One-parameter Case Research Articles

Bayesian Inference for the Gamma Zero-Truncated Poisson Distribution with an Application to Real Data

This article presents Bayesian estimation methods applied to the gamma zero-truncated Poisson (GZTP) and the complementary gamma zero-truncated Poisson (CGZTP) distributions, encompassing both one-parameter and two-parameter models. These distributions are notably flexible and useful for modeling lifetime data. In the one-parameter model case, the Jeffreys prior is mathematically derived. The use of informative and noninformative priors, combined with the random walk Metropolis algorithm within a Bayesian framework, generates samples from the posterior distributions. Bayesian estimators’ effectiveness is examined through extensive simulation studies, in comparison with the maximum likelihood method. Results indicate that Bayesian estimators provide more precise parameter estimates, even with smaller sample sizes. Furthermore, the study and comparison of the coverage probabilities (CPs) and average lengths (ALs) of the credible intervals with those from Wald intervals suggest that Bayesian credible intervals typically yield shorter ALs and higher CPs, thereby demonstrating the effectiveness of Bayesian inference in the context of GZTP and CGZTP distributions. Lastly, Bayesian inference is applied to real data.

Optimization of Supply Air Parameters Control Based on Gappy POD Method for Creating Non-Uniform Temperature Fields

This paper explores the implementation and application of non-uniform ambient fields from two perspectives: constant heat source and abrupt heat source variation. On the one hand, the proper orthogonal decomposition (POD) method is used to find the optimum air supply parameters for different target temperatures at multiple demand points. The one-parameter, two-parameter and three-parameter cases were considered, respectively, and the parameters obtained from the search for optimisation were verified, yielding mean deviations of 0.405 K, 0.368 K and 0.380 K and mean errors of 1.48%, 1.61% and 1.68%, respectively. The accuracy of the reconstructed results of the POD method and the reliability of the POD method for finding the best results are verified step by step with the help of the experimental platform. The validation results show that the average error between the reconstructed data and the experimental data for the POD method does not exceed 5%; the average errors between the measured and set demand temperatures at the target point are 1.2% and 0.8%. On the other hand, the gappy POD method is used to accurately fill in the elements of the system with arbitrary missing data and to reconstruct the flow field in the presence of missing (gappy) data with a limited number of sensors combined with POD. The errors of the gappy POD method reconstruction are 0.54% and 1.75%. POD and gappy POD methods can better create non-uniform temperature fields in practical scenarios according to actual needs. The results of the study can provide a methodological reference for real-time reconstruction and real-time control of the indoor flow field environment. It also serves as a help and suggestion for the actual project in terms of end-regulation system and reverse design method.

Cover your bases: asymptotic distributions of the profile likelihood ratio when constraining effective field theories in high-energy physics

We investigate the asymptotic distribution of the profile likelihood ratio (PLR) when constraining effective field theories (EFTs) and show that Wilks’ theorem is often violated, meaning that we should not assume the PLR to follow a \chi^2χ2-distribution. We derive the correct asymptotic distributions when either one or two real EFT couplings modulate observable cross sections with a purely linear or quadratic dependence. We then discover that when both the linear and quadratic terms contribute, the PLR distribution does not have a simple form. In this case we provide a partly-numerical solution for the one-parameter case. Using a novel approach, we find that the constants which define our asymptotic distributions may be obtained experimentally using a profile of the Asimov likelihood contour. Our results may be immediately used to obtain the correct coverage when deriving real-world EFT constraints using the PLR as a test-statistic.

A modification of the Prudnikov and Laguerre polynomials

A two-parameter sequence of orthogonal polynomials with respect to the weight function , where is the modified Bessel function, is investigated. The case corresponds to the Prudnikov polynomials and t = 0 is related to the Laguerre polynomials. A special one-parameter case is analysed as well.

Moment matching priors for non-regular models

This paper presents moment matching priors for non-regular models whose supports depend on an unknown parameter. Both one-parameter and multi-parameter cases are considered. The resulting priors are given by matching the posterior mean and bias-adjusted maximum likelihood estimator up to the higher order. Some examples of proposed priors are also given.

On interpolatory subdivision symbol formulation and parameter convergence intervals

This paper is concerned with general symmetric 2 n -point interpolatory subdivision scheme (ISS) with polynomial reproduction of arbitrary order m ≤ n . An explicit formulation is derived for the corresponding refinement symbol, and convergence intervals are obtained for the one-parameter case, the left hand endpoints of which improve on previous such lower parameter convergence bounds.

