Asymptotic properties of M-estimators in linear and nonlinear multivariate regression models

We consider the (possibly nonlinear) regression model in \(\mathbb{R }^q\) with shift parameter \(\alpha \) in \(\mathbb{R }^q\) and other parameters \(\beta \) in \(\mathbb{R }^p\) . Residuals are assumed to be from an unknown distribution function (d.f.). Let \(\widehat{\phi }\) be a smooth \(M\) -estimator of \(\phi = {{\beta }\atopwithdelims (){\alpha }}\) and \(T(\phi )\) a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of \(T(\widehat{\phi })\) and an estimator of \(T(\phi )\) with bias \(\sim n^{-2}\) requiring \(\sim n\) calculations. (In contrast, the jackknife and bootstrap estimators require \(\sim n^2\) calculations.) For a linear regression with random covariates of low skewness, if \(T(\phi ) = \nu \beta \) , then \(T(\widehat{\phi })\) has bias \(\sim n^{-2}\) (not \(n^{-1}\) ) and skewness \(\sim n^{-3}\) (not \(n^{-2}\) ), and the usual approximate one-sided confidence interval (CI) for \(T(\phi )\) has error \(\sim n^{-1}\) (not \(n^{-1/2}\) ). These results extend to random covariates.     

Random weighting estimation of stable exponent

This paper presents a new random weighting method to estimation of the stable exponent. Assume that $X_1, X_2, \ldots ,X_n$ is a sequence of independent and identically distributed random variables with $\alpha $ -stable distribution G, where $\alpha \in (0,2]$ is the stable exponent. Denote the empirical distribution function of G by $G_n$ and the random weighting estimation of $G_n$ by $H_n$ . An empirical distribution function $\widetilde{F}_n$ with U-statistic structure is defined based on the sum-preserving property of stable random variables. By minimizing the Cramer-von-Mises distance between $H_n$ and ${\widetilde{F}}_n$ , the random weighting estimation of $\alpha $ is constructed in the sense of the minimum distance. The strong consistency and asymptotic normality of the random weighting estimation are also rigorously proved. Experimental results demonstrate that the proposed random weighting method can effectively estimate the stable exponent, resulting in higher estimation accuracy than the Zolotarev, Press, Fan and maximum likelihood methods.     

Consistency of a nonparametric estimation of a density functional

LetX be a random variable with distribution functionF and density functionf. Let ? and ψ be known measurable functions defined on the real lineR and the closed interval [0, 1], respectively. This paper proposes a smooth nonparametric estimate of the density functional \(\theta = \int\limits_R \phi (x) \psi \left[ {F (x)} \right]f^2 (x) dx\) based on a random sampleX ₁, ...,X _n fromF using a kernel functionk. The proposed estimate is given by \(\hat \theta = (n^2 a_n )^{ - 1} \mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^n \phi (X_i ) \psi \left[ {\hat F (X_i )} \right]k\left[ {(X_i - X_j )/a_n } \right]\) , where \(\hat F(x) = n^{ - 1} \mathop \sum \limits_{i = 1}^n K\left[ {(x - X_i )/a_n } \right]\) with \(K (w) = \int\limits_{ - \infty }^w {k (u) } du\) . The estimate \(\hat \theta \) is shown to be consistent both in the weak and strong sense and is used to estimate the asymptotic relative efficiency of various nonparametric tests, with particular reference to those using the Chernoff-Savage statistic.     

Estimation of the scale parameter of a log-logistic distribution

In this work the ranked set sampling technique has been applied to estimate the scale parameter $\alpha $ of a log-logistic distribution under a situation where the units in a sample can be ordered by judgement method without any error. We have evaluated the Fisher information contained in the order statistics arising from this distribution and observed that median of a random sample contains the maximum information about the parameter $\alpha $ . Accordingly we have used median ranked set sampling to estimate $\alpha $ . We have further carried out the multistage median ranked set sampling to estimate $\alpha $ with improved precision. Suppose it is not possible to rank the units in a sample according to judgement method without error but the units can be ordered based on an auxiliary variable $Z$ such that $(X, Z)$ has a Morgenstern type bivariate log-logistic distribution (MTBLLD). In such a situation we have derived the Fisher information contained in the concomitant of rth order statistic of a random sample of size $n$ from MTBLLD and identified those concomitants among others which possess largest amount of Fisher information and defined an unbalanced ranked set sampling utilizing those units in the sample and thereby proposed an estimator of $\alpha $ using the measurements made on those units in this ranked set sample.     

