Exponential Estimates Research Articles

A simulation based optimization of factor-type exponential estimators in sample surveys with coefficients of variation and kurtosis

In the context of sample surveys, this article presents factor-type exponential ratio estimators as a technique for estimating population means. These estimators use supplementary information for an auxiliary variable, such as coefficients of variation and coefficients of kurtosis. The research includes a range of alpha values from −1 to +1 to improve estimate.Following that, the study rigorously develops formulas for both bias and mean square error for the suggested estimators. Up to the first degree of large sample approximation, these formulas are produced. The empirical findings we present are renowned in that they show that the suggested estimators consistently beat existing exponential estimators with significantly lower mean squared errors.We calculate the suggested estimator’s percent relative efficiency in comparison to the traditional mean estimator to put a number on how effective it is. We show persuasive proof of the significant improvement offered by our estimators over existing exponential estimators by a complete numerical illustration and a rigorously conducted simulations exercise. Summary:We present factor-type exponential ratio estimators for estimating population means in sample surveys. Key points:We take into account auxiliary information, different alpha values, and various correlation circumstances. Objective:We want to improve population mean estimation precision. Method:To assess our estimators, we use numerical examples and simulations. Results:Our estimators typically outperform competitors, with smaller mean squared errors. Advantage:We demonstrate their robustness and efficiency in improving population mean estimate through rigorous simulations.

Distributed exponential state estimation of linear systems over jointly connected switching networks

Recently, the distributed state estimation problem for continuous-time linear systems over jointly connected switching networks was solved. Due to the use of the generalized Barbalat’s Lemma, it can only show that the estimation errors asymptotically converge to the origin. By a completely different approach, this paper further studies the distributed state estimation problem with two new features. First, the asymptotic convergence is strengthened to the exponential convergence. This strengthened result not only offers a guaranteed convergence rate, but also renders the error system total stability and thus the capability to withstand small disturbances. Second, the coupling gains of our local observers can be different from each other and thus offers greater design flexibility, while the coupling gains in the existing result were required to be identical. These two new features are achieved by establishing exponential stability for two classes of linear time-varying systems, which may have other applications.

Modified Exponential Ratio-Type Estimator of Population Mean in Stratified Sampling using Calibration Approach

In this paper, the problem of estimation of finite population mean in stratified random sampling is considered. Two improved exponential logarithmic type calibration estimators for finite population mean have been proposed for stratified random sampling when auxiliary information related to variable under study is available for each stratum. To judge the performance of the proposed estimators, a simulation study has been carried out in R-software using two datasets, one real and another one artificial generated population. The proposed estimators have also been compared with the estimators developed by Bahl and Tuteja [1] and Singh [17] in case of stratified random sampling.

Imperfect Debugging-Based Generalized Exponential Testing Effort Estimation for Software Fault Detection and Removal

In the current technological era, the reliability growth model depicts the failure in software that is employed for estimating the reliability of software. Conventional software reliability growth model (SRGMs) usually presume that faults in software are immediately corrected once a software fault is found and no new faults are introduced. This consideration can be impractical. However, introduction of several new faults is possible throughout debugging, which is called imperfect debugging. During such a process of debugging and introduction of new faults, consumption of testing effort also plays an important role. Hence, in this paper, two models are proposed that incorporate testing effort along with imperfect debugging during fault correction. Additionally, the combined testing effort is the addition of two efforts, specifically detection effort as well as correction effort. The formulated models are validated for real data set and contrasted with additional famous SRGMs. These inferences indicate that Model 2 performs best in terms of fitting and prediction capability.

