Notes on consistency of some minimum distance estimators with simulation results

We focus on the minimum distance density estimators \({\widehat{f}}_n\) of the true probability density \(f_0\) on the real line. The consistency of the order of \(n^{-1/2}\) in the (expected) L\(_1\)-norm of Kolmogorov estimator (MKE) is known if the degree of variations of the nonparametric family \(\mathcal {D}\) is finite. Using this result for MKE we prove that minimum Lévy and minimum discrepancy distance estimators are consistent of the order of \(n^{-1/2}\) in the (expected) L\(_1\)-norm under the same assumptions. Computer simulation for these minimum distance estimators, accompanied by Cramér estimator, is performed and the function \(s(n)=a_0+a_1\sqrt{n}\) is fitted to the L\(_1\)-errors of \({\widehat{f}}_n\) leading to the proportionality constant \(a_1\) determination. Further, (expected) L\(_1\)-consistency rate of Kolmogorov estimator under generalized assumptions based on asymptotic domination relation is studied. No usual continuity or differentiability conditions are needed.     

Ordering properties of the smallest order statistics from generalized Birnbaum–Saunders models with associated random shocks

Let \(X_{1},\ldots , X_{n}\) be lifetimes of components with independent non-negative generalized Birnbaum–Saunders random variables with shape parameters \(\alpha _{i}\) and scale parameters \(\beta _{i},~ i=1,\ldots ,n\), and \(I_{p_{1}},\ldots , I_{p_{n}}\) be independent Bernoulli random variables, independent of \(X_{i}\)’s, with \(E(I_{p_{i}})=p_{i},~i=1,\ldots ,n\). These are associated with random shocks on \(X_{i}\)’s. Then, \(Y_{i}=I_{p_{i}}X_{i}, ~i=1,\ldots ,n,\) correspond to the lifetimes when the random shock does not impact the components and zero when it does. In this paper, we discuss stochastic comparisons of the smallest order statistic arising from such random variables \(Y_{i},~i=1,\ldots ,n\). When the matrix of parameters \((h({\varvec{p}}), {\varvec{\beta }}^{\frac{1}{\nu }})\) or \((h({\varvec{p}}), {\varvec{\frac{1}{\alpha }}})\) changes to another matrix of parameters in a certain mathematical sense, we study the usual stochastic order of the smallest order statistic in such a setup. Finally, we apply the established results to two special cases: classical Birnbaum–Saunders and logistic Birnbaum–Saunders distributions.     

Blocked factor aliased effect-number pattern and column rank of blocked regular designs

In factorial experiments, estimation precision of specific factor effects depends not only on design selection but also on factor assignments to columns of selected designs. Usually, different columns in a design play different roles when estimating factor effects. Zhou et al. (Can J Stat 41:540-555, 2013) introduced a factor aliased effect-number pattern (F-AENP) and proposed a column ranking scheme for all the GMC \(2^{n-m}\) designs with \(5N/16+1\le n\le N-1\), where \(N=2^{n-m}\). In this paper, we first introduce a blocked factor aliased effect-number pattern (B-F-AENP) for blocked regular designs as an extension of the F-AENP. Then, by using the B-F-AENP, we propose a column ranking scheme for all the B\(^1\)-GMC \(2^{n-m}:2^s\) designs with \(5N/16+1\le n\le N-1\), as well as an assignment strategy for important factors.     

Approximate maximum likelihood estimation for stochastic differential equations with random effects in the drift and the diffusion

Consider N independent stochastic processes \((X_i(t), t\in [0,T])\), \(i=1,\ldots , N\), defined by a stochastic differential equation with random effects where the drift term depends linearly on a random vector \(\Phi _i\) and the diffusion coefficient depends on another linear random effect \(\Psi _i\). For these effects, we consider a joint parametric distribution. We propose and study two approximate likelihoods for estimating the parameters of this joint distribution based on discrete observations of the processes on a fixed time interval. Consistent and \(\sqrt{N}\)-asymptotically Gaussian estimators are obtained when both the number of individuals and the number of observations per individual tend to infinity. The estimation methods are investigated on simulated data and show good performances.     