Reliability Estimation for the Two-Parameter Exponential Family Based on Progressively Type-II Censored Data

In this paper, we deal with the problem of estimating the reliability function of the two-parameter exponential distribution. Classical Maximum likelihood and Bayes estimates for one and two parameters and the reliability function are obtained on the basis of progressively type-II censored samples. The inverted gamma conjugate prior density is assumed for the one-parameter case, whereas the joint prior density of the two-parameter case is composed of the inverted gamma and the uniform densities. A comparison between the obtained estimators is made through a Monte Carlo simulation study. A real example is used to illustrate the proposed methods.

Two-parameter quantum affine algebra of type [formula omitted], Drinfeld realization and vertex representation

The two-parameter quantum vertex operator representation of level-one is explicitly constructed for Ur,s(Cn(1)) based on its two-parameter Drinfeld realization we give. This construction will be degenerated to the one-parameter case of Jing–Koyama–Misra [27] when rs=1.

Thermodynamics of charged black holes in Einstein-Horndeski-Maxwell theory

We extend an earlier investigation of the thermodynamics of static black holes in an Einstein-Horndeski theory of gravity coupled to a scalar field, by including now an elec- tromagnetic field as well. By studying the two-parameter families of charged static black holes, we obtain much more powerful constraints on the thermodynamics since, unlike in the uncharged one-parameter case, now the right-hand side of the first law is not automatically integrable. In fact, this allows us to demonstrate that there must be an additional contribution in the first law, over and above the usual terms expected for charged black holes. The origin of the extra contribution can be attributed to the behaviour of the scalar field on the horizon of the black hole. We carry out the calculations in four dimensions and also in general dimensions. We also derive the ratio of viscosity to entropy for the dual boundary field theory, showing that the usual viscosity bound for isotropic solutions can be violated, with the ratio depending on the mass and charge.

Dyadic structure theorems for multiparameter function spaces

We prove that the multiparameter (product) space BMO of functions of bounded mean oscillation can be written as the intersection of finitely many dyadic product BMO spaces, with equivalent norms, generalizing the one-parameter result of T. Mei. We establish the analogous dyadic structure theorems for the space VMO of functions of vanishing mean oscillation, for A_p weights, for reverse-Hölder weights and for doubling weights. We survey several definitions of VMO and prove their equivalences, in the continuous, dyadic, one-parameter and product cases. In particular, we introduce the space of dyadic product VMO functions. We show that the weighted product Hardy space H^1_{\omega} is the sum of finitely many translates of dyadic weighted H^1_{\omega} , for each A_{\infty} weight \omega , and that the weighted strong maximal function is pointwise comparable to the sum of finitely many dyadic weighted strong maximal functions, for each doubling weight \omega . Our results hold in both the compact and non-compact cases.

AN IMPROVED PREDICTION OF DCT-BASED IMAGE FILTERS EFFICIENCY USING REGRESSION ANALYSIS

Efficiency of DCT-based filters for a wide-class of images is investigated. The study is carried out for additive white Gaussian noise (AWGN) case with several intensity levels. Local DCT-based filter is used as basic denoising technique. Nonlocal BM3D filter known as the state-of-the-art technique for AWGN removal is also exploited. A precise prediction method of denoising efficiency for several quality metrics is proposed. It is shown that statistics of DCT coefficients provides useful information. Regression models for analyzed filters and metrics are presented. The obtained dependence approximations of quality metrics on DCT statistics have high goodness of fit. One-parameter and multi-parameter fitting cases are considered. The most valuable DCT statistics are found.

Two-Parameter Quantum Algebras, Canonical Bases, and Categorifications

A theory of canonical basis for a two-parameter quantum algebra is developed in parallel with the one in one-parameter case. A geometric construction of the negative part of a two-parameter quantum algebra is given by using mixed perverse sheaves and Deligne's weight theory based on Lusztig's work. A categorification of the negative part of a two-parameter quantum algebra is provided. A two-parameter quantum algebra is shown to be a two-cocycle deformation, depending only on the second parameter, of its one-parameter analogue.

Vector Polynomials and a Matrix Weight Associated to Dihedral Groups

The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case for even dihedral groups). The matrix weight function for the Gaussian form is found explicitly by solving a boundary value problem, and then computing the normalizing constant. An orthogonal basis for the homogeneous harmonic polynomials is constructed. The coefficients of these polynomials are found to be balanced terminating $_4F_3$-series.