Thermal relaxation toward equilibrium and periodically pulsating spherical bubbles in an incompressible liquid

We study the radial relaxation dynamics toward equilibrium and time-periodic pulsating spherically symmetric gas bubbles in an incompressible liquid due to thermal effects. The asymptotic model (Prosperetti, 1991; Biro and Velazquez, 2000) is one where the pressure within the gas bubble is spatially uniform and satisfies an ideal gas law, relating the pressure, density and temperature of the gas. The temperature of the surrounding liquid is taken to be constant and the behavior of the liquid pressure at infinity is prescribed to be constant or periodic in time. In (Lai and Weinstein, 2023), for the case where the liquid pressure at infinity is a positive constant, we proved the existence of a one-parameter manifold of spherical equilibria, parametrized by the bubble mass, and further proved that it is a nonlinearly and exponentially asymptotically stable center manifold. In the present article, we first refine the exponential time-decay estimates, via a study of the linearized dynamics subject to the constraint of fixed mass. We then study the nonlinear radial dynamics of the bubble–fluid system subject to a small-amplitude, time-periodic far-field pressure. We prove that nonlinearly and exponentially asymptotically stable time-periodically pulsating solutions of the nonlinear model exist for all sufficiently small forcing amplitudes. The existence of such states is formulated as a fixed point problem for the Poincaré return map, and the existence of a fixed point makes use of our exponential time-decay estimates of the linearized problem.

Sharp asymptotic estimates for expectations, probabilities, and mean first passage times in stochastic systems with small noise

AbstractFreidlin‐Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well‐posed matrix Riccati equations involving the minimizer of the Freidlin‐Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction‐advection‐diffusion type.

Exponential decay estimates for the damped defocusing Schrödinger equation in exterior domains

Exponential decay estimates for the damped defocusing Schrödinger equation in exterior domains

Adaptive Fuzzy Boundary Observer Design for Uncertain Linear Coupled Hyperbolic Partial Differential Equation Systems

Joint uncertainties and state estimation of a class of linear coupled hyperbolic partial differential equation systems in the presence of unstructured and structured uncertainties are studied in this paper. For unstructured uncertainties which are completely unknown, by employing Takagi-Sugeno fuzzy logic system to approximate the unstructured uncertainties, a novel adaptive fuzzy boundary observer is developed to estimate both unknown system states as well as unknown weights in the fuzzy logic system, and the estimation errors are ultimately bounded. Therein, in the design of the proposed observer, a set of swapping filters and infinite dimensional backstepping technique are combined. On the other hand, for structured uncertainties that can be described in a concrete parameterized form, the proposed method can easily achieve the exact estimation of weights and states to their true values. The rigorous proof is provided to show that the ultimately bounded estimation errors for the case of unstructured uncertainties and the exponential convergent estimation errors for the case of structured uncertainties can be realized. Finally, three illustrative simulations are carried out to show the feasibility and effectiveness of the developed methods in this paper.

The spectrum theory of the discrete Schrödinger operator and its application

This paper introduces the spectrum theory of discrete Schrödinger operators with different kinds of potentials, including bounded, unbounded, periodic, or complex potentials. The paper also provides exponential estimates of the Green's function and eigenfunctions of the discrete Schrödinger operators. As an application, I review some of our results on standing wave solutions of discrete Schrödinger equations.

Exponentially quantile regression-ratio-type estimators for robust mean estimation

Traditional ordinary least square (OLS) regression is commonly utilized to develop regression-ratio type estimators with traditional and non traditional measures of location. However, when data are contaminated by outliers, the ordinary least square estimates become inappropriate, and the alternative approach is to use the robust regression method. To solve this issue, the use of robust regression tools for mean estimation is a commonly settled practice. In the present study, we have proposed an efficient family of exponential quantile regression-ratio type estimators by using the auxiliary information for estimating the finite population mean under simple random sampling scheme. Here it is worth noting that quantile regression is robust to outliers. Mathematical expressions such as bias, mean squared error (MSE), and minimum mean squared error are derived up to first order of approximation. To support theoretical findings, two real data collections originating from different sources are used for numerical illustration. The results are showing the superiority of proposed exponential quantile robust regression family of estimators as compared to the existing estimators under simple random sampling scheme.

