Almost sure central limit theorem for the products of <Emphasis Type="Italic">U</Emphasis>-statistics

Let (X _n ) be a sequence of i.i.d random variables and U _n a U-statistic corresponding to a symmetric kernel function h, where h ₁(x ₁) = Eh(x ₁, X ₂, X ₃, . . . , X _m ), μ = E(h(X ₁, X ₂, . . . , X _m )) and ? ₁ = Var(h ₁(X ₁)). Denote \({\gamma=\sqrt{\varsigma_{1}}/\mu}\), the coefficient of variation. Assume that P(h(X ₁, X ₂, . . . , X _m ) > 0) = 1, ? ₁ > 0 and E|h(X ₁, X ₂, . . . , X _m )|³ < ∞. We give herein the conditions under which

$\lim_{N\rightarrow\infty}\frac{1}{\log N}\sum_{n=1}^{N}\frac{1}{n}g\left(\left(\prod_{k=m}^{n}\frac{U_{k}}{\mu}\right)^{\frac{1}{m\gamma\sqrt{n}}}\right) =\int\limits_{-\infty}^{\infty}g(x)dF(x)\quad {\rm a.s.}$

for a certain family of unbounded measurable functions g, where F(·) is the distribution function of the random variable \({\exp(\sqrt{2} \xi)}\) and ξ is a standard normal random variable.

     

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.     

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.     

Dispersive ordering of fail-safe systems with heterogeneous exponential components

Let X ₁, . . . , X _n be independent exponential random variables with respective hazard rates λ₁, . . . , λ _n , and Y ₁, . . . , Y _n be independent and identically distributed random variables from an exponential distribution with hazard rate λ. Then, we prove that X _2:n , the second order statistic from X ₁, . . . , X _n , is larger than Y _2:n , the second order statistic from Y ₁, . . . , Y _n , in terms of the dispersive order if and only if

$\lambda\geq \sqrt{\frac{1}{{n\choose 2}}\sum_{1\leq i < j\leq n}\lambda_i\lambda_j}.$

We also show that X _2:n is smaller than Y _2:n in terms of the dispersive order if and only if

$ \lambda\le\frac{\sum^{n}_{i=1} \lambda_i-{\rm max}_{1\leq i\leq n} \lambda_i}{n-1}. $

Moreover, we extend the above two results to the proportional hazard rates model. These two results established here form nice extensions of the corresponding results on hazard rate, likelihood ratio, and MRL orderings established recently by Pǎltǎnea (J Stat Plan Inference 138:1993–1997, 2008), Zhao et al. (J Multivar Anal 100:952–962, 2009), and Zhao and Balakrishnan (J Stat Plan Inference 139:3027–3037, 2009), respectively.

     

R-optimal designs for multi-factor models with heteroscedastic errors

In this paper, we consider the R-optimal design problem for multi-factor regression models with heteroscedastic errors. It is shown that a R-optimal design for the heteroscedastic Kronecker product model is given by the product of the R-optimal designs for the marginal one-factor models. However, R-optimal designs for the additive models can be constructed from R-optimal designs for the one-factor models only if sufficient conditions are satisfied. Several examples are presented to illustrate and check optimal designs based on R-optimality criterion.     

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.     

Relations for product moments and covariances of <Emphasis Type="Italic">k</Emphasis>th records from discrete distributions

The aim of this paper is to establish recurrence relations satisfied by product moments and covariances of kth records arising from discrete distributions. They will be evaluated for geometric underlying distribution. Then we use these results to obtain formulas for correlation coefficients of geometric kth records. We consider all three known types of kth records: strong, ordinary, and weak.     

New axioms for immediate acceptance

Following Kojima and Ünver (Econ Theory 55(3):515–544, 2014) and Afacan (Math Soc Sci 66(2):176–179, 2013), this paper provides two characterizations of the Boston school choice mechanism determined by the student-proposing immediate acceptance algorithm. A mechanism is the Boston mechanism if and only if it satisfies one of the following two groups of axioms: favoring higher ranks and weak fairness; favoring higher ranks, rank monotonicity, and rank rationality.     

Revisiting the difference between mixed methods and multimethods: Is it all in the name?

The literature on mixed methods and multimethods has burgeoned over the last 20 years, and researchers from a growing number and diversity of fields have progressively embraced these approaches. However, rapid growth in any movement inevitably gives rise to gaps or shortcomings, such as “identity crises” or divergent conceptual views. Although some authors draw a clear and sometimes opinionated distinction between mixed methods and multimethods, for others, they are synonymous. The concepts underlying both terms therefore have become blurred and generated much confusion. The aim of this article is to explore the origins of the confusion, describe our view of mixed methods and multimethod studies, and by doing so, help to clearly delineate the two concepts. The authors have presented their opinion of how these terms and concepts should be distinguished and call for a constructive debate of the issues involved in the mixed methods and multimethod literature. This is a way truly to propel the field forward.     

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.     

Bayesian estimation based on ranked set sample from Morgenstern type bivariate exponential distribution when ranking is imperfect

In this paper we consider Bayes estimation based on ranked set sample when ranking is imperfect, in which units are ranked based on measurements made on an easily and exactly measurable auxiliary variable X which is correlated with the study variable Y. Bayes estimators under squared error loss function and LINEX loss function for the mean of the study variate Y, when (X, Y) follows a Morgenstern type bivariate exponential distribution, are obtained based on both usual ranked set sample and extreme ranked set sample. Estimation procedures developed in this paper are illustrated using simulation studies and a real data.     

