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.     

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.     

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.     

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 \(r=2^m-3,2^m-2\) .     

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.     

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 $W^{\prime }_s\stackrel{d}{=} W_s$ .     

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.     

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.     

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.     

On Kullback–Leibler information of order statistics in terms of the relative risk

The representation of the entropy in terms of the hazard function and its extensions have been studied by many authors including Teitler et al. (IEEE Trans Reliab 35:391–395, 1986). In this paper, we consider a representation of the Kullback–Leibler information of the first \(r\) order statistics in terms of the relative risk (Park and Shin in Statistics, 2012), the ratio of hazard functions, and extend it to the progressively Type II censored data. Then we study the change in Kullback–Leibler information of the first \(r\) order statistics according to \(r\) and discuss its relation with Fisher information in order statistics.     

Double symmetry model and its decompositions in square contingency tables having ordered categories

In this paper, we have employed the non-standard log-linear models to fit the double symmetry models and some of its decompositions to square contingency tables having ordered categories. SAS PROC GENMOD was employed to fit these models although we could similarly have used GENLOG in SPSS or GLM in STATA. A SAS macro generates the factor or scalar variables required to fit these models. Two sets of \(4 \times 4\) unaided distance vision data that have been previously analyzed in (Tahata and Tomizawa, Journal of the Japan Statistical Society 36:91–106, 2006) were employed for verification of results. We also extend the approach to the Danish \(5 \times 5\) Mobility data as well as to the \(3 \times 3\) Danish longitudinal study data of subjective health, firstly reported in (Andersen, The Statistical Analysis of Categorical Data, Springer:Berlin, 1994) and analyzed in (Tahata and Tomizawa, Statistical Methods and Applications 19:307–318, 2010). Results obtained agree with those published in previous literature on the subject. The approaches suggest here eliminate any programming that might be required in order to apply these class of models to square contingency tables.     

D-optimal chemical balance weighing designs with autoregressive errors

In this paper, we consider the estimation problem of individual weights of three objects. For the estimation we use the chemical balance weighing design and the criterion of D-optimality. We assume that the error terms ${\varepsilon_{i},\ i=1,2,\dots,n,}$ are a first-order autoregressive process. This assumption implies that the covariance matrix of errors depends on the known parameter ρ. We present the chemical balance weighing design matrix ${\widetilde{\bf X}}$ and we prove that this design is D-optimal in certain classes of designs for ${\rho\in[0,1)}$ and it is also D-optimal in the class of designs with the design matrix ${{\bf X} \in M_{n\times 3}(\pm 1)}$ for some ρ ≥ 0. We prove also the necessary and sufficient conditions under which the design is D-optimal in the class of designs ${M_{n\times 3}(\pm 1)}$ , if ${\rho\in[0,1/(n-2))}$ . We present also the matrix of the D-optimal factorial design with 3 two-level factors.     

Asymptotic behavior of the hazard rate in systems based on sequential order statistics

The limiting behavior of the hazard rate of coherent systems based on sequential order statistics is examined. Related results for the survival function of the system lifetime are also considered. For deriving the results, properties of limits involving a relevation transform are studied in detail. Then, limits of characteristics in sequential \(k\) -out-of- \(n\) systems and general coherent systems with failure-dependent components are obtained. Applications to the comparison of different systems based on their long run behavior and to limits of coefficients in a signature-based representation of the residual system lifetime are given.     

{\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 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.     

On the choice of measures of reliability and validity in the content-analysis of texts

The paper discusses several reliability measures: Scott’s pi, Krippendorff’s alpha, free marginal adjustment (Bennett, Alpert and Goldstein’s \(S\) ), Cohen’s kappa, and Perreault and Leigh’s \(I\) and the assumptions on which they are based. It is suggested that correlation coefficients between, on one hand, the distribution of qualitative codes and, on the other hand, word co-occurrences and the distribution of the categories identified with the help of the dictionary based on substitution complement the other reliability measures. The paper shows that the choice of the reliability measure depends on the format of the text (stylistic versus rhetorical) and the type of reading (comprehension versus interpretation). Namely, Cohen’s kappa and Bennett, Alpert and Goldstein’s \(S\) emerge as reliability measures particularly suited for perspectival reading of rhetorical texts. Outcomes of the content analysis of 57 texts performed by four coders with the help of computer program QDA Miner inform the analysis.