Semiparametric analysis based on weighted estimating equations for transformation models with missing covariates

Missing covariate data are very common in regression analysis. In this paper, the weighted estimating equation method (Qi et al., 2005) [25] is used to extend the so-called unified estimation procedure (Chen et al., 2002) [4] for linear transformation models to the case of missing covariates. The non-missingness probability is estimated nonparametrically by the kernel smoothing technique. Under missing at random, the proposed estimators are shown to be consistent and asymptotically normal, with the asymptotic variance estimated consistently by the usual plug-in method. Moreover, the proposed estimators are more efficient than the weighted estimators with the inverse of true non-missingness probability as weight. Finite sample performance of the estimators is examined via simulation and a real dataset is analyzed to illustrate the proposed methods.     

Testing the no-treatment effect based on a possibly misspecified accelerated failure time model

The accelerated failure time model is a useful alternative to the Cox proportional hazard model. We investigate whether or not a misspecified accelerated failure time model provides a valid test of the no-treatment effect in randomized clinical trials. We show that the minimum dispersion statistic based on rank regression by Wei et al. (1990) must be modified in order to conduct valid tests under misspecification, whereas the resampling-based methods by Jin et al. (2003) are valid without any modification. Numerical studies are conducted to examine the small sample behavior of the modified minimum dispersion statistic and the resampling-based method. Finally, an illustration is given with a dataset from a clinical trial.     

Multivariate measures of skewness for the skew-normal distribution

The main objective of this work is to calculate and compare different measures of multivariate skewness for the skew-normal family of distributions. For this purpose, we consider the Mardia (1970) [10], Malkovich and Afifi (1973) [9], Isogai (1982) [17], Srivastava (1984) [15], Song (2001) [14], Móri et al. (1993) [11], Balakrishnan et al. (2007) [3] and Kollo (2008) [7] measures of skewness. The exact expressions of all measures of skewness, except for Song’s, are derived for the family of skew-normal distributions, while Song’s measure of shape is approximated by the use of delta method. The behavior of these measures, their similarities and differences, possible interpretations, and their practical use in testing for multivariate normal are studied by evaluating their power in the case of some specific members of the multivariate skew-normal family of distributions.     

Case-based belief formation under ambiguity

In this paper, we consider a decision maker who tries to learn the distribution of outcomes from previously observed cases. For each observed database of cases the decision maker predicts a set of priors expressing his beliefs about the underlying probability distribution. We impose a version of the concatenation axiom introduced in Billot et al. (2005) which ensures that the sets of priors can be represented as a weighted sum of the observed frequencies of cases. The weights are the uniquely determined similarities between the observed cases and the case under investigation. The predicted probabilities, however, may vary with the number of observations. This generalization of Billot et al. (2005) allows one to model learning processes.     

Bayes minimax estimators of the mean of a scale mixture of multivariate normal distributions

Bayes estimation of the mean of a variance mixture of multivariate normal distributions is considered under sum of squared errors loss. We find broad class of priors (also in the variance mixture of normal class) which result in proper and generalized Bayes minimax estimators. This paper extends the results of Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264] in a manner similar to that of Maruyama [Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distribution, J. Multivariate Anal. 21 (2003) 69-78] but somewhat more in the spirit of Fourdrinier et al. [On the construction of bayes minimax estimators, Ann. Statist. 26 (1998) 660-671] for the normal case, in the sense that we construct classes of priors giving rise to minimaxity. A feature of this paper is that in certain cases we are able to construct proper Bayes minimax estimators satisfying the properties and bounds in Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264]. We also give some insight into why Strawderman's results do or do not seem to apply in certain cases. In cases where it does not apply, we give minimax estimators based on Berger's [Minimax estimation of location vectors for a wide class of densities, Ann. Statist. 3 (1975) 1318-1328] results. A main condition for minimaxity is that the mixing distributions of the sampling distribution and the prior distribution satisfy a monotone likelihood ratio property with respect to a scale parameter.     

On pricing arithmetic average reset options with multiple reset dates in a lattice framework

We develop a straightforward algorithm to price arithmetic average reset options with multiple reset dates in a Cox et al. (CRR) (1979) [10] framework. The use of a lattice approach is due to its adaptability and flexibility in managing arithmetic average reset options, as already evidenced by Kim et al. (2003) [9]. Their model is based on the Hull and White (1993) [5] bucketing algorithm and uses an exogenous exponential function to manage the averaging feature, but their choice of fictitious values does not guarantee the algorithm’s convergence (cfr., Forsyth et al. (2002) [11]). We propose to overcome this drawback by selecting a limited number of trajectories among the ones reaching each node of the lattice, where we compute effective averages. In this way, the computational cost of the pricing problem is reduced, and the convergence of the discrete time model to the corresponding continuous time one is guaranteed.     

