Classical versus Bayesian risks in acceptance sampling: a sensitivity analysis

Assuming a beta prior distribution on the fraction defective, $p$ , failure-censored sampling plans for Weibull lifetime models using classical (or average) and Bayesian (or posterior) producer’s and consumer’s risks are designed to determine the acceptability of lots of a given product. The average risk criterion provides a certain assurance that good (bad) lots will be accepted (rejected), whereas the posterior risk criterion provides a determined confidence that an accepted (rejected) lot is indeed good (bad). The performance of classical and Bayesian risks are analyzed in developing sampling plans when the lifetime variable follows the Weibull distribution. Several figures and tables illustrate the sensitivity of the risks and optimal sample sizes for selected censoring levels and specifications according to the available prior information on $p$ . The analysis clarifies the distinction among the different risks for a given sampling plan, and the effect of the prior knowledge on the required sample size. The study shows that, under uncertainty in the prior variance of $p$ , the designs using Bayesian risks are more appropriate.     

{l_p}-Recovery of the Most Significant Subspace Among Multiple Subspaces with Outliers

We assume data sampled from a mixture of \(d\) -dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity, we may assume that all distributions are uniform on their corresponding unit spheres). We also assume mixture weights for the different components. We say that one of the underlying subspaces of the model is most significant if its mixture weight is higher than the sum of the mixture weights of all other subspaces. We study the recovery of the most significant subspace by minimizing the \(l_p\) -averaged distances of data points from \(d\) -dimensional subspaces of \(\mathbb R^D\) , where \(0 < p \in \mathbb R\) . Unlike other \(l_p\) minimization problems, this minimization is nonconvex for all \(p>0\) and thus requires different methods for its analysis. We show that if \(0 , then for any fraction of outliers, the most significant subspace can be recovered by \(l_p\) minimization with overwhelming probability (which depends on the generating distribution and its parameters). We show that when adding small noise around the underlying subspaces, the most significant subspace can be nearly recovered by \(l_p\