scholarly journals Bootstrapping M-estimators in generalized autoregressive conditional heteroscedastic models

Biometrika ◽  
2020 ◽  
Vol 107 (3) ◽  
pp. 753-760
Author(s):  
K Mukherjee
Keyword(s):  
Consistent Estimate ◽  
Bootstrap Approximation ◽  
Weighted Bootstrap ◽  
Asymptotically Normal ◽  
Special Cases ◽  
M Estimator ◽  
Heteroscedastic Model ◽  
Coverage Rates ◽  
M Estimates ◽  
The Given

Summary We consider the weighted bootstrap approximation to the distribution of a class of M-estimators for the parameters of the generalized autoregressive conditional heteroscedastic model. We prove that the bootstrap distribution, given the data, is a consistent estimate in probability of the distribution of the M-estimator, which is asymptotically normal. We propose an algorithm for the computation of M-estimates which at the same time is useful for computing bootstrap replicates from the given data. Our simulation study indicates superior coverage rates for various weighted bootstrap schemes compared with the rates based on the normal approximation and existing bootstrap methods for the generalized autoregressive conditional heteroscedastic model, such as percentile $t$-subsampling schemes. Since some familiar bootstrap schemes are special cases of the weighted bootstrap, this paper thus provides a unified theory and algorithm for bootstrapping in generalized autoregressive conditional heteroscedastic models.

scholarly journals Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dafang Zhao ◽  
Muhammad Aamir Ali ◽  
Artion Kashuri ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Definition Of ◽  
The Given ◽  
Class Of Functions ◽  
Interval Valued ◽  
New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

scholarly journals A New Class of Estimators Based on a General Relative Loss Function

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1138
Author(s):  
Tao Hu ◽  
Baosheng Liang
Keyword(s):  
Loss Function ◽  
Asymptotically Normal ◽  
New Class ◽  
Statistical Inferences ◽  
Relative Loss ◽  
Special Cases ◽  
Class Of Estimators ◽  
Box Cox Transformation ◽  
Loss Estimator ◽  
Prostate Cancer Study

Motivated by the relative loss estimator of the median, we propose a new class of estimators for linear quantile models using a general relative loss function defined by the Box–Cox transformation function. The proposed method is very flexible. It includes a traditional quantile regression and median regression under the relative loss as special cases. Compared to the traditional linear quantile estimator, the proposed estimator has smaller variance and hence is more efficient in making statistical inferences. We show that, in theory, the proposed estimator is consistent and asymptotically normal under appropriate conditions. Extensive simulation studies were conducted, demonstrating good performance of the proposed method. An application of the proposed method in a prostate cancer study is provided.

Boundary Integral Equation Formulation in Generalized Linear Thermo-Viscoelasticity With Rheological Volume

2003 ◽  
Vol 70 (5) ◽  
pp. 661-667 ◽  
Author(s):  
A. S. El-Karamany
Keyword(s):  
Integral Equation ◽  
Rheological Properties ◽  
Boundary Integral Equation ◽  
Relaxation Times ◽  
Fundamental Solutions ◽  
Boundary Integral ◽  
Phase Lag ◽  
Integral Equation Formulation ◽  
Special Cases ◽  
The Given

A general model of generalized linear thermo-viscoelasticity for isotropic material is established taking into consideration the rheological properties of the volume. The given model is applicable to three generalized theories of thermoelasticity: the generalized theory with one (Lord-Shulman theory) or with two relaxation times (Green-Lindsay theory) and with dual phase-lag (Chandrasekharaiah-Tzou theory) as well as to the dynamic coupled theory. The cases of thermo-viscoelasticity of Kelvin-Voigt model or thermoviscoelasticity ignoring the rheological properties of the volume can be obtained from the given model. The equations of the corresponding thermoelasticity theories result from the given model as special cases. A formulation of the boundary integral equation (BIE) method, fundamental solutions of the corresponding differential equations are obtained and an example illustrating the BIE formulation is given.