Principal points of a multivariate mixture distribution

A set of n-principal points of a distribution is defined as a set of n points that optimally represent the distribution in terms of mean squared distance. It provides an optimal n-point-approximation of the distribution. However, it is in general difficult to find a set of principal points of a multivariate distribution. Tarpey et al. [T. Tarpey, L. Li, B. Flury, Principal points and self-consistent points of elliptical distributions, Ann. Statist. 23 (1995) 103-112] established a theorem which states that any set of n-principal points of an elliptically symmetric distribution is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem, called a “principal subspace theorem”, is a strong tool for the calculation of principal points. In practice, we often come across distributions consisting of several subgroups. Hence it is of interest to know whether the principal subspace theorem remains valid even under such complex distributions. In this paper, we define a multivariate location mixture model. A theorem is established that clarifies a linear subspace in which n-principal points exist.     

On sparse estimation for semiparametric linear transformation models

Semiparametric linear transformation models have received much attention due to their high flexibility in modeling survival data. A useful estimating equation procedure was recently proposed by Chen et al. (2002) [21] for linear transformation models to jointly estimate parametric and nonparametric terms. They showed that this procedure can yield a consistent and robust estimator. However, the problem of variable selection for linear transformation models has been less studied, partially because a convenient loss function is not readily available under this context. In this paper, we propose a simple yet powerful approach to achieve both sparse and consistent estimation for linear transformation models. The main idea is to derive a profiled score from the estimating equation of Chen et al. [21], construct a loss function based on the profile scored and its variance, and then minimize the loss subject to some shrinkage penalty. Under regularity conditions, we have shown that the resulting estimator is consistent for both model estimation and variable selection. Furthermore, the estimated parametric terms are asymptotically normal and can achieve a higher efficiency than that yielded from the estimation equations. For computation, we suggest a one-step approximation algorithm which can take advantage of the LARS and build the entire solution path efficiently. Performance of the new procedure is illustrated through numerous simulations and real examples including one microarray data.     

Remarks on between estimator in the intraclass correlation model with missing data

In this paper, we consider the between estimator under the intraclass correlation model with missing data. We give a necessary and sufficient condition for existing exact simultaneous confidence intervals for all contrasts in the means under the between transformed model, which indicates the F-test statistic and simultaneous confidence intervals, constructed by Seo et al. [T. Seo, J. Kikuchi, K. Koizumi, On simultaneous confidence intervals for all contracts in the means of the intraclass correlation model with missing data, J. Multivariate Anal. 97 (2006) 1976–1983] based on the between estimator, is invalid. Furthermore, using the distribution of the between estimator, we present the exact test statistics and confidence intervals for partial contrasts.     

Low dimensional semiparametric estimation in a censored regression model

A new estimation procedure for a partial linear additive model with censored responses is proposed. To this aim, ideas of Lewbel and Linton [A. Lewbel, O. Linton, Nonparametric censored and truncated regression, Econometrica 70 (2002) 765-779] on censored model regression are combined with those of Kim et al. [W. Kim, O. Linton, N.W. Hengartner, A computationally efficient estimator for additive nonparametric regression with bootstrap confidence intervals, Journal of Computational and Graphical Statistics, 8 (1999) 278-297] on marginal integration and those on average derivatives. This allows for dimension reduction, interpretability and — depending on the context — for weights yielding computationally attractive estimates. Asymptotic behavior is provided for all proposed estimators.     

A new model transformation method and its application to extending a class of stability criteria of neutral type systems

This paper proposes a generalized equivalent model transformation method, which can include methods proposed by Fridman et al. and Bellen et al., for the stability analysis of a class of neutral type systems. By using the proposed model transformation method, a class of existing stability criteria derived by the Lyapunov functional approach can be extended to less conservative ones in terms of nonlinear matrix inequalities. Furthermore, procedures to solve these nonlinear matrix inequalities are also proposed. Illustrative examples are presented to demonstrate the effectiveness of the proposed model transformation method.     

A new dependence ordering with applications

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159-171]. Applications of this partial order to order statistics, k-record values and frailty models are given.     

On the comparison of the pre-test and shrinkage phi-divergence test estimators for the symmetry model of categorical data

The estimation problem of the parameters in a symmetry model for categorical data has been considered for many authors in the statistical literature (for example, Bowker (1948) [1], Ireland et al. (1969) [2], Quade and Salama (1975) [3], Cressie and Read (1988) [4], Menéndez et al. (2005) [5]) without using uncertain prior information. It is well known that many new and interesting estimators, using uncertain prior information, have been studied by a host of researchers in different statistical models, and many papers have been published on this topic (see Saleh (2006) [9] and references therein). In this paper, we consider the symmetry model of categorical data and we study, for the first time, some new estimators when non-sample information about the symmetry of the probabilities is considered. The decision to use a “restricted” estimator or an “unrestricted” estimator is based on the outcome of a preliminary test, and then a shrinkage technique is used. It is interesting to note that we present a unified study in the sense that we consider not only the maximum likelihood estimator and likelihood ratio test or chi-square test statistic but we consider minimum phi-divergence estimators and phi-divergence test statistics. Families of minimum phi-divergence estimators and phi-divergence test statistics are wide classes of estimators and test statistics that contain as a particular case the maximum likelihood estimator, likelihood ratio test and chi-square test statistic. In an asymptotic set-up, the biases and the risk under the squared loss function for the proposed estimators are derived and compared. A numerical example clarifies the content of the paper.     

