Moment Inequalities Research Articles

Asymptotic properties of recursive kernel density estimation for long-span high-frequency data

. This article mainly studies the asymptotic properties of recursive kernel density estimation for high-frequency data with a long time span. Under appropriate conditions, the variance, bias, mean squared error, and optimal bandwidth are obtained. The asymptotic normality is proven by using moment inequalities for ρ-mixing high-frequency data. Numerical simulations show that the recursive kernel density estimation provides good fitting results for the invariant density function of the Vasicek diffusion process. Empirical analysis reveals that recursive kernel density estimation not only reflects the distribution characteristics of financial data but also allows us to fit parameter models by minimizing the squared deviation between the recursive kernel density estimation and the parameter model. It provides more applications for the financial field.

A convexity-constrained parameterization of the random effects generalized partial credit model.

An alternative closed-form expression for the marginal joint probability distribution of item scores under the random effects generalized partial credit model is presented. The closed-form expression involves a cumulant generating function and is therefore subjected to convexity constraints. As a consequence, complicated moment inequalities are taken into account in maximum likelihood estimation of the parameters of the model, so that the estimation solution is always proper. Another important favorable consequence is that the likelihood function has a single local extreme point, the global maximum. Furthermore, attention is paid to expected a posteriori person parameter estimation, generalizations of the model, and testing the goodness-of-fit of the model. Procedures proposed are demonstrated in an illustrative example.

Correcting for Misclassified Binary Regressors Using Instrumental Variables

Estimators that exploit an instrumental variable to correct for misclassification in a binary regressor typically assume that the misclassification rates are invariant across all values of the instrument. We show this assumption is invalid in routine empirical settings. We derive a new estimator which allows misclassification rates to vary across values of the instrumental variable. Our key identifying assumption, that the sum of misclassification rates remains constant across instrument values, follows from the empirical examples we present. We also show this assumption can be relaxed using moment inequalities that arise from our model. We demonstrate the usefulness of our estimator through Monte Carlo simulations and a re-analysis of the extent to which Medicaid eligibility crowds out other forms of health insurance. Correcting for measurement error substantially reduces estimates of crowd out and the extent to which Medicaid eligibility lowers the share of the uninsured.

Isoperimetric inequalities for inner parallel curves

We prove weighted isoperimetric inequalities for smooth, bounded, and simply-connected domains. More precisely, we show that the moment of inertia of inner parallel curves for domains with fixed perimeter attains its maximum for a disk. This inequality, which was previously only known for convex domains, allows us to extend an isoperimetric inequality for the magnetic Robin Laplacian to non-convex centrally symmetric domains. Furthermore, we extend our isoperimetric inequality for moments of inertia, which are second moments, to p -th moments for all p smaller than or equal to two. We also show that the disk is a strict local maximiser in the nearly circular, centrally symmetric case for all p strictly less than three, and that the inequality fails for all p strictly bigger than three.

Moment inequalities for sums of weakly dependent random fields

Moment inequalities for sums of weakly dependent random fields

Φ-moment inequalities for noncommutative differentially subordinate martingales

We establish some Φ \Phi -moment inequalities for noncommutative differentially subordinate martingales. Let Φ \Phi be a p p -convex and q q -concave Orlicz function with 1 > p ≤ q > 2 1>p\leq q>2 . Suppose that x x and y y are two self-adjoint martingales such that y y is weakly differentially subordinate to x x . We show that, for N ≥ 0 N\geq 0 , τ [ Φ ( | y N | ) ] ≤ c p , q τ [ Φ ( | x N | ) ] , \begin{equation*} \tau \big [\Phi (|y_N|)\big ]\leq c_{p,q}\tau \big [\Phi (|x_N|)\big ], \end{equation*} where the constant c p , q c_{p,q} is of the best order when p = q p=q . The Φ \Phi -moment estimates for square functions of noncommutative differentially subordinate martingales are also obtained in this article. Our approach provides constructive proofs of noncommutative Φ \Phi -moment Burkholder–Gundy inequalities and Burkholder inequalities.