Species Dynamics in the Two-Parameter Poisson-Dirichlet Diffusion Model

The recently introduced two-parameter infinitely-many-neutral-alleles model extends the celebrated one-parameter version (which is related to Kingman's distribution) to diffusive two-parameter Poisson-Dirichlet frequencies. In this paper we investigate the dynamics driving the species heterogeneity underlying the two-parameter model. First we show that a suitable normalization of the number of species is driven by a critical continuous-state branching process with immigration. Secondly, we provide a finite-dimensional construction of the two-parameter model, obtained by means of a sequence of Feller diffusions of Wright-Fisher flavor which feature finitely many types and inhomogeneous mutation rates. Both results provide insight into the mathematical properties and biological interpretation of the two-parameter model, showing that it is structurally different from the one-parameter case in that the frequency dynamics are driven by state-dependent rather than constant quantities.

Species Dynamics in the Two-Parameter Poisson-Dirichlet Diffusion Model

The recently introduced two-parameter infinitely-many-neutral-alleles model extends the celebrated one-parameter version (which is related to Kingman's distribution) to diffusive two-parameter Poisson-Dirichlet frequencies. In this paper we investigate the dynamics driving the species heterogeneity underlying the two-parameter model. First we show that a suitable normalization of the number of species is driven by a critical continuous-state branching process with immigration. Secondly, we provide a finite-dimensional construction of the two-parameter model, obtained by means of a sequence of Feller diffusions of Wright-Fisher flavor which feature finitely many types and inhomogeneous mutation rates. Both results provide insight into the mathematical properties and biological interpretation of the two-parameter model, showing that it is structurally different from the one-parameter case in that the frequency dynamics are driven by state-dependent rather than constant quantities.

BMO from dyadic BMO via expectations on product spaces of homogeneous type

Using the random dyadic lattices developed by Hytönen and Kairema, we build up a bridge between BMO and dyadic BMO, and hence one between VMO and dyadic VMO, via expectations over dyadic lattices on spaces of homogeneous type, including both the one-parameter and product cases. We also obtain a similar relationship between Ap and dyadic Ap, as well as one between the reverse Hölder class RHp and dyadic RHp, via geometric–arithmetic expectations. These results extend the earlier theory along this line, developed by Garnett, Jones, Pipher, Ward, Xiao and Treil, to the more general setting of spaces of homogeneous type in the sense of Coifman and Weiss.

Two-parameter Quantum Affine Algebra $U_{r,s}(\widehat{\frak{sl}}_2)$

We give two realizations of the two-parameter quantum affine algebra [Formula: see text] and prove that its structure is isomorphic to the one-parameter case. In particular, the evaluation representation of [Formula: see text] is obtained.

Singular integrals on Carleson measure spaces ${CMO}^p$ on product spaces of homogeneous type

In the setting of product spaces M of homogeneous type, we prove that every product non-isotropic smooth (NIS) operator T is bounded on the generalized Carleson measure space CMOp( M) of Han, Li and Lu for p0 < p < 1. Here p0 depends on the homogeneous dimensions of the measures on factors of the product space M and on the regularity of the quasi-metrics on factors of M . The Lp boundedness for 1 < p < ∞ of the class of NIS operators was developed in both the one-parameter case and the multiparameter case by Nagel and Stein, and the Hp boundedness was established in the multiparameter case by Han, Li and Lu.

INTERACTING TIME-FRACTIONAL AND Δν PDES SYSTEMS VIA BROWNIAN-TIME AND INVERSE-STABLE-LÉVY-TIME BROWNIAN SHEETS

Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto–Sivashinsky PDE, and is not shared with its one-time-parameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a feature exhibited in well known fluids dynamics models. On the other hand, the memory-preserving interaction between the PDE solution and the initial data is common to both the single- and the multi-parameter Brownian-time PDEs. Here, we introduce a new — even in the one-parameter case — proof that combines stochastic analysis with analysis and fractional calculus to simultaneously link BTBS to a new system of temporally half-derivative interacting PDEs as well as to the fourth-order system proved earlier and differently by Allouba. We then introduce a general class of random fields we call inverse-stable-Lévy-time Brownian sheets (ISLTBSs), and we link them to β-fractional-time-derivative systems of interacting PDEs for 0 &lt; β &lt; 1. When β = 1/ν, ν ∈ {2, 3, …}, our proof also connects an ISLTBS to a system of memory-preserving ν-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled k-Laplacians of the initial data, k = 1, …, ν - 1. Using a Fourier–Laplace-transform-fractional-calculus approach, we give a conditional equivalence result that gives a necessary and sufficient condition for the equivalence between the fractional and the high order systems. In the one-parameter case this condition automatically holds.

Special features of strongly coupled systems of convection–diffusion equations with two small parameters

Strong coupling of convection–diffusion equations with two small parameters generates a solution decomposition which differs significantly from that for the one-parameter case. We explain the basic features and prove pointwise estimates for the first-order derivatives which allow us to analyze the upwind finite difference scheme on layer-adapted meshes.