Estimation of the regression slope by means of Gini’s cograduation index

The simple linear model \(Y_i = \alpha + \beta \, x_i + \epsilon _i\) \((i=1,2, \ldots ,N \ge 2)\) is considered, where the \(x_i\)’s are given constants and \(\epsilon _1, \epsilon _2 , \ldots , \epsilon _N\) are independent identically distributed (iid) with continuous distribution function F. An estimator \(\tilde{\beta }\) of the slope parameter is proposed, based on a stochastic process which makes use of Gini’s cograduation index. The properties of \(\tilde{\beta }\) and of the related confidence interval are studied. Some comparisons are given, in terms of asymptotic relative efficiency, with other estimators of \(\beta \) including that obtained with the method of least squares.     

On consistency of the weighted least squares estimators in a semiparametric regression model

This paper is concerned with the semiparametric regression model \(y_i=x_i\beta +g(t_i)+\sigma _ie_i,~~i=1,2,\ldots ,n,\) where \(\sigma _i^2=f(u_i)\), \((x_i,t_i,u_i)\) are known fixed design points, \(\beta \) is an unknown parameter to be estimated, \(g(\cdot )\) and \(f(\cdot )\) are unknown functions, random errors \(e_i\) are widely orthant dependent random variables. The p-th (\(p>0\)) mean consistency and strong consistency for least squares estimators and weighted least squares estimators of \(\beta \) and g under some more mild conditions are investigated. A simulation study is also undertaken to assess the finite sample performance of the results that we established. The results obtained in the paper generalize and improve some corresponding ones of negatively associated random variables.     

Stochastic comparisons of distorted distributions,coherent systems and mixtures with ordered components

A distribution function F is a generalized distorted distribution of the distribution functions \(F_1,\ldots ,F_n\) if \(F=Q(F_1,\ldots ,F_n)\) for an increasing continuous distortion function Q such that \(Q(0,\ldots ,0)=0\) and \(Q(1,\ldots ,1)=1\). In this paper, necessary and sufficient conditions for the stochastic (ST) and the hazard rate (HR) orderings of generalized distorted distributions are provided when the distributions \(F_1,\ldots ,F_n\) are ordered. These results are used to obtain distribution-free ordering properties for coherent systems with heterogeneous components. In particular, we determine all the ST and HR orderings for coherent systems with 1–3 independent components. We also compare systems with dependent components. The results on distorted distributions are also used to get comparisons of finite mixtures.     

Asymptotic inference for a one-dimensional simultaneous autoregressive model

A nonstationary simultaneous autoregressive model \({X^{(n)}_k=\alpha \Big(X^{(n)}_{k-1}+X^{(n)}_{k+1}\Big)+\varepsilon_k, k=1, 2, \ldots , n-1}\), is investigated, where \({X^{(n)}_0}\) and \({X^{(n)}_n}\) are given random variables. It is shown that in the unstable case α = 1/2 the least squares estimator of the autoregressive parameter converges to a functional of a standard Wiener process with a rate of convergence n ², while in the stable situation |α| < 1/2 the estimator is biased but asymptotically normal with a rate n ^1/2.     

A new measure of association between random variables

We propose a new measure of association between two continuous random variables X and Y based on the covariance between X and the log-odds rate associated to Y. The proposed index of correlation lies in the range [\(-1\), 1]. We show that the extremes of the range, i.e., \(-1\) and 1, are attainable by the Fr\(\acute{\mathrm{e}}\)chet bivariate minimal and maximal distributions, respectively. It is also shown that if X and Y have bivariate normal distribution, the resulting measure of correlation equals the Pearson correlation coefficient \(\rho \). Some interpretations and relationships to other variability measures are presented. Among others, it is shown that for non-negative random variables the proposed association measure can be represented in terms of the mean residual and mean inactivity functions. Some illustrative examples are also provided.     