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.     

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.     

Organizational Socialization and Psychological Contract: the Vulnerability of Temporary Newcomers. A Case Study from an Italian <Emphasis Type="Italic">Call Center</Emphasis>

This study aims to analyze the link between the construction of an effective psychological contract with the organization and the success of the socialization process. To this purpose 241 employees of a Call Center organization have been contacted. A questionnaire composed by measures of Organizational Socialization (Haueter et al. Journal of Vocational Behavior, 63, 20–39, 2003), Psychological Contract (Rousseau 1995), Job Satisfaction (Wanous et al. Journal of Applied Psychology, 82, 247–252, 1997) and Organizational Committment (Allen and Meyer 1990) was administered. Results have underlined that organizational socialization may influence the development of the psychological contract thus determining job satisfaction and organizational commitment. This research has been developed in an interdisciplinary perspective, taking into account the peculiarity of the Italian legal framework. In this regard, the analysis has been focused on how the E.U. flexicurity strategy has been implemented in Italy, according to the recent reform of labour market regulation (2012–13) and on the specific regulations introduced for call centres.     

Asymptotic non-deficiency of the Bayes sequential estimation in a family of transformed Chi-square distributions

Bayes sequential estimation in a family of transformed Chi-square distributions using a linex loss function and a cost c > 0 for each observation is considered in this paper. It is shown that an asymptotic pointwise optimal rule (A.P.O.) is asymptotically non-deficient, i.e., the difference between the Bayes risk of the A.P.O. rule and the Bayes risk of the optimal procedure is of smaller order of magnitude than c, the cost of single observation, as c → 0.     

Humiliation at Work and Union Interest: An Empirical Analysis of Community Union Satisfaction as Related to Status Enhancement

How noneconomic benefits claimed by labor unions relate to union interest is not well articulated. Based on Torres and Bergner’s (Journal of the American Academy of Psychiatry and the Law, 38, 195–204, 2010; Psychotherapy, 49, 492–501, 2012) analysis of severe public humiliation, in which status enhancement underlies recovery, we examined an augmented relationship between humiliation at work (the underside of dignity at work) and willingness to join a union. As hypothesized, nonunion employees who were less detached from work showed more willingness to join when presented with evidence that members of a union were satisfied with community aspects of membership related to status enhancement above and beyond their satisfaction with economic aspects. Implications for union interest research and applications are discussed.     

“Speculative Influence Network” during financial bubbles: application to Chinese stock markets

We introduce the Speculative Influence Network (SIN) to decipher the causal relationships between sectors (and/or firms) during financial bubbles. The SIN is constructed in two steps. First, we develop a Hidden Markov Model (HMM) of regime-switching between a normal market phase represented by a geometric Brownian motion and a bubble regime represented by the stochastic super-exponential Sornette and Andersen (Int J Mod Phys C 13(2):171–188, 2002) bubble model. The calibration of the HMM provides the probability at each time for a given security to be in the bubble regime. Conditional on two assets being qualified in the bubble regime, we then use the transfer entropy to quantify the influence of the returns of one asset i onto another asset j, from which we introduce the adjacency matrix of the SIN among securities. We apply our technology to the Chinese stock market during the period 2005–2008, during which a normal phase was followed by a spectacular bubble ending in a massive correction. We introduce the Net Speculative Influence Intensity variable as the difference between the transfer entropies from i to j and from j to i, which is used in a series of rank ordered regressions to predict the maximum loss (%MaxLoss) endured during the crash. The sectors that influenced other sectors the most are found to have the largest losses. There is some predictability obtained by using the transfer entropy involving industrial sectors to explain the %MaxLoss of financial institutions but not vice versa. We also show that the bubble state variable calibrated on the Chinese market data corresponds well to the regimes when the market exhibits a strong price acceleration followed by clear change of price regimes. Our results suggest that SIN may contribute significant skill to the development of general linkage-based systemic risks measures and early warning metrics.     

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.     

Multivariate saddlepoint tests on the mean direction of the von Mises–Fisher distribution

This article provides P values for two new tests on the mean direction of the von Mises–Fisher distribution. The test statistics are obtained from the exponent of the saddlepoint approximation to the density of M-estimators, as suggested by Robinson et al. (Ann Stat 31:1154–1169, 2003). These test statistics are chi-square distributed with asymptotically small relative errors. Despite the high dimensionality of the problem, the proposed P values are accurate and simple to compute. The numerical precision of the P values of the new tests is illustrated by some simulation studies.     

On the welfare effects of affirmative actions in school choice

Recent evidence, from both academia and practice, indicates that implementing affirmative action policies in school choice problems may induce substantial welfare losses on the intended beneficiaries. This paper addresses the following two questions: what are the causes of such perverse consequences, and when we can effectively implement affirmative action policies without unsatisfactory outcomes. Using the minority reserve policy in the student optimal stable mechanism as an example, I show that two acyclicity conditions, type-specific acyclicity and strongly type-specific acyclicity, are crucial for the effective implementations of minority reserve policies. I further illustrate how restrictive these two acyclicity conditions are, and the intrinsic difficulty of embedding diversity goals into stable matching mechanisms. Under some regularity conditions, I demonstrate that the minority reserve policy is very unlikely to cause welfare losses on any minority students when the number of schools is sufficiently large.