On iterative procedures of asymptotic inference

Abstract  An informal discussion is given on performing an unconstrained maximization or solving non-linear equations of statistics by iterative methods with the quadratic termination property. It is shown that if a miximized function, e.g. likelihood, is asymptotically quadratic, then for asymptotically efficient inference finitely many iterations are needed.     

Local minimax pointwise estimation of a multivariate density

We consider the problem of the nonparametric minimax estimation of a multivariate density at a given point. A concept of smoothness classes in nonparametric minimax estimation problems is proposed. The smoothness of a function is characterized by the approximability of the function at a point by an integral of the product of this function with an approximate identity. We propose a singular integral estimator, an integral of this approximate identity with respect to the empirical distribution function. Under some assumptions on the approximate identity, the bias of the estimator is shown to be of smaller order asymptotically than the variance, and the estimator itself is shown to be asymptotically locally minimax with respect to the quadratic risk in a proper topology.     

Large-sample estimation strategies for eigenvalues of a Wishart matrix

The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.     

Quasi-maximum likelihood estimation of volatility with high frequency data

This paper investigates the properties of the well-known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise, by extending the classic asymptotic results of quasi-maximum likelihood estimation. When trying to estimate the integrated volatility and the variance of noise, this parametric approach remains consistent, efficient and robust as a quasi-estimator under misspecified assumptions. Moreover, it shares the model-free feature with nonparametric alternatives, for instance realized kernels, while being advantageous over them in terms of finite sample performance. In light of quadratic representation, this estimator behaves like an iterative exponential realized kernel asymptotically. Comparisons with a variety of implementations of the Tukey–Hanning₂ kernel are provided using Monte Carlo simulations, and an empirical study with the Euro/US Dollar future illustrates its application in practice.     

Distribution-free tests for time series models specification

We consider a class of time series specification tests based on quadratic forms of weighted sums of residuals autocorrelations. Asymptotically distribution-free tests in the presence of estimated parameters are obtained by suitably transforming the weights, which can be optimally chosen to maximize the power function when testing in the direction of local alternatives. We discuss in detail an asymptotically optimal distribution-free alternative to the popular Box–Pierce when testing in the direction of AR or MA alternatives. The performance of the test with small samples is studied by means of a Monte Carlo experiment.     

The solution of the infinite horizon tracking problem for discrete time systems possessing an exogenous component

In this paper we derive an algorithm that yields, for a discrete-time system, a control minimizing a quadratic cost functional. The system considered is linear and possesses an exogenous component. The cost functional is a quadratic tracking equation over an infinite time horizon with positive semi-definite weighting matrices such that a weighted sum of these matrices is positive definite. The infinite planning horizon Minimum Variance cost criterion and the Linear Quadratic regulator are special cases. For stabilizable systems we give a characterization of the asymptotically admissible reference trajectories.     

Some tests of alternative urban population density functions

The purpose of this study is to test the usefulness of some alternative gross population density functions for urbanized areas. Two evaluation criteria are used; maximum explanatory power in standard regression analysis and accuracy in predicting total population in the urbanized area. It is concluded that the explanatory power of the negative exponential function can be improved upon in some cases by adding a quadratic distance term, but that population can be predicted more accurately if the quadratic term is omitted. Also, it is found that constraining the negative exponential function to predict population exactly reduces explanatory power by an insignificant amount.     

Semiparametric inference in a GARCH-in-mean model

A new semiparametric estimator for an empirical asset pricing model with general nonparametric risk-return tradeoff and GARCH-type underlying volatility is introduced. Based on the profile likelihood approach, it does not rely on any initial parametric estimator of the conditional mean function, and it is under stated conditions consistent, asymptotically normal, and efficient, i.e., it achieves the semiparametric lower bound. A sampling experiment provides finite sample comparisons with the parametric approach and the iterative semiparametric approach with parametric initial estimate of Conrad and Mammen (2008). An application to daily stock market returns suggests that the risk-return relation is indeed nonlinear.     

Jackknife model averaging

We consider the problem of obtaining appropriate weights for averaging M approximate (misspecified) models for improved estimation of an unknown conditional mean in the face of non-nested model uncertainty in heteroskedastic error settings. We propose a “jackknife model averaging” (JMA) estimator which selects the weights by minimizing a cross-validation criterion. This criterion is quadratic in the weights, so computation is a simple application of quadratic programming. We show that our estimator is asymptotically optimal in the sense of achieving the lowest possible expected squared error. Monte Carlo simulations and an illustrative application show that JMA can achieve significant efficiency gains over existing model selection and averaging methods in the presence of heteroskedasticity.     

Optimal estimation under nonstandard conditions

We analyze optimality properties of maximum likelihood (ML) and other estimators when the problem does not necessarily fall within the locally asymptotically normal (LAN) class, therefore covering cases that are excluded from conventional LAN theory such as unit root nonstationary time series. The classical Hájek–Le Cam optimality theory is adapted to cover this situation. We show that the expectation of certain monotone “bowl-shaped” functions of the squared estimation error are minimized by the ML estimator in locally asymptotically quadratic situations, which often occur in nonstationary time series analysis when the LAN property fails. Moreover, we demonstrate a direct connection between the (Bayesian property of) asymptotic normality of the posterior and the classical optimality properties of ML estimators.     

Might randomization in queue discipline be useful when waiting cost is a concave function of waiting time?

This paper suggests that introducing randomization in queue discipline might be welfare enhancing in certain queues for which the cost of waiting is a concave function of waiting time. Concavity can make increased variability in waiting times good not bad for aggregate customer welfare. Such concavity may occur if the costs of waiting asymptotically approach some maximum or if the customer incurs a fixed cost if there is any wait at all. As examples, cost might asymptotically approach a maximum for patients seeking organ transplants who will not live beyond a certain threshold time, and fixed costs could pertain for knowledge workers seeking a piece of information that is required to proceed with their current task, so any delay creates a “set up charge” associated with switching tasks.     