Every member of the core is as respectful as any other

We strategically separate different core outcomes. The natural counterparts of a core allocation in a strategic environment are the α-core, the β-core and the strong equilibrium, modified by assuming that utility is transferable in a strategic context as well. Given a core allocation ω of a convex transferable utility (TU) game \(v\), we associate a strategic coalition formation game with \( \left( {v, \omega } \right) \) in which ω survives, while most other core allocations are eliminated. If the TU game is strictly convex, the core allocations respected by the TU-α-core, the TU-β-core and the TU-strong equilibrium shrink to ω only in the canonical family of coalition formation games associated with \( \left( {v, \omega } \right) \). A mechanism, which strategically separates core outcomes from noncore outcomes for each convex TU game according to the TU-strong equilibrium notion is reported.     

A new approach to distribution free tests in contingency tables

We suggest an extremely wide class of asymptotically distribution free goodness of fit tests for testing independence in two-way contingency tables, or equivalently, independence of two discrete random variables. The nature of these tests is that the test statistics can be viewed as definite functions of the transformation of \(\widehat{T}_n = (\widehat{T}_{ij})=\Big (\frac{\nu _{ij}- n\hat{a}_i\hat{b}_j}{\sqrt{n\hat{a}_i\hat{b}_j}}\Big )\) where \(\nu _{ij}\) are frequencies and \(\hat{a}_i, \hat{b}_j\) are estimated marginal distributions. Our method is also applicable for testing independence of two discrete random vectors. We make some comparisons on statistical powers of the new tests with the conventional chi-square test and suggest some cases in which this class is significantly more powerful.     

Let the data speak: semiparametric evidence on the environmental Kuznets curve in Africa

This paper revisits the environmental Kuznets curve hypothesis and applies a flexible semi-parametric panel fixed effects technique to identify the true shape of the income-pollution relationship for a sample of 49 African countries for the period 1990–2010. Compared with standard panel data techniques which yields different conclusions, the former show that the income-pollution nexus is non-monotonically increasing and decreasing for \(\hbox {CO}_2\) and \(\hbox {PM}_{10}\) emissions respectively. Hence, the effect of economic growth differs with each atmospheric pollutants and is not sufficient for improving environmental quality. There is need for policies that emphasizes sustainable economic growth and the use of cleaner energy sources.     

Unbiased estimates for a lognormal regression problem and a nonparametric alternative

Given a normal sample with means \({{\bf x}_{1}^{\prime} {\bf \varphi}, \ldots, {\bf x}_{n}^{\prime} {\bf \varphi}}\) and variance v, minimum variance unbiased estimates are given for the moments of L, where log L is normal with mean \({{\bf x}^{\prime} {\bf \varphi}}\) and variance v. These estimates converge to wrong values if the normality assumption is false. In the latter case estimates based on any M-estimate of \({{\bf \varphi}}\) are available of bias \({O\left(n^{-1}\right)}\) and \({O\left(n^{-2}\right)}\). More generally, these are given for any smooth function of \({\left({\bf \varphi}, F\right)}\), where F is the unknown distribution of the residuals. The regression functions need not be linear.     

The variance of the discrepancy distribution of rounding procedures,and sums of uniform random variables

When \(\ell \) probabilities are rounded to integer multiples of a given accuracy n, the sum of the numerators may deviate from n by a nonzero discrepancy. It is proved that, for large accuracies \(n \rightarrow \infty \), the limiting discrepancy distribution has variance \(\ell /12\). The relation to the uniform distribution over the interval \([-1/2, 1/2]\), whose variance is 1 / 12, is explored in detail.     

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.     

