Asymptotic Theorem Research Articles

On a composition of Ismail–May operator

In this paper, we address a discrete operator based on the composition of Ismail–May operator and Szász–Mirakjan operator and study its approximation properties for various classes of continuous functions using Taylor's formula. We estimate its moments and further give direct theorems employing Peetre's K‐functional and the modulus of continuity. Furthermore, Voronovskaja‐type asymptotic theorems are given. Additionally, we provide further compositions and give a comparison among their approximation properties.

Sequential monitoring for conditional quantiles of general conditional heteroscedastic time series models

AbstractIn this study, we introduce an online monitoring procedure designed to sequentially detect change points in the conditional quantiles of location‐scale time series models. This statistical process control issue holds great significance in risk management, particularly in measuring the value‐at‐risk or expected shortfall of financial assets. Our approach employs suitable detectors, including cumulative sum statistics. We then define a stopping rule and determine control limits based on asymptotic theorems to signal an anomaly. To further evaluate the proposed methods, we conduct a comprehensive empirical study analyzing various aspects of our monitoring procedures when applied to location‐scale time series models. Additionally, we perform a real data analysis using the daily returns of the Korea Composite Stock Price Index (KOSPI) and EuroStoxx 50 indices to affirm the adequacy of the proposed monitoring procedures in real‐world applications.

Optimal Calibration of Items for Multidimensional Achievement Tests

AbstractMultidimensional achievement tests are recently gaining more importance in educational and psychological measurements. For example, multidimensional diagnostic tests can help students to determine which particular domain of knowledge they need to improve for better performance. To estimate the characteristics of candidate items (calibration) for future multidimensional achievement tests, we use optimal design theory. We generalize a previously developed exchange algorithm for optimal design computation to the multidimensional setting. We also develop an asymptotic theorem saying which item should be calibrated by examinees with extreme abilities. For several examples, we compute the optimal design numerically with the exchange algorithm. We see clear structures in these results and explain them using the asymptotic theorem. Moreover, we investigate the performance of the optimal design in a simulation study.

Bid-ask spread dynamics: large upward jump with geometric catastrophes

We propose a simple continuous-time stochastic model for capturing the dynamics of a limit order book in the presence of liquidity fluctuations, manifested by gaps in filled price levels within the OB. Inspired by [D. Farmer, L. Gillemot, F. Lillo, S. Mike and A. Sen, Quant. Finance 4 (2004) 383–397.], we define a model for the dynamics of spread that incorporates liquidity fluctuations and undertake a comprehensive theoretical study of the model’s properties, providing rigorous proofs of several key asymptotic theorems. Furthermore, we show how large deviations manifest in the spread under this regime.

Asymptotic Theorems for Discrete Markov Chains

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Factor-augmented Model for Functional Data

We propose modeling raw functional data as a mixture of a smooth function and a high-dimensional factor component. The conventional approach to retrieving the smooth function from the raw data is through various smoothing techniques. However, the smoothing model is inadequate to recover the smooth curve or capture the data variation in some situations. These include cases where there is a large amount of measurement error, the smoothing basis functions are incorrectly identified, or the step jumps in the functional mean levels are neglected. A factor-augmented smoothing model is proposed to address these challenges, and an iterative numerical estimation approach is implemented in practice. Including the factor model component in the proposed method solves the aforementioned problems since a few common factors often drive the variation that cannot be captured by the smoothing model. Asymptotic theorems are also established to demonstrate the effects of including factor structures on the smoothing results. Specifically, we show that the smoothing coefficients projected on the complement space of the factor loading matrix are asymptotically normal. As a byproduct of independent interest, an estimator for the population covariance matrix of the raw data is presented based on the proposed model. Extensive simulation studies illustrate that these factor adjustments are essential in improving estimation accuracy and avoiding the curse of dimensionality. The superiority of our model is also shown in modeling Australian temperature data.

Approximation by Modified Szasz Operators with a new Modification of Brenke Type Polynomials

In the present article we study the approximation properties of modified Szász operators with a new modification of Brenke type polynomials. First, we estimate the rate of convergence, for the newly defined operators, by means of modulus of smoothness, Peetre’s K-functional and Lipschitz type functions. Furthermore, we also prove a Voronovskaja type asymptotic theorem.

Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials

<p style='text-indent:20px;'>Karaisa [<xref ref-type="bibr" rid="b29">29</xref>] presented Jakimovski- Leviatan- Durrmeyer type operators by means of Appell polynomials. In a similar manner, Wani et al. [<xref ref-type="bibr" rid="b43">43</xref>] proposed a sequence of Jakimovski-Leviatan-Durrmeyer type operators involving Brenke type polynomials which include Appell polynomials and Hermite polynomials. We note that the definitions of the operators given in both these papers are not correct. In the present article, we introduce a Stancu variant of the operators considered in [<xref ref-type="bibr" rid="b43">43</xref>] after correcting their definition. The definition of the operator proposed in [<xref ref-type="bibr" rid="b29">29</xref>] may be similarly corrected. We establish the Korovkin type approximation theorem and the rate of convergence by means of the usual modulus of continuity, Peetre's K-functional and the class of Lipschitz type functions for our operators. Next, we discuss the Voronovskaja and Gr<inline-formula><tex-math id="M1">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type asymptotic theorems. Finally, we study the convergence of these operators in a weighted space and the Korovkin type weighted statistical approximation theorem.</p>

Approximation by half Bernstein polynomials

This paper defines and studies the half Bernstein polynomials to approximate functions in C[0,1]. The pointwise convergence in ordinary approximation is given for these polynomials. Also, the order of approximation, the Voronovskaya-type asymptotic theorem and the error estimate are introduced. Finally, to show the ability convergence of these polynomials we provide some examples to approximate these polynomials to test functions and compare numerical results with the results corresponding to classical Bernstein polynomials.

Factor Overnight GARCH-Itô Models

Abstract This article introduces a unified factor overnight GARCH-Itô model for large volatility matrix estimation and prediction. To account for whole-day market dynamics, the proposed model has two different instantaneous factor volatility processes for the open-to-close and close-to-open periods, while each embeds the discrete-time multivariate GARCH model structure. To estimate latent factor volatility, we assume the low rank plus sparse structure and employ nonparametric estimation procedures. Then, based on the connection between the discrete-time model structure and the continuous-time diffusion process, we propose a weighted least squares estimation procedure with the non-parametric factor volatility estimator and establish its asymptotic theorems.

Semi-discrete Voronovskaya-type theorem for positive linear operators based on Hermite interpolation with two double knots

Since the classical asymptotic theorems of Voronovskaya-type for positive and linear operators are in fact based on the Taylor’s formula which is a very particular case of Lagrange-Hermite interpolation formula, in the recent paper Gal [3], I have obtained semi-discrete quantitative Voronovskaya-type theorems based on other Lagrange-Hermite interpolation formulas. These include Lagrange interpolation on two and three simple knots and Hermite interpolation on two knots, one simple and the other one double. In the present paper we obtain a semi-discrete quantitative Voronovskaya-type theorems based on Hermite interpolation on two double knots.

A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: Scalar-valued kernels

We characterize Fredholm determinants of a class of Hankel composition operators via matrix-valued Riemann-Hilbert problems, for additive and multiplicative compositions. The scalar-valued kernels of the underlying integral operators are not assumed to display the integrable structure known from the seminal work of Its, Izergin, Korepin and Slavnov [42]. Yet we are able to describe the corresponding Fredholm determinants through a naturally associated Riemann-Hilbert problem of Zakharov-Shabat type by solely exploiting the kernels' Hankel composition structures. We showcase the efficiency of this approach through a series of examples, we then compute several rank one perturbed determinants in terms of Riemann-Hilbert data and finally derive Akhiezer-Kac asymptotic theorems for suitable kernel classes.

Higher-Spin Currents, Operator Mixing and UV Asymptotics in Large-N QCD-like Theories

We extend to operator mixing—specifically, to higher-spin twist-2 operators—the asymptotic theorem on the ultraviolet asymptotics of the spectral representation of 2-point correlators of multiplicatively renormalizable operators in large-N confining QCD-like theories. The extension is based on a recent differential geometric approach to operator mixing that involves the Poincaré-Dulac theorem and allows us to reduce generically the operator mixing to the multiplicatively renormalizable case, provided that γ0β0 is diagonalizable and a certain nonresonant condition for its eigenvalues holds according to the Poincaré-Dulac theorem, with γ0 and β0 the one-loop coefficients of the anomalous dimension matrix and beta function respectively. Relatedly, we solve a conundrum about the generic nonconservation of higher-spin currents versus the conservation—up to contact terms—of the corresponding free propagators in the spectral representation of 2-point correlators of higher-spin operators of pure integer spin to the leading large-N order.