On the maxima of heterogeneous gamma variables with different shape and scale parameters

In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let \(X_{1}\) and \(X_{2}\) be two independent gamma random variables with \(X_{i}\) having shape parameter \(r_{i}>0\) and scale parameter \(\lambda _{i}\) , \(i=1,2\) , and let \(X^{*}_{1}\) and \(X^{*}_{2}\) be another set of independent gamma random variables with \(X^{*}_{i}\) having shape parameter \(r_{i}^{*}>0\) and scale parameter \(\lambda _{i}^{*}\) , \(i=1,2\) . Denote by \(X_{2:2}\) and \(X^{*}_{2:2}\) the corresponding maxima, respectively. It is proved that, among others, if \((r_{1},r_{2})\) majorize \((r_{1}^{*},r_{2}^{*})\) and \((\lambda _{1},\lambda _{2})\) weakly majorize \((\lambda _{1}^{*},\lambda _{2}^{*})\) , then \(X_{2:2}\) is stochastically larger that \(X^{*}_{2:2}\) in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature.     

Bayesian prediction in doubly stochastic Poisson process

A stochastic marked point process model based on doubly stochastic Poisson process is considered in the problem of prediction for the total size of future marks in a given period, given the history of the process. The underlying marked point process \((T_{i},Y_{i})_{i\ge 1}\) , where \(T_{i}\) is the time of occurrence of the \(i\) th event and the mark \(Y_{i}\) is its characteristic (size), is supposed to be a non-homogeneous Poisson process on \(\mathbb {R}_{+}^{2}\) with intensity measure \(P\times \varTheta \) , where \(P\) is known, whereas \(\varTheta \) is treated as an unknown measure of the total size of future marks in a given period. In the problem of prediction considered, a Bayesian approach is used assuming that \(\varTheta \) is random with prior distribution presented by a gamma process. The best predictor with respect to this prior distribution is constructed under a precautionary loss function. A simulation study for comparing the behavior of the predictors under various criteria is provided.     

Estimation and inference in parametric deterministic frontier models

In this paper we consider parametric deterministic frontier models. For example, the production frontier may be linear in the inputs, and the error is purely one-sided, with a known distribution such as exponential or half-normal. The literature contains many negative results for this model. Schmidt (Rev Econ Stat 58:238–239, 1976) showed that the Aigner and Chu (Am Econ Rev 58:826–839, 1968) linear programming estimator was the exponential MLE, but that this was a non-regular problem in which the statistical properties of the MLE were uncertain. Richmond (Int Econ Rev 15:515–521, 1974) and Greene (J Econom 13:27–56, 1980) showed how the model could be estimated by two different versions of corrected OLS, but this did not lead to methods of inference for the inefficiencies. Greene (J Econom 13:27–56, 1980) considered conditions on the distribution of inefficiency that make this a regular estimation problem, but many distributions that would be assumed do not satisfy these conditions. In this paper we show that exact (finite sample) inference is possible when the frontier and the distribution of the one-sided error are known up to the values of some parameters. We give a number of analytical results for the case of intercept only with exponential errors. In other cases that include regressors or error distributions other than exponential, exact inference is still possible but simulation is needed to calculate the critical values. We also discuss the case that the distribution of the error is unknown. In this case asymptotically valid inference is possible using subsampling methods.     

{\varvec{P}}-value model selection criteria for exponential families of increasing dimension

Let $\mathcal{M }_{\underline{i}}$ be an exponential family of densities on $[0,1]$ pertaining to a vector of orthonormal functions $b_{\underline{i}}=(b_{i_1}(x),\ldots ,b_{i_p}(x))^\mathbf{T}$ and consider a problem of estimating a density $f$ belonging to such family for unknown set ${\underline{i}}\subset \{1,2,\ldots ,m\}$ , based on a random sample $X_1,\ldots ,X_n$ . Pokarowski and Mielniczuk (2011) introduced model selection criteria in a general setting based on p-values of likelihood ratio statistic for $H_0: f\in \mathcal{M }_0$ versus $H_1: f\in \mathcal{M }_{\underline{i}}\setminus \mathcal{M }_0$ , where $\mathcal{M }_0$ is the minimal model. In the paper we study consistency of these model selection criteria when the number of the models is allowed to increase with a sample size and $f$ ultimately belongs to one of them. The results are then generalized to the case when the logarithm of $f$ has infinite expansion with respect to $(b_i(\cdot ))_1^\infty $ . Moreover, it is shown how the results can be applied to study convergence rates of ensuing post-model-selection estimators of the density with respect to Kullback–Leibler distance. We also present results of simulation study comparing small sample performance of the discussed selection criteria and the post-model-selection estimators with analogous entities based on Schwarz’s rule as well as their greedy counterparts.     