Bounds for novel extended beta and hypergeometric functions and related results

We introduce a new unified extension of the integral form of Euler’s beta function with a MacDonald function in the integrand and establish functional upper bounds for it. We use this definition to extend as well the Gaussian and Kummer’s confluent hypergeometric functions, for which we provide bounding inequalities. Moreover, we use our extension of the beta function to define a new probability distribution, for which we establish raw moments and moment inequalities and, as by-products, Turán inequalities for the initially defined extended beta function.

On consistency for wavelet estimator of regression function based on biased samples under extended negatively dependence

ABSTRACT For a nonparametric regression model with biased samples under extended negatively dependence, we concern with the estimation problem of the regression function over Sobolev space. On the basis of the wavelet kernel, we propose a wavelet estimator and investigate its consistency properties under mild conditions. For the proposed wavelet estimator, we establish the pth mean consistency based on moment inequality and get the complete consistency by virtue of exponential inequality and truncation technique. It should be pointed out that the complete consistency of wavelet estimator is a novel theoretical result in the nonparametric regression model with biased samples. In addition, the effectiveness of the proposed wavelet estimator is demonstrated on some numerical experiments.

Second-order moments of the size of randomly induced subgraphs of given order

For a graph G and a positive integer c, let Mc(G) be the size of a subgraph of G induced by a randomly sampled subset of c vertices. Second-order moments of Mc(G) encode part of the structure of G. We use this fact, coupled to classical moment inequalities, to prove graph theoretical results, to give combinatorial identities, to bound the size of the c-densest subgraph from below and the size of the c-sparsest subgraph from above, and to provide bounds for approximate enumeration of trivial subgraphs.

On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence

Abstract In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than that obtained for the real-valued random variables by Chen et al. [The von Bahr–Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl. 419 (2014), 1290–1302], and Chen and Sung [Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications, J. Math. Inequal. 10(3) (2016), 837–848]. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended.

Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains

Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains

Asymptotic normality of kernel density estimation for mixing high-frequency data

High-frequency data is widely used and studied in many fields. In this paper, the asymptotic normality of kernel density estimator under ρ-mixing high-frequency data is studied. We first derive some moment inequalities for mixing high-frequency data, and then use them to study the asymptotic normality of the kernel density estimator, and give Berry-Esseen upper bounds. The numerical simulations report that the kernel density estimation of high-frequency data has asymptotic normality, and the result is consistent with the theoretical conclusions. The actual data analysis shows that the kernel density estimation can well capture the characteristics of the distribution, and can use these features and the least square deviation principle to fit the parameter model, which is more convenient for further theoretical analysis and application analysis.

Asymptotic normality of Nadaraya–Waton kernel regression estimation for mixing high-frequency data

High-frequency data is widely used and studied in many fields, especially in the econometrics and statistics. In this paper, the asymptotic normality of Nadaraya–Waton (NW) kernel regression estimator under ρ-mixing high-frequency data is studied. We first derive some moment inequalities for ρ-mixing high-frequency data, and then use them to study the asymptotic normality of NW kernel regression estimator, and give Berry–Esseen upper bounds. The numerical simulations report that the kernel regression estimator of high-frequency data has good asymptotic normality. Our empirical analysis is to fit the correlation between the five minute interval price increment and the corresponding trading volume for the Shanghai Stock Exchange Index, Entrepreneurship Index and Real Estate Index. These kernel regression curves better reflect people's investment behaviour.

Moment inequalities for multinomial choice with fixed effects

This paper proposes a new approach to identification of the semiparametric multinomial choice model with fixed effects. The framework employed is the semiparametric version of the traditional multinomial logit with the fixed‐effects model (Chamberlain (1980)). This semiparametric multinomial choice model places no restrictions on either the joint distribution of the random utility disturbances across choices or their within group (or across time) correlations. We show that a novel within‐group comparison leads to a set of conditional moment inequalities. Our main finding shows that the derived conditional moment inequalities yield the sharp identified set for the random utility covariate index, while avoiding the incidental parameter problem. Specializing this result to the binary choice case shows that Manski (1987)'s conditional moment inequalities still lead to sharp bounds without restrictions on covariates.

