheeeler's propagators and the convolution of two complex mass Wheeler's propagators.

Two characterizations based on order statistics and records

Two characterizations are established in this paper. The first one is a characterization of the uniform distribution based on order statistics. The second characterization makes use of the conditional distribution of the second record value, given the maximum, for characterizing the distributions of exponential type.