The dynamics of social agreement according to Conceptual Agreement Theory

Many social phenomena can be viewed as processes in which individuals in social groups develop agreement (e.g., public opinion, the spreading of rumor, the formation of social and linguistic conventions). Conceptual Agreement Theory (CAT) models social agreement as a simplified communicational event in which an Observer \((O)\) and Actor \((A)\) exchange ideas about a concept \(C\) , and where \(O\) uses that information to infer whether \(A\) ’s conceptual state is the same as its own (i.e., to infer agreement). Agreement may be true (when \(O\) infers that \(A\) is thinking \(C\) and this is in fact the case, event \(a1\) ) or illusory (when \(O\) infers that \(A\) is thinking \(C\) and this is not the case, event \(a2\) ). In CAT, concepts that afford \(a1\) or \(a2\) become more salient in the minds of members of social groups. Results from an agent-based model (ABM) and probabilistic model that implement CAT show that, as our conceptual analyses suggested would be the case, the simulated social system selects concepts according to their usefulness to agents in promoting agreement among them (Experiment 1). Furthermore, the ABM exhibits more complex dynamics where similar minded agents cluster and are able to retain useful concepts even when a different group of agents discards them (Experiment 2). We discuss the relevance of CAT and the current findings for analyzing different social communication events, and suggest ways in which CAT could be put to empirical test.     

Asymptotic behaviour of near-maxima of Gaussian sequences

Let \((X_1,X_2,\ldots ,X_n)\) be a Gaussian random vector with a common correlation coefficient \(\rho _n,\,0\le \rho _n<1\) , and let \(M_n= \max (X_1,\ldots , X_n),\,n\ge 1\) . For any given \(a>0\) , define \(T_n(a)= \left\{ j,\,1\le j\le n,\,X_j\in (M_n-a,\,M_n]\right\} ,\,K_n(a)= \#T_n(a)\) and \(S_n(a)=\sum \nolimits _{j\in T_n(a)}X_j,\,n\ge 1\) . In this paper, we obtain the limit distributions of \((K_n(a))\) and \((S_n(a))\) , under the assumption that \(\rho _n\rightarrow \rho \) as \(n\rightarrow \infty ,\) for some \(\rho \in [0,1)\) .     

The effect of the regularity of the error process on the performance of kernel regression estimators

This article considers estimation of regression function $f$ in the fixed design model $Y(x_i)=f(x_i)+ \epsilon (x_i), i=1,\ldots ,n$ , by use of the Gasser and Müller kernel estimator. The point set $\{ x_i\}_{i=1}^{n}\subset [0,1]$ constitutes the sampling design points, and $\epsilon (x_i)$ are correlated errors. The error process $\epsilon $ is assumed to satisfy certain regularity conditions, namely, it has exactly $k$ ( $=\!0, 1, 2, \ldots $ ) quadratic mean derivatives (q.m.d.). The quality of the estimation is measured by the mean squared error (MSE). Here the asymptotic results of the mean squared error are established. We found that the optimal bandwidth depends on the $(2k+1)$ th mixed partial derivatives of the autocovariance function along the diagonal of the unit square. Simulation results for the model of $k$ th order integrated Brownian motion error are given in order to assess the effect of the regularity of this error process on the performance of the kernel estimator.     

An additive property of weak records from geometric distributions

Let $\{W_m\}{_{m\ge 1}}$ be the sequence of weak records from a discrete parent random variable, $X$ , supported on the non-negative integers. We obtain a new characterization of geometric distributions based on an additive property of weak records: $X$ follows a geometric distribution if and only if for certain integers, $n,\, s\ge 1, W_{n+s}\stackrel{d}{=}W_n+W^{\prime }_s$ , with $W^{\prime }_s$ independent of $W_n$ and $W^{\prime }_s\stackrel{d}{=} W_s$ .     

Some results on constructing general minimum lower order confounding 2^{n-m} designs for n\le 2^{n-m-2}

Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect-number pattern for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. All the GMC \(2^{n-m}\) designs with \(N/4+1\le n\le N-1\) were constructed by Li et al. (Stat Sinica 21:1571–1589, 2011), Zhang and Cheng (J Stat Plan Inference 140:1719–1730, 2010) and Cheng and Zhang (J Stat Plan Inference 140:2384–2394, 2010), where \(N=2^{n-m}\) is run number and \(n\) is factor number. In this paper, we first study some further properties of GMC design, then we construct all the GMC \(2^{n-m}\) designs respectively with the three parameter cases of \(n\le N-1\) : (i) \(m\le 4\) , (ii) \(m\ge 5\) and \(n=(2^m-1)u+r\) for \(u>0\) and \(r=0,1,2\) , and (iii) \(m\ge 5\) and \(n=(2^m-1)u+r\) for \(u\ge 0\) and \(r=2^m-3,2^m-2\) .     