A Cramér-type large deviation theorem for sums of functions of higher order non-overlapping spacings

Let U ₁, U ₂, . . . , U _n–1 be an ordered sample from a Uniform [0,1] distribution. The non-overlapping uniform spacings of order s are defined as \({G_{i}^{(s)} =U_{is} -U_{(i-1)s}, i=1,2,\ldots,N^\prime, G_{N^\prime+1}^{(s)} =1-U_{N^\prime s}}\) with notation U ₀ = 0, U _n = 1, where \({N^\prime=\left\lfloor n/s\right\rfloor}\) is the integer part of n/s. Let \({ N=\left\lceil n/s\right\rceil}\) be the smallest integer greater than or equal to n/s, f _m (u), m = 1, 2, . . . , N, be a sequence of real-valued Borel-measurable functions. In this article a Cramér type large deviation theorem for the statistic \({f_{1,n} (nG_{1}^{(s)})+\cdots+f_{N,n} (nG_{N}^{(s)} )}\) is proved.     

An iterative computational scheme for solving the coupled Hamilton–Jacobi–Isaacs equations in nonzero-sum differential games of affine nonlinear systems

In this paper, we present iterative or successive approximation methods for solving the coupled Hamilton–Jacobi–Isaacs equations (HJIEs) arising in nonzero-sum differential game for affine nonlinear systems. We particularly consider the ones arising in mixed \({\mathcal H}_{2}/{\mathcal H}_{\infty }\) control. However, the approach is perfectly general and can be applied to any others including those arising in the N-player case. The convergence of the method is established under fairly mild assumptions, and examples are solved to demonstrate the utility of the method. The results are also specialized to the coupled algebraic Riccati equations arising typically in mixed \({\mathcal H}_{2}/{\mathcal H}_{\infty }\) linear control. In this case, a bound within which the optimal solution lies is established. Finally, based on the iterative approach developed, a local existence result for the solution of the coupled-HJIEs is also established.     

Some general points on the $$I^2$$-measure of heterogeneity in meta-analysis

Meta-analysis has developed to be a most important tool in evaluation research. Heterogeneity is an issue that is present in almost any meta-analysis. However, the magnitude of heterogeneity differs across meta-analyses. In this respect, Higgins’ \(I^2\) has emerged to be one of the most used and, potentially, one of the most useful measures as it provides quantification of the amount of heterogeneity involved in a given meta-analysis. Higgins’ \(I^2\) is conventionally interpreted, in the sense of a variance component analysis, as the proportion of total variance due to heterogeneity. However, this interpretation is not entirely justified as the second part involved in defining the total variation, usually denoted as \(s^2\), is not an average of the study-specific variances, but in fact some other function of the study-specific variances. We show that \(s^2\) is asymptotically identical to the harmonic mean of the study-specific variances and, for any number of studies, is at least as large as the harmonic mean with the inequality being sharp if all study-specific variances agree. This justifies, from our point of view, the interpretation of explained variance, at least for meta-analyses with larger number of component studies or small variation in study-specific variances. These points are illustrated by a number of empirical meta-analyses as well as simulation work.     

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.     

Empirical likelihood for heteroscedastic partially linear single-index models with growing dimensional data

In this paper, we propose a new approach to the empirical likelihood inference for the parameters in heteroscedastic partially linear single-index models. In the growing dimensional setting, it is proved that estimators based on semiparametric efficient score have the asymptotic consistency, and the limit distribution of the empirical log-likelihood ratio statistic for parameters \((\beta ^{\top },\theta ^{\top })^{\top }\) is a normal distribution. Furthermore, we show that the empirical log-likelihood ratio based on the subvector of \(\beta \) is an asymptotic chi-square random variable, which can be used to construct the confidence interval or region for the subvector of \(\beta \). The proposed method can naturally be applied to deal with pure single-index models and partially linear models with high-dimensional data. The performance of the proposed method is illustrated via a real data application and numerical simulations.