Bessel Distribution Research Articles

From Monotonicity of a Class of Bessel Distribution Functions to new Bounds for Related Functionals

In this note we prove a monotonicity result with respect to the parameter ν of the cumulative distribution function for the McKay Iν Bessel distribution and uniform upper bounds for a bilinear expression containing modified Bessel function of the first kind Iν. Certain implications, among others with the Horn function Φ2 and for the Gaussian hypergeometric function close the exposition

Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions

The class of natural exponential families (NEFs) of distributions having power variance functions (NEF-PVFs) is huge (uncountable), with enormous applications in various fields. Based on a characterization property that holds for the cumulants of the members of this class, we developed a novel goodness-of-fit (gof) test for testing whether a given random sample fits fixed members of this class. We derived the asymptotic null distribution of the test statistic and developed an appropriate bootstrap scheme. As the content of the paper is mainly theoretical, we exemplify its applicability to only a few elements of the NEF-PVF class, specifically, the gamma and modified Bessel-type NEFs. A Monte Carlo study was executed for examining the performance of both—the asymptotic test and the bootstrap counterpart—in controlling the type I error rate and evaluating their power performance in the special case of gamma, while real data examples demonstrate the applicability of the gof test to the modified Bessel distribution.

Probabilistic and Analytical Aspects of the Symmetric and Generalized Kaiser–Bessel Window Function

The generalized Kaiser–Bessel window function is defined via the modified Bessel function of the first kind and arises frequently in tomographic image reconstruction. In this paper, we study in details the properties of the Kaiser–Bessel distribution, which we define via the symmetric form of the generalized Kaiser–Bessel window function. The Kaiser–Bessel distribution resembles to the Bessel distribution of McKay of the first type, it is a platykurtic or sub-Gaussian distribution, it is not infinitely divisible in the classical sense and it is an extension of the Wigner’s semicircle, parabolic and n-sphere distributions, as well as of the ultra-spherical (or hyper-spherical) and power semicircle distributions. We deduce the moments and absolute moments of this distribution and we find its characteristic and moment generating function in two different ways. In addition, we find its cumulative distribution function in three different ways and we deduce a recurrence relation for the moments and absolute moments. Moreover, by using a formula of Ismail and May on quotient of modified Bessel functions of the first kind, we deduce a closed-form expression for the differential entropy. We also prove that the Kaiser–Bessel distribution belongs to the family of log-concave and geometrically concave distributions, and we study in details the monotonicity and convexity properties of the probability density function with respect to the argument and each of the parameters. In the study of the monotonicity with respect to one of the parameters we complement a known result of Gronwall concerning the logarithmic derivative of modified Bessel functions of the first kind. Finally, we also present a modified method of moments to estimate the parameters of the Kaiser–Bessel distribution, and by using the classical rejection method we present two algorithms for sampling independent continuous random variables of Kaiser–Bessel distribution. The paper is closed with conclusions and proposals for future works.

The Bayesian Estimate of Vector Autoregressive Model Parameters Adopt Informative Prior Information

This research included the bayesian estimate for vector Autoregressive model with rank (p) in addition to statistical tests and predict Bayesian when the random error of model followed generalized multivariate modified Bessel distribution. The prior information about the parameters of model is represented by probability distributions belong to conjugate families. It found that the posterior marginal probability distribution for parameters matrix (Φ) is a Matrix t distribution, The posterior marginal probability distribution of covariance matrix (Σ) is uncommon and the predictive probability distribution of future observations vector is multivariate t distribution.

Temperature and friction fluctuations inside a harmonic potential.