A Bayesian learning procedure for the.(s, Q) inventory policy

We present an asymptotically optimal Bayesian learning procedure for the ( s, Q ) inventory policy, for the case when the probability distribution of lead time demand is unknown. This distribution is not required to be a member of a certain family, and the maximal lead time demand is also allowed to be unknown. The algorithm developed for this purpose Is an extension of a standard iterative procedure, which in its original form -in spite of claims to the contrary-might produce solution values that are arbitrarily far away from the optimal one.     

An efficient GMM estimator of spatial autoregressive models

In this paper, we consider GMM estimation of the regression and MRSAR models with SAR disturbances. We derive the best GMM estimator within the class of GMM estimators based on linear and quadratic moment conditions. The best GMM estimator has the merit of computational simplicity and asymptotic efficiency. It is asymptotically as efficient as the ML estimator under normality and asymptotically more efficient than the Gaussian QML estimator otherwise. Monte Carlo studies show that, with moderate-sized samples, the best GMM estimator has its biggest advantage when the disturbances are asymmetrically distributed. When the diagonal elements of the spatial weights matrix have enough variation, incorporating kurtosis of the disturbances in the moment functions will also be helpful.     

Asymptotic properties of minimum contrast estimators for parameters of boolean models

Summary This paper presents a method for the estimation of parameters of random closed sets (racs’s) in ℝ ^d based on a single realization within a (large) convex sampling window. The essential idea first applied by Diggle (1981) in a special case consists in defining the estimation by minimizing a suitably defined distance (called contrast function) between the true and the empirical contact distribution function of the racs under consideration, where the most relevant case of Boolean models is discussed in details. The resulting estimates are shown to be strongly consistent (if the racs is ergodic) and asymptotically normal (if the racs is Boolean) when the sampling window expands unboundedly.     

The local stability of an open-loop Nash equilibrium in a finite horizon differential game

We propose a finite time differential game as a model for some economic processes and derive conditions for the Nash equilibrium solution to be locally asymptotically stable. We adopt the traditional ‘Cournot-reaction function’ notion of stability, which in our (continuous time) model becomes a function-to-function, or trajectory-to-trajectory, mapping. The conditions for stability seem to make economic sense. The equilibrium is less stable if the interaction terms in each period are large, if the game has a long duration, and if the discount rate is small.     

Minimum distance estimators for count data based on the probability generating function with applications

This paper studies properties of parameter estimators obtained by minimizing a distance between the empirical probability generating function and the probability generating function of a model for count data. Specifically, it is shown that, under certain not restrictive conditions, the resulting estimators are consistent and, suitably normalized, asymptotically normal. These properties hold even if the model is misspecified. Three applications of the obtained results are considered. First, we revisit the goodness-of-fit problem for count data and propose a weighted bootstrap estimator of the null distribution of test statistics based on the above cited distance. Second, we give a probability generating function version of the model selection test problem for separate, overlapping and nested families of distributions. Finally, we provide an application to the problem of testing for separate families of distributions. All applications are illustrated with numerical examples.     

The Misspecification of Arma Models

The object of this paper is to assess the effects of fitting a model of the wrong order to a time series which is generated by an autoregressive moving–average process. The method is to examine the spectral density functions which are indicated by the probability limits of the least–squares estimators of the misspecified models. The least–squares estimates are asymptotically equivalent to the maximum–likelihood estimates.
The experiments reported in this paper suggest that, if the spectral density function of the data–generating process displays prominent modes, then there is a danger of being seriously misled about the nature of the process whenever one fits a model with too few parameters. It also transpires that, in such cases, the criterion function is liable to have several local minima.
Autoregressive moving–average models are being used increasingly in object detection applications where the spectrum of a signal serves to identify the nature of its source. Our results suggest that radical misidentifications can result from the use of incorrectly parametrised models.     

The Engel curve for alcohol and the rank of demand systems

This paper shows that the quadratic shape of the Engel curve for alcohol is induced by preference heterogeneity between drinkers and abstainers in a Japanese data set. With controlling the heterogeneity, it is shown that the Engel curve for alcohol slopes monotonically downwards for drinkers, and that the probability of being a drinker is an increasing function of total expenditure. These two relationships generate a quadratic shape for the Engel curve for alcohol. Other goods in this data set appear to have nearly linear Engel curves, so if the alcohol Engel curve for drinkers is also linear, then after controlling for this preference heterogeneity the rank of this demand system would be two. Copyright © 2006 John Wiley & Sons, Ltd.     

GMM estimation of social interaction models with centrality

This paper considers the specification and estimation of social interaction models with network structures and the presence of endogenous, contextual, correlated, and group fixed effects. When the network structure in a group is captured by a graph in which the degrees of nodes are not all equal, the different positions of group members as measured by the Bonacich (1987) centrality provide additional information for identification and estimation. In this case, the Bonacich centrality measure for each group can be used as an instrument for the endogenous social effect, but the number of such instruments grows with the number of groups. We consider the 2SLS and GMM estimation for the model. The proposed estimators are asymptotically efficient, respectively, within the class of IV estimators and the class of GMM estimators based on linear and quadratic moments, when the sample size grows fast enough relative to the number of instruments.     

基于C语言数学计算器设计算法探讨

本文讲述了运用C语言来编写求一元二次、一元三次和多元线性方程的解的思路方法。首先对每个功能方法的原理进行阐述,最后我们对计算出来的结果进行了检验。在方程中,我们用到公式法、牛顿切线法、雅可比迭代法和高斯-赛德尔迭代法。