Calibration estimation of subpopulation total for direct and indirect situations

Sample surveys are conducted to estimate parameters of subpopulation such as mean, variance and totals, etc. The utility of the supporting information lies in the calibration technique, we use to estimate the subpopulation parameter. In this article, we propose calibration-based estimators for a total of the subpopulations. The defined estimators are based on gamma and exponential models. Since both models have right-skewed or positively skewed forms, the defined estimators are appropriate for right-skewed or positively skewed supportive variables. Two situations are considered in which the total of the supporting variable hold for all subpopulations: (i) the total of subpopulations of the supporting variable is known in advance and (ii) the total subpopulations of the supportive variable are not known. Due to difficulty arising in the calibration weight, we have carried out a simulation study in terms of absolute relative bias and simulated relative standard error using Sweden Municipalities 1984 (Sarndal, Swensson, and Wretman, 1992, Appendix B). A simulation study shows that the estimator based on the gamma function is more efficient than ratio, generalized regression, and exponential estimators for all subpopulations in the considered situations.

Neutrosophic ratio-type exponential estimators for estimation of population mean

The current work is one step in filling a large void in the research left by the advent of neutrosophic Statistics (NS), a philosophized variant of classical statistics (CS). The philosophy of NS deals with techniques for investigating data that is ambiguous, hazy, or uncertain. The traditional techniques of estimation utilizing auxiliary information work under specific determinate data, which in the case of neutrosophic data may lead to mistakes (over/ under-estimation). This study presents a generalized neutrosophic ratio-type exponential estimator (NRTEE) for estimating location parameters and achieving the lowest mean square error (MSE) possible for interval neutrosophic data (IND). The offered NRTEE helps to deal with the uncertainty and ambiguity of data. Unlike typical estimators, its findings are not single-valued but rather in interval form, which reduces the possibility of over-or under-estimation caused by single crisp outcomes and also increases the likelihood of the parameter dwelling in the interval. It improves the efficiency of the estimator since we have an estimated interval that contains the unknown value of the population mean with a minimal MSE. The suggested NRTEE’s efficiency is further addressed by utilizing real-life IND of temperature and simulations. A comparison is also performed to establish the superiority of the proposed estimator over the traditional estimators. The limits are calculated and discussed in cases when our suggested estimator is always efficient. The suggested estimator is the most efficient of all estimators and outperformed all others on both IND and classical data.

Exponential method of estimation in sampling theory under robust quantile regression methods

–In the regression analysis, ordinary least square techniques is commonly used. However, the data’s outcomes may be untrustworthy if there is an outliers in it. In order to deal with the outliers problem, robust quantile regression methods have been frequently presented as alternatives to OLS for a long time. In this article, primarily a exponential ratio-type estimators is suggested. After that, robust quantile regression estimators are proposed, that is a useful strategy. The application of robust quantile regression empowered the efficiency of the estimators especially for outliers in the data. The MSE equations of the various estimators are computed and compared to OLS approaches. Numerical illustration and simulations studies are performed to support our theoretical findings.

New Criteria of Event-triggered Exponential State Estimation for Delayed semi-Markovian Memristor-based Neural Networks

New Criteria of Event-triggered Exponential State Estimation for Delayed semi-Markovian Memristor-based Neural Networks

Exponential Stability and Estimates of Solutions to Systems of Functional Differential Equations

Exponential Stability and Estimates of Solutions to Systems of Functional Differential Equations

Stability and Stabilization of Nonlinear Stochastic Systems With Synchronous and Asynchronous Switching Parameters to the States.

This article is concerned with the stability and stabilization of nonlinear hybrid stochastic systems. First, the concepts of synchronous and asynchronous switching parameters to the system states are proposed for a generalization of the asynchronous control. As preliminaries, a hybrid nonlinear differential inequality is established for coping with the synchronous models, and the -continuity of the system parameters driven by Markov chains is established for coping with the asynchronous models by the attribute of Markov chains for less conservativeness. Second, stability criteria with moment exponential estimates are established for two kinds of system models under the nonlinear growth condition. Third, the stabilization problem is illustrated based on the stability criteria and Riccati like matrix equations. A corollary is given to improve some stability theorems obtained in the related literature. New initial condition is formally proposed and explained for the nonlinear stochastic systems. Finally, a numerical example with simulation is proposed to illustrate the method, verify the conclusions of this article, and show the superiority of this work.