In this article we study the trapped motion of a molecule undergoing diffusivity fluctuations inside a harmonic potential. For the same diffusing-diffusivity process, we investigate two possible interpretations. Depending on whether diffusivity fluctuations are interpreted as temperature or friction fluctuations, we show that they display drastically different statistical properties inside the harmonic potential. We compute the characteristic function of the process under both types of interpretations and analyze their limit behavior. Based on the integral representations of the processes we compute the mean-squared displacement and the normalized excess kurtosis. In the long-time limit, we show for friction fluctuations that the probability density function (PDF) always converges to a Gaussian whereas in the case of temperature fluctuations the stationary PDF can display either Gaussian distribution or generalized Laplace (Bessel) distribution depending on the ratio between diffusivity and positional correlation times.

Tuning Axial Resolution Independent of Lateral Resolution in a Computational Imaging System Using Bessel Speckles.

Speckle patterns are formed by random interferences of mutually coherent beams. While speckles are often considered as unwanted noise in many areas, they also formed the foundation for the development of numerous speckle-based imaging, holography, and sensing technologies. In the recent years, artificial speckle patterns have been generated with spatially incoherent sources using static and dynamic optical modulators for advanced imaging applications. In this report, a basic study has been carried out with Bessel distribution as the fundamental building block of the speckle pattern (i.e., speckle patterns formed by randomly interfering Bessel beams). In general, Bessel beams have a long focal depth, which in this scenario is counteracted by the increase in randomness enabling tunability of the axial resolution. As a direct imaging method could not be applied when there is more than one Bessel beam, an indirect computational imaging framework has been applied to study the imaging characteristics. This computational imaging process consists of three steps. In the first step, the point spread function (PSF) is calculated, which is the speckle pattern formed by the random interferences of Bessel beams. In the next step, the intensity distribution for an object is obtained by a convolution between the PSF and object function. The object information is reconstructed by processing the PSF and the object intensity distribution using non-linear reconstruction. In the computational imaging framework, the lateral resolution remained a constant, while the axial resolution improved when the randomness in the system was increased. Three-dimensional computational imaging with statistical averaging for different cases of randomness has been synthetically demonstrated for two test objects located at two different distances. The presented study will lead to a new generation of incoherent imaging technologies.

Observations on the McKay Iν Bessel distribution

The main aim of this article is to derive new representation formulae for CDF of random variable distributed according to the McKay Iν Bessel law: first set of formulae are given in terms of incomplete generalized Fox–Wright function while the other expressions include Exton generalized hypergeometric X function of two variables, and also the incomplete Lipschitz–Hankel integral for the modified Bessel function of the first kind. Also we establish the connection between CDF of the McKay Iν Bessel rv and another important CDF which belongs to the non–central chi–square law. Another goal is to give answers to the moment determinacy problem, unimodality and some mode results for this distribution.

Bayesian Inference of a Non normal Multivariate Partial Linear Regression Model

This research includes the Bayesian estimation of the parameters of the multivariate partial linear regression model when the random error follows the matrix-variate generalized modified Bessel distribution and found the statistical test of the model represented by finding the Bayes factor criterion, the predictive distribution under assumption that the shape parameters are known. The prior distribution about the model parameters is represented by non-informative information, as well as the simulate on the generated data from the model by a suggested way based on different values ​​of the shape parameters, the kernel function used in the generation was a Gaussian kernel function, the bandwidth (Smoothing) parameter was according to the rule of thumb. It found that the posterior marginal probability distribution of the location matrix and the predictive probability distribution is a matrix-t distribution with different parameters, the posterior marginal probability distribution of the scale matrix is proper distribution but it does not belong to the conjugate family, Through the Bayes factor criterion, it was found that the sample that was used in the generation process was drawn from a population that does not belong to the generalized modified Bessel population.

Modeling the temperature profile of an extrudate in material extrusion additive manufacturing

A simple model is developed to find the temperature profile of an extrudate in material extrusion (MatEx) additive manufacturing. This model predicts that the temperature has Bessel distribution over the extrudate’s cross-section. For a considered case, the maximum temperature variation over the cross-section is found to be 20 °C, with a diminishing rate of 20 °C/s, and it appears when the extrudate just leaves its extruder.