Nonparametric rank-based tests of bivariate extreme-value dependence

A new class of tests of extreme-value dependence for bivariate copulas is proposed. It is based on the process comparing the empirical copula with a natural nonparametric rank-based estimator of the unknown copula under extreme-value dependence. A multiplier technique is used to compute approximate p-values for several candidate test statistics. Extensive Monte Carlo experiments were carried out to compare the resulting procedures with the tests of extreme-value dependence recently studied in Ben Ghorbal et al. (2009) [1] and Kojadinovic and Yan (2010) [19]. The finite-sample performance study of the tests is complemented by local power calculations.     

The wavelet transform method applied to a coupled chaotic system with asymmetric coupling schemes

The wavelet transform method originated by Wei et al. (2002) [19] is an effective tool for enhancing the transverse stability of the synchronous manifold of a coupled chaotic system. Much of the theoretical study on this matter is centered on networks that are symmetrically coupled. However, in real applications, the coupling topology of a network is often asymmetric; see Belykh et al. (2006)  [23], [24], Chavez et al. (2005)  [25], Hwang et al. (2005)  [26], Juang et al. (2007)  [17], and Wu (2003)  [13]. In this work, a certain type of asymmetric sparse connection topology for networks of coupled chaotic systems is presented. Moreover, our work here represents the first step in understanding how to actually control the stability of global synchronization from dynamical chaos for asymmetrically connected networks of coupled chaotic systems via the wavelet transform method. In particular, we obtain the following results. First, it is shown that the lower bound for achieving synchrony of the coupled chaotic system with the wavelet transform method is independent of the number of nodes. Second, we demonstrate that the wavelet transform method as applied to networks of coupled chaotic systems is even more effective and controllable for asymmetric coupling schemes as compared to the symmetric cases.     

A mathematical modeling for the lookback option with jump-diffusion using binomial tree method

The binomial tree method (BTM), first proposed by Cox et al. (1979) [4] in diffusion models and extended by Amin (1993) [9] to jump-diffusion models, is one of the most popular approaches to pricing options. In this paper, we present a binomial tree method for lookback options in jump-diffusion models and show its equivalence to certain explicit difference scheme. We also prove the existence and convergence of the optimal exercise boundary in the binomial tree approximation to American lookback options and give the terminal value of the genuine exercise boundary. Further, numerical simulations are performed to illustrate the theoretical results.     

A new updating method for the damped mass-spring systems

In this paper, we concern the inverse problem of constructing a monic quadratic pencil which possesses the prescribed partial eigendata, and the damping matrix and stiffness matrix are symmetric tridiagonal. Furthermore, the stiffness matrix is positive semi-definite and weakly diagonally dominant, which has positive diagonal elements and negative off-diagonal elements. Based on the solution of the inverse eigenvalue problem, we apply the alternating direction method with multiplier to solve the finite element model updating problem for the serially linked mass-spring system. The positive semi-definiteness of stiffness matrix, nonnegativity of stiffness and the physical connectivity of the original model are preserved. Numerical results show that our proposed method works well.     

Bias-reduced estimators for bivariate tail modelling

Ledford and Tawn (1997) introduced a flexible bivariate tail model based on the coefficient of tail dependence and on the dependence of the extreme values of the random variables. In this paper, we extend the concept by specifying the slowly varying part of the model as done by Hall (1982) with the univariate case. Based on Beirlant et al. (2009), we propose a bias-reduced estimator for the coefficient of tail dependence and for the estimation of small tail probabilities. We discuss the properties of these estimators via simulations and a real-life example. Furthermore, we discuss some theoretical asymptotic aspects of this approach.     

An adaptive estimator of the memory parameter and the goodness-of-fit test using a multidimensional increment ratio statistic

The increment ratio (IR) statistic was first defined and studied in Surgailis et al. (2007) [19] for estimating the memory parameter either of a stationary or an increment stationary Gaussian process. Here three extensions are proposed in the case of stationary processes. First, a multidimensional central limit theorem is established for a vector composed by several IR statistics. Second, a goodness-of-fit χ²-type test can be deduced from this theorem. Finally, this theorem allows to construct adaptive versions of the estimator and the test which are studied in a general semiparametric frame. The adaptive estimator of the long-memory parameter is proved to follow an oracle property. Simulations attest to the interesting accuracies and robustness of the estimator and the test, even in the non Gaussian case.     

A note on using periodogram-based distances for comparing spectral densities

Motivated by a recent paper of Caiado et al. (2009), we investigate testing problems for spectral densities of time series with unequal sample sizes. We thereby focus on analyzing their mathematical properties and illustrate our results in a small simulation study.