Approximation by modified Bernstein polynomials based on real parameters

In this paper, we introduce a modified Bernstein-type operators based on two real parameters and study its various approximation properties. We derive some direct results e.g. Voronovkaja type asymptotic theorem, an estimate of error in ordinary as well as in Ditzian Totik modulus of smoothness and an error estimate for functions belonging to the Lipschitz type space. Further, we examine the rate of approximation for a Kirov and Popova type generalization of these operators.

VORONOVSKAYA-TYPE THEOREM FOR POSITIVE LINEAR OPERATORS BASED ON LAGRANGE INTERPOLATION

Since the classical asymptotic theorems of Voronovskaya-type for positive and linear operators are in fact based on the Taylor’s formula which is a very particular case of Lagrange-Hermite interpolation for­mula, in the recent paper Gal [3], I have obtained semi-discrete quanti­tative Voronovskaya-type theorems based on other Lagrange-Hermite interpolation formulas, like Lagrange interpolation on two and three simple knots and Hermite interpolation on two knots, one simple and the other one double. In the present paper we obtain a semi-discrete quantitative Voronovskaya-type theorem based on Lagrange interpola­tion on arbitrary p + 1 simple distinct knots.

Some approximation results on Chlodowsky type q−Bernstein-Schurer operators

The main concern of this article is to obtain several approximation features of the new Chlodowsky type q-Bernstein-Schurer operators. We prove the Korovkin type approximation theorem and discuss the order of convergence with regard to the ordinary modulus of continuity, an element of Lipschitz type and Peetre?s K-functional, respectively. In addition, we derive the Voronovskaya type asymptotic theorem. Finally, using of Maple software, we present the comparison of the convergence of Chlodowsky type q-Bernstein-Schurer operators to the certain functions with some graphical illustrations and error estimation tables.

Approximation Properties of the Blending-Type Bernstein–Durrmeyer Operators

We construct the blending-type modified Bernstein–Durrmeyer operators and investigate their approximation properties. First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. Finally, we derive the rate of convergence for functions with a derivative of bounded variation. The results show that the new operators have good approximation properties.

Data analysis of progressive‐stress accelerated life tests with group effects

ABSTRACT Progressive-stress accelerated life testing (PSALT) is a special type of experiment that tests the lifetime of a product with continuously varying stress levels. Due to the limitations of testing equipments and costs, the lifetime data collected by PSALT are usually censored and have group effects. In order to deal with the two characteristics in the data, this paper presents a novel PSALT model with group effects under progressive censoring. Two-stage and Gauss-Hermite quadrature methods are proposed to estimate the model parameters, while the interval estimates are constructed by bootstrap and the asymptotic theorem, respectively. Simulation studies are conducted to compare the proposed model with the traditional models without group effects in terms of the relative bias and root mean squared error under different scenarios. The results show that the proposed model can detect group-to-group variation, and that the models without group effects will result in large biases for estimating the characteristic lifetime of the product. Finally, the proposed model is validated by a real dataset.

On the rate of convergence of modified \(\alpha\)-Bernstein operators based on q-integers

In the present paper we define a q-analogue of the modified a-Bernstein operators introduced by Kajla and Acar (Ann. Funct. Anal. 10 (4) 2019, 570-582). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre's \(K\)-functional. Next, we investigate the Gruss- Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre's K- functional , the Voronovskaja and Gruss Voronovskaja type theorems are also investigated. Lastly, we construct the associated GBS (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness.

Precise Asymptotics for the Uniform Empirical Process and the Uniform Sample Quantile Process

One of the sources of “invariance principle” is that the limit properties of the uniform empirical process coincide with that of a Brownian bridge. The deep discussion of limit theorem of the uniform empirical process gathered wild interest of the researchers. In this paper, the precise convergence rate of the uniform empirical process is considered. As is well-known, when ε tends to 0, the precise asymptotic theorems can be demonstrated by referring to the classical method of Gut and Spǎtaru, by using some nice probability inequalities and so on. However, if ε tends to a positive constant, other powerful methods and tools are needed. The method of strong approximation is used in this paper. The main theorems are proved by using the Brownian bridge B t to approximate the uniform empirical process α n t . The relevant results for the uniform sample quantile process are also presented.