Binomial acceptance corrections for particle number distributions in high-energy reactions

The binomial acceptance correction procedure is studied for particle number distributions detected in high energy reactions in finite regions of the momentum space. We present acceptance correction formulas for scaled variance, skewness, and kurtosis. Our considerations include various specific types of particles - positively or negatively charged, baryons and antibaryons - as well as conserved charges, namely, the net baryon number and electric charge. A simple model with effects of exact charge conservation, namely the Bessel distribution, is studied in some detail where effects of multi-particle correlations are present. The accuracy of the binomial filter is studied with UrQMD model simulations of inelastic proton-proton reactions. Binomial acceptance correction procedure works well when used inside a small region of phase space as well as for certain efficiency corrections, in particular for constructing net proton fluctuation from net baryon ones. Its performance is less accurate when applied to obtain UrQMD fluctuations in a finite rapidity window from fluctuations in the full $4\pi$ space.

Approximate estimator for parameters of non-normal VMA(1) model

In many applications of time series , the white noise does not follow the normal distribution but follows one of the heavy tailed distributions. When using normal models in estimation and forecasting, these phenomena they will produce far from reality and inefficient estimators and predictions. The generalized multivariate modified Bessel distribution belongs to the potentially heavy-tailed distribution family and has wide applications in events that change over time. On this basis, this paper concerned with the study of vector moving average model of the first order (VMA(1)), is the white noise error term of this model follows GMMB. The non linear VMA(1) model was approximated to a linear model. The parameters of approximated model was estimated by Bayesian technique when non-informative priors. We supposed that some parameters of the distribution ( ) known. Different loss functions has been used in Bayesian analysis, We proposed two positive weight functions in weighted balanced loss functions . Some of theoretical results were applied on empirical sample generated from VMA(1) model. It is concluded that the estimators under proposed weighted balanced loss functions are better.

Fabrication of parabolic cylindrical microlens array by shaped femtosecond laser

A simple and efficient technique for fabricating parabolic cylindrical microlens arrays (CMLAs) on the surface of fused silica by shaped femtosecond (fs) laser direct-writing is demonstrated. By means of spatially shaping of a Gaussian fs laser beam to a Bessel distribution, an inversed cylindrical shape laser intensity profile is formed in a specific cross-sectional plane among the shaped optical field. Applying it to experiments, large area close-packed parabolic CMLAs with line-width of 37.5 μm and array size of about 5 × 5 mm are produced. The cross-sectional outline of obtained lenslets has a satisfied parabolic profile and the numerical aperture (NA) of lenslets is more than 0.35. Furthermore, the focusing performance of the fabricated CMLA is also tested in this work and it has been demonstrated that the focusing power of the CMLA with a parabolic profile is better than that with a semi-circular one.

Analysis of survival data by a Weibull–Bessel distribution

ABSTRACTIn survival analysis and reliability studies, problems with random sample size arise quite frequently. More specifically, in cancer studies, the number of clonogens is unknown and the time to relapse of the cancer is defined by the minimum of the incubation times of the various clonogenic cells. In this article, we have proposed a new model where the distribution of the incubation time is taken as Weibull and the distribution of the random sample size as Bessel, giving rise to a Weibull–Bessel distribution. The maximum likelihood estimation of the model parameters is studied and a score test is developed to compare it with its special submodel, namely, exponential–Bessel distribution. To illustrate the model, two real datasets are examined, and it is shown that the proposed model, presented here, fits better than several other existing models in the literature. Extensive simulation studies are also carried out to examine the performance of the estimates.

Comparison between broadband Bessel beam launchers based on either Bessel or Hankel aperture distribution for millimeter wave short pulse generation.