On Some New Exponential Ratio Estimator of Population Mean in Two Phase Sampling

In this paper, we suggest employing the exponential ratio estimator to estimate the mean of the study variable using a two-phase sample strategy with two modified auxiliary variables. Several researchers discussed the properties of the estimators they proposed and discovered that the estimators in their studies were relatively efficient. The estimators previously studied are listed chronologically in the appendix to this paper. In two phase sampling, the estimator's mean square errors and relative efficiencies are calculated using auxiliary variable information. To assess the properties of our proposed estimator, we noticed that it has a lower mean square error (MSE) than the classical ratio estimator and some other exponential ratio estimators. The estimator is more useful than other estimators in solving real-world issues, notably in engineering, environmental science, management, and biological sciences. The proposed estimator has been applied to real-world data sets such as BRICS, Son's Head Measurement, Number of Hospital Beds, Sale Price of Residence, Ambient Pressure (AP), and Heating Load. In survey research, our suggested estimator has also been demonstrated to be more effective.

Acetabular fractures from Judet and Letournel to the present: Research trends and global outcomes with bibliometric analysis during 1980 to 2022.

Fractures of the acetabulum are one of the most challenging injuries treated by orthopedic surgeons. However, a bibliometric analysis has not been performed in the literature on acetabular fractures, which seriously affect the quality of life of patients. The aim of this study was to summarize the bibliometric and intellectual structure, and determine and map the most recent trends on the topic of acetabular fractures by analyzing the social and structural relationships between the different research components of articles published on the acetabular fractures. 1599 articles on acetabular fractures published between 1980 and 2022 were extracted from the Web of Science (WoS) database and analyzed. Bibliometric visualization maps were used to reveal trending topics, citation analyses, and international collaborations. Spearman correlation analysis was performed for correlation investigations. The trend in the expected number of articles to be published over the next few years was displayed using the exponential smoothing estimator. The top 3 contributing countries to the literature were United States of America (USA) (551, 34.4%), China (170, 10.6%), and Germany (160, 10%). The most active author was Berton R. Moed (n = 29) and the most active institution was the University of California System (n = 41). A high-level statistically significant correlation was found between the number of articles on the topic of acetabular fractures published by nations and the gross domestic product (GDP) and GDP per capita values of those countries (R = 0.719, P < .001; R = 0.701, P < .001, respectively). The trending topics researched in recent years were 3D printing, 3-dimensional printing, outcomes, Open Reduction and Internal Fixation (ORIF), mortality, Kocher-Langenbeck, Pararectus approach, tranexamic acid, transfusion, epidemiology, fracture mapping, modified Stoppa approach, post-traumatic osteoarthritis, pelvis fracture, pelvic trauma, fracture reduction, and pelvic ring injury. The leading countries in research on the subject of acetabular fractures were seen to be western countries with large economies (especially the USA, European countries, and Canada) and China, India and Turkey.

Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups

AbstractGiven a Gromov‐hyperbolic group endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of . More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry–Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates address a question of Kaimanovich–Kapovich–Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson–Sullivan measures on the boundary of the group.

Lyapunov functionals for linear continuous-time difference systems: A single delay case

This paper focuses on the Lyapunov stability problem of linear continuous-time difference systems. We show that although the stability of these systems has been extensively investigated, the problem of determining suitable quadratic Lyapunov functionals is not completely solved yet, even for the simplest system with a single delay. Herein, using the single delay case as an example, we construct new quadratic Lyapunov functionals. The functionals, which depend on a matrix-valued function strongly related to the discrete-time Lyapunov matrix equation, allow deriving a converse Lyapunov functional result. We show how the functionals can be used to solve problems such as the computation of exponential estimates for solutions and robustness bounds of perturbed systems.