Circles on the Complex Plane

2021 ◽  
pp. 3-12
Author(s):  
A. Girsh
Keyword(s):  
Euclidean Space ◽  
Complex Plane ◽  
Euclidean Plane ◽  
Complex Space ◽  
Radical Center ◽  
Special Cases ◽  
Different Types ◽  
Different Dimensions ◽  
The Given ◽  
Made In

The Euclidean plane and Euclidean space themselves do not contain imaginary elements by definition, but are inextricably linked with them through special cases, and this leads to the need to propagate geometry into the area of imaginary values. Such propagation, that is adding a plane or space, a field of imaginary coordinates to the field of real coordinates leads to various variants of spaces of different dimensions, depending on the given axiomatics. Earlier, in a number of papers, were shown examples for solving some urgent problems of geometry using imaginary geometric images [2, 9, 11, 13, 15]. In this paper are considered constructions of orthogonal and diametrical positions of circles on a complex plane. A generalization has been made of the proposition about a circle on the complex plane orthogonally intersecting three given spheres on the proposition about a sphere in the complex space orthogonally intersecting four given spheres. Studies have shown that the diametrical position of circles on the Euclidean E-plane is an attribute of the orthogonal position of the circles’ imaginary components on the pseudo-Euclidean M-plane. Real, imaginary and degenerated to a point circles have been involved in structures and considered, have been demonstrated these circles’ forms, properties and attributes of their orthogonal position. Has been presented the construction of radical axes and a radical center for circles of the same and different types. A propagation of 2D mutual orthogonal position of circles on 3D spheres has been made. In figures, dashed lines indicate imaginary elements.

scholarly journals Parametric Extended General Mixed Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Complementarity Problems ◽  
Equivalent Formulation ◽  
Resolvent Equations ◽  
Special Cases ◽  
General Variational Inequalities ◽  
Significant Extension ◽  
Mixed Variational Inequalities ◽  
The Given

It is well known that the resolvent equations are equivalent to the extended general mixed variational inequalities. We use this alternative equivalent formulation to study the sensitivity of the extended general mixed variational inequalities without assuming the differentiability of the given data. Since the extended general mixed variational inequalities include extended general variational inequalities, quasi (mixed) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. In fact, our results can be considered as a significant extension of previously known results.

Planar Tooth Profile Synthesis for Relative Curvature

2017 ◽  
Author(s):  
Zhiyuan Yu ◽  
Kwun-Lon Ting
Keyword(s):  
Contact Stress ◽  
Tooth Profile ◽  
Hertzian Contact ◽  
Rolling Motion ◽  
Special Cases ◽  
Systematic Methodology ◽  
Hertzian Contact Stress ◽  
Curvature Theory ◽  
The Given ◽  
Tooth Pair

This paper is the first that uses the new conjugation curvature theory [1] to directly synthesize conjugate tooth profiles with the given relative curvature that determines the Hertzian contact stress. Conjugation curvature theory offers a systematic methodology to synthesize the relative curvature for a tooth pair. For any given relative curvature between the contact tooth profiles, a generating point can be located on an auxiliary body. Under the rolling motion among the pinion pitch, the gear pitch and the pitch on the auxiliary body, the generating point will trace fully conjugate profiles on the pinion and gear bodies with the given relative curvature at the instant of the contact. Full conjugation throughout the contact of the profiles is guaranteed with the three instant centers remaining coincident [1]. The methodology is demonstrated with a planar tooth profile synthesis with given relative curvature. One may find that the Wildhaber-Novikov tooth profile, which is known to have low relative curvature and Hertzian contact stress, and its variations become special cases under such methodology.