In this paper, a comparison is presented between Bessel beam launchers at millimeter waves based on either a cylindrical standing wave (CSW) or a cylindrical inward traveling wave (CITW) aperture distribution. It is theoretically shown that CITW launchers are better suited for the generation of electromagnetic short pulses because they maintain their performances over a larger bandwidth than those realizing a CSW aperture distribution. Moreover, the wavenumber dispersion of both the launchers is evaluated both theoretically and numerically. To this end, two planar Bessel beam launchers, one enforcing a CSW and the other enforcing a CITW aperture distribution, are designed at millimeter waves with a center operating frequency of f¯=60GHz and analyzed in the bandwidth 50 - 70 GHz by using an in-house developed numerical code to solve Maxwell's equations based on the method of moments. It is shown that a monochromatic Bessel beam can be efficiently generated by both the launchers over a wide fractional bandwidth. Finally, we investigate the generation of limited-diffractive electromagnetic pulses at millimeter waves, up to a certain non-diffractive range. Namely, it is shown that by feeding the launcher with a Gaussian short pulse, a spatially confined electromagnetic pulse can be efficiently generated in front of the launcher.

Femtosecond laser one-step direct-writing cylindrical microlens array on fused silica.

We demonstrate an efficient method for fabricating high-quality cylindrical microlens arrays (CMLAs) on the surface of fused silica, fully based on spatially shaping of a femtosecond laser beam from Gaussian to Bessel distribution. As the envelope of shaped spatial intensity distribution matches the profile of cylindrical microlens perfectly, a CMLA with more than 50 uniform microlenses is fabricated by simple line scanning. The radius and height of these microlens units can be finely controlled by adjusting the power of laser pulses. Excellent optical imaging and high-speed fabrication performances are also demonstrated by our fabricated CLMA.

A weak kernel formula for Bessel functions

In this paper, we prove a weak kernel formula of Bessel functions attached to irreducible generic representations of p p -adic G L ( n ) GL(n) . As an application, we show that the Bessel function defined by Bessel distribution coincides with the Bessel function defined via uniqueness of Whittaker models on the open Bruhat cell.

Symmetric form-invariant dual Pearcey beams.

We introduce another type of Pearcey beam, namely, dual Pearcey (DP) beams, based on the Pearcey function of catastrophe theory. DP beams are experimentally generated by applying Fresnel diffraction of bright elliptic rings. Form-invariant Bessel distribution beams can be regarded as a special case of DP beams. Subsequently, the basic propagation characteristics of DP beams are identified. DP beams are the result of the interference of two half DP beams instead of two classical Pearcey beams. Moreover, we also verified that half DP beams (including special-case parabolic-like beams) generated by half elliptical rings (circular rings) are a new member of the family of form-invariant beams.

Arago-Poisson diffraction spot observed in the shadow area of an axicon lens

In this paper we examined both theoretically and experimentally the diffraction field along the axis of propagation of the zero order Bessel distribution produced by an axicon. An axicon is typically used to obtain quasi non-diffracting beams. We found that an Arago-Poisson diffraction spot, normally only observed behind an opaque circular obstacle, can also be observed in the shadow area of an axicon lens. Both theoretical and experimental analysis indicate that the intensity distribution of the diffraction spot follows a zero order Bessel function in the paraxial region. The diffraction spot diameter increases linearly with the propagation distance in the shadow area while its energy decreases with propagation distance. The illuminated region and Arago-Poisson diffraction spot never overlap because the diameters of either increase linearly with propagation distance. Therefore, in theory, the Arago-Poisson diffraction spot could propagate to infinity. Our observation may be useful to produce an accurate straight line reference in commercial alignment systems.

Distribution of the product of Bessel distribution of first kind and gamma distribution-properties and characterization

In this paper, a new distribution is introduced by taking the product of the probability density functions of Bessel distribution of first kind and gamma distribution. Some distributional properties and a characterization of the proposed product distribution are provided. The percentile points for some selected values of parameters are also presented. It is hoped that the proposed product distribution will be useful in many applied research.

Distribution of the product of Bessel distribution of second kind and gamma distribution-properties and characterization

Distribution of the product of Bessel distribution of second kind and gamma distribution-properties and characterization