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.     

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.     

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.     

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.     

On purely sequential estimation of an inverse Gaussian mean

The first part of this paper deals with developing a purely sequential methodology for the point estimation of the mean \(\mu \) of an inverse Gaussian distribution having an unknown scale parameter \(\lambda \). We assume a weighted squared error loss function and aim at controlling the associated risk function per unit cost by bounding it from above by a known constant \(\omega \). We also establish first-order and second-order asymptotic properties of our stopping rule. The second part of this paper deals with obtaining a purely sequential fixed accuracy confidence interval for the unknown mean \(\mu \), assuming that the scale parameter \(\lambda \) is known. First-order asymptotic efficiency and asymptotic consistency properties are also built of our proposed procedures. We then provide extensive sets of simulation studies and real data analysis using data from fatigue life analysis to show encouraging performances of our proposed stopping strategies.     

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.     

Proper strong-Fibonacci games

We define proper strong-Fibonacci (PSF) games as the subset of proper homogeneous weighted majority games which admit a Fibonacci representation. This is a homogeneous, type-preserving representation whose ordered sequence of type weights and winning quota is the initial string of Fibonacci numbers of the one-step delayed Fibonacci sequence. We show that for a PSF game, the Fibonacci representation coincides with the natural representation of the game. A characterization of PSF games is given in terms of their profile. This opens the way up to a straightforward formula which gives the number \(\varPsi (t)\) of such games as a function of t, number of non-dummy players’ types. It turns out that the growth rate of \(\varPsi (t)\) is exponential. The main result of our paper is that, for two consecutive t values of the same parity, the ratio \(\varPsi (t+2)/\varPsi (t)\) converges toward the golden ratio \({\varPhi }\).     

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.     

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.     

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.     

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.     

Test by adaptive LASSO quantile method for real-time detection of a change-point

This article proposes a test statistic based on the adaptive LASSO quantile method to detect in real-time a change in a linear model. The model can have a large number of explanatory variables and the errors don’t satisfy the classical assumptions for a statistical model. For the proposed test statistic, the asymptotic distribution under \(H_0\) is obtained and the divergence under \(H_1\) is shown. It is shown via Monte Carlo simulations, in terms of empirical sizes, of empirical powers and of stopping time detection, that the useful test statistic for applications is better than other test statistics proposed in literature. Two applications on the air pollution and in the health field data are also considered.     

Is the service sector different in size heterogeneity?

We examined whether significant differences in size heterogeneity exist between the service and the manufacturing industries by using PL exponents as the proxy for intra-industry size heterogeneity. For the purpose, we analyzed firm size distribution (FSD) and estimated the PL exponents, on the right tails of FSD, of the service and manufacturing industries in Korea for the period 2008–2012 using the Business Activity Survey dataset created by the Korean National Statistical Office As a result, we observed that the estimates of the PL exponents for the service industry are lower than those for the manufacturing industry (\(\upalpha _\mathrm{Service}<\upalpha _\mathrm{Manufacturing}\)) regardless of size variable, year, and dataset. This relationship may be related to the weaker negative relationship between the size and growth of the service industry, which made the slope of the PL distribution in the right tail of the FSD smoother. This finding implies that size heterogeneity may be more distinctive in the service industry than in the manufacturing industry. In addition, the PL exponents of sales were larger than those of assets and smaller than those of employees (\(\upalpha _\mathrm{Asset}<\upalpha _\mathrm{Sales}<\upalpha _\mathrm{Employee}\)) regardless of industry, year, and dataset. We also observed the PL exponents in the survived-firm dataset to decrease, compared to those in the all-firm dataset.     

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.     

Linear estimation of location and scale parameters using partial maxima

Consider an i.i.d. sample \({X^*_{1},X^*_{2},\ldots,X^*_{n}}\) from a location-scale family, and assume that the only available observations consist of the partial maxima (or minima) sequence, \({X^*_{1:1},X^*_{2:2},\ldots,X^*_{n:n}}\), where \({X^*_{j:j}=\max\{ X^*_1, \ldots,X^*_j \}}\). This kind of truncation appears in several circumstances, including best performances in athletics events. In the case of partial maxima, the form of the BLUEs (best linear unbiased estimators) is quite similar to the form of the well-known Lloyd’s (in Biometrica 39:88–95, 1952) BLUEs, based on (the sufficient sample of) order statistics, but, in contrast to the classical case, their consistency is no longer obvious. The present paper is mainly concerned with the scale parameter, showing that the variance of the partial maxima BLUE is at most of order O(1/ log n), for a wide class of distributions.     

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.