The propagation of electromagnetic waves in a stratified medium

1929 ◽  
Vol 25 (1) ◽  
pp. 97-120 ◽  
Author(s):  
D. R. Hartree
Keyword(s):  
Refractive Index ◽  
Reflection Coefficient ◽  
Electromagnetic Waves ◽  
Stratified Medium ◽  
Simple Extension ◽  
Angle Of Incidence ◽  
First Approximation ◽  
Special Cases ◽  
Cartesian Coordinate ◽  
The Given

The equations of propagation of electromagnetic waves in a stratified medium (i.e. a medium in which the refractive index is a function of one Cartesian coordinate only—in practice the height) are obtained first from Maxwell's equations for a material medium, and secondly from the treatment of the refracted wave as the sum of the incident wave and the wavelets scattered by the particles of the medium. The equations for the propagation in the presence of an external magnetic field are also derived by a simple extension of the second method.The significance of a reflection coefficient for a layer of stratified medium is discussed and a general formula for the reflection coefficient is found in terms of any two independent solutions of the equations of propagation in a given stratified medium.Three special cases are worked out, for waves with the electric field in the plane of incidence, viz.(1) A finite, sharply bounded, medium which is “totally reflecting” at the given angle of incidence.(2) Two media of different refractive index with a transition layer in which μ2 varies linearly from the value in one to the value in the other.(3) A layer in which μ2 is a minimum at a certain height and increases linearly to 1 above and below, at the same rate.For cases (2) and (3) curves are drawn showing the variation of reflection coefficient with thickness of the stratified layer.Case (3) may be of some importance as a first approximation to the conditions in the Heaviside layer.

Phase and signal velocities of eigenwaves in a binary dissipative plasma

1974 ◽  
Vol 11 (2) ◽  
pp. 269-298 ◽  
Author(s):  
M. Kranyš ◽  
J. Teichmann
Keyword(s):  
Electromagnetic Wave ◽  
Bulk Viscosity ◽  
Present Theory ◽  
Speed Of Light ◽  
Longitudinal Diffusion ◽  
Front Speed ◽  
Phase Velocities ◽  
Special Cases ◽  
Complete Set ◽  
The Given

The signal and phase velocities of the possible waves (eigenwaves) in a dissipative binary plasma have been studied on the basis of the linearized thirteenmoment theory. The complete set of eigenmodes for the given system is formed using approximations in the range of frequencies high compared with typical collision or plasma frequencies. There are twenty non-trivial propagation modes. Four of them are modes with signal-front speed c associated with the induced electromagnetic wave. Eight modes belong to each of the gas constituents, among them two longitudinal diffusion modes (±WI), four transverse viscosity modespolarized respectively along two transverse axes ±W⊥ in the direction x1 and ±W⊥ in the direction x2) and two longitudinal thermal modes (±WII). The signalfront speed of all eigenwaves is lower than the speed of light, so that the system is hyperbolic. The ultra-relativistic plasma is also briefly discussed. In this case, all the mixture constituents will have the same set of signal-front speeds as a single gas, irrespective of the bulk viscosity. Comparison is made of the present theory and results with other theories (for some special cases).

On Three and Two-Dimensional Disturbances of Pipe Flow

1960 ◽  
Vol 27 (3) ◽  
pp. 381-389 ◽  
Author(s):  
Kurt Spielberg ◽  
Hans Timan
Keyword(s):  
Order Differential Equation ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Main Flow ◽  
Two Dimensional ◽  
Straight Pipe ◽  
Special Cases ◽  
Set Up ◽  
The Stability ◽  
The Given

A system of ordinary, coupled differential equations is set up for three-dimensional disturbances of Poiseuille flow in a straight pipe of circular cross section. The commonly treated equations are shown to be special cases arising from particular assumptions. It is shown that in the nonviscous, and therefore also in the general case, there exists, in contrast to the analogous problem in Cartesian co-ordinates, no transformation reducing the given problem to a two-dimensional one. A fourth-order differential equation is derived for disturbances independent of the direction of the main flow. The solutions, which are obtained, show that those two-dimensional disturbances, termed cross disturbances, decay with time and do therefore not disturb the stability of the main flow. Explicit expressions for the cross disturbances are obtained and a discussion of their nature is given.

Sign in / Sign up

Export Citation Format

Share Document