On the Inequalities for Moments of Branching Random Processes

On the Inequalities for Moments of Branching Random Processes

Discordant relaxations of misspecified models

In many set‐identified models, it is difficult to obtain a tractable characterization of the identified set. Therefore, researchers often rely on nonsharp identification conditions, and empirical results are often based on an outer set of the identified set. This practice is often viewed as conservative yet valid because an outer set is always a superset of the identified set. However, this paper shows that when the model is refuted by the data, two sets of nonsharp identification conditions derived from the same model could lead to disjoint outer sets and conflicting empirical results. We provide a sufficient condition for the existence of such discordancy, which covers models characterized by conditional moment inequalities and the Artstein (1983) inequalities. We also derive sufficient conditions for the nonexistence of discordant submodels, therefore providing a class of models for which constructing outer sets cannot lead to misleading interpretations. In the case of discordancy, we follow Masten and Poirier (2021) by developing a method to salvage misspecified models, but unlike them, we focus on discrete relaxations. We consider all minimum relaxations of a refuted model that restores data consistency. We find that the union of the identified sets of these minimum relaxations is robust to detectable misspecifications and has an intuitive empirical interpretation.

Partial identification and inference in duration models with endogenous censoring

AbstractThis paper studies identification and inference in transformation models with endogenous censoring. Many kinds of duration models, such as the accelerated failure time model, proportional hazard model, and mixed proportional hazard model, can be viewed as transformation models. We allow the censoring of a duration outcome to be arbitrarily correlated with observed covariates and unobserved heterogeneity. We impose no parametric restrictions on either the transformation function or the distribution function of the unobserved heterogeneity. In this setting, we develop bounds on the regression parameters and the transformation function, which are characterized by conditional moment inequalities involving U‐statistics. Subsequently, we provide inference methods for them by constructing an inference approach for conditional moment inequality models in which the sample analogs of moments are U‐statistics. We apply the proposed inference methods to evaluate the effect of unemployment insurance on duration of joblessness using data from the Current Population Survey's Displaced Workers Supplements.

Convergence properties for randomly weighted sums of ρ-mixing sequences with related statistical applications

In this paper, we investigate some convergence properties, such as complete convergence, complete moment convergence, complete f-moment convergence and strong law of large numbers, for partial sums of randomly weighted sums of -mixing sequences by using the moment inequality, which extend and improve the corresponding ones in the literature. As applications, we obtain the complete consistency for the randomly weighted estimator in a nonparametric regression models and the estimators for the unknown parameters in the simple linear errors-in-variables regression models based on -mixing errors.

Moment inequalities for mixing long-span high-frequency data and strongly consistent estimation of OU integrated diffusion process

Mixing is not much used in the high-frequency literature so far. However, mixing is a common weakly dependent property of continuous and discrete stochastic processes, such as Gaussian, Ornstein–Uhlenberck (OU), Vasicek, CIR, CKLS, logistic diffusion, generalized logistic diffusion, and double-well diffusion processes. So, long-span high-frequency data typically have weak dependence, and using mixing to study them is also an alternative approach. In this paper, we give some moment inequalities for long-span high-frequency data with ϕ-mixing, ρ-mixing, and α-mixing. These inequalities are effective tools for studying asymptotic properties. Applying these inequalities, we investigate the strong consistency of parameter estimation for the OU-integrated diffusion process. We also derive the mean square error of the estimation of the OU process and the optimal interval for the drift parameter estimator.

Local linearization based subvector inference in moment inequality models

This paper introduces a bootstrap-based profiling inference method for subvectors in moment inequality models following insights from Bugni et al. (2017). Compared to their paper, the new method calculates the critical value by searching over a local neighborhood of a pre-estimator, instead of the whole null parameter space, to profile out nuisance parameters. In this way, non-linear moment conditions are simplified by linear expansion and the bootstrap iterates over quadratic programming problems, which significantly simplifies and accelerates computation. This method controls asymptotic size uniformly over a large class of data generating processes. In the Monte Carlo simulations, the new procedure improves upon the computing time of Bugni et al. (2017) and Kaido et al. (2019) significantly. I provide an empirical illustration estimating an airline entry game.