Asymptotic infimum coverage probability for interval estimation of proportions

In this paper, we discuss asymptotic infimum coverage probability (ICP) of eight widely used confidence intervals for proportions, including the Agresti–Coull (A–C) interval (Am Stat 52:119–126, 1998) and the Clopper–Pearson (C–P) interval (Biometrika 26:404–413, 1934). For the A–C interval, a sharp upper bound for its asymptotic ICP is derived. It is less than nominal for the commonly applied nominal values of 0.99, 0.95 and 0.9 and is equal to zero when the nominal level is below 0.4802. The \(1-\alpha \) C–P interval is known to be conservative. However, we show through a brief numerical study that the C–P interval with a given average coverage probability \(1-\gamma \) typically has a similar or larger ICP and a smaller average expected length than the corresponding A–C interval, and its ICP approaches to \(1-\gamma \) when the sample size goes large. All mathematical proofs and R-codes for computation in the paper are given in Supplementary Materials.     

Stochastic orderings of convolution residuals

In this paper we study convolution residuals, that is, if $X_1,X_2,\ldots ,X_n$ are independent random variables, we study the distributions, and the properties, of the sums $\sum _{i=1}^lX_i-t$ given that $\sum _{i=1}^kX_i>t$ , where $t\in \mathbb R $ , and $1\le k\le l\le n$ . Various stochastic orders, among convolution residuals based on observations from either one or two samples, are derived. As a consequence computable bounds on the survival functions and on the expected values of convolution residuals are obtained. Some applications in reliability theory and queueing theory are described.     

Hermite ranks and U-statistics

We focus on the asymptotic behavior of $U$ -statistics of the type $$\begin{aligned} \sum _{1\le i\ne j\le n} h(X_i,X_j)\\ \end{aligned}$$ in the long-range dependence setting, where $(X_i)_{i\ge 1}$ is a stationary mean-zero Gaussian process. Since $(X_i)_{i\ge 1}$ is Gaussian, $h$ can be decomposed in Hermite polynomials. The goal of this paper is to compare the different notions of Hermite rank and to provide conditions for the remainder term in the decomposition to be asymptotically negligeable.     

Testing for the shape parameter of generalized extreme value distribution based on the L_q-likelihood ratio statistic

This paper studies the applications of extreme value theory on analysis for closing price data of the Dow-Jones industrial index and Danish fire insurance claims data. The generalized extreme value (GEV) distribution is considered in analyzing the real data, and the hypothesis testing problem for the shape parameter of GEV distribution is investigated based on a new test statistic—the $L_q$ -likelihood ratio ( $L_q$ R) statistic. The $L_q$ R statistic can be treated as a generalized form of the classical likelihood ratio (LR) statistic. We show that the asymptotic behavior of proposed statistic is characterized by the degree of distortion $q$ . For small and modest sample sizes, the $L_q$ R statistic is still available when $q$ is properly chosen. By simulation studies, the proposed statistic not only performs the asymptotic properties, but also outperforms the classical LR statistic as the sample sizes are modest or even small. Meanwhile, the test power based on the new statistic is also simulated by Monte Carlo methods. At last, the models are diagnosed by graphical methods.     

An extremal property of the generalized arcsine distribution

The main result of the paper is the following characterization of the generalized arcsine density p _γ (t) = t ^γ?1(1 ? t) ^γ?1/B(γ, γ)   with ${t \in (0, 1)}$ and ${\gamma \in(0,\frac12) \cup (\frac12,1)}$ : a r.v. ξ supported on [0, 1] has the generalized arcsine density p _γ (t) if and only if ${ {\mathbb E} |\xi- x|^{1-2 \gamma}}$ has the same value for almost all ${x \in (0,1)}$ . Moreover, the measure with density p _γ (t) is a unique minimizer (in the space of all probability measures μ supported on (0, 1)) of the double expectation ${ (\gamma-\frac12 ) {\mathbb E} |\xi-\xi^{\prime}|^{1-2 \gamma}}$ , where ξ and ξ′ are independent random variables distributed according to the measure μ. These results extend recent results characterizing the standard arcsine density (the case ${\gamma=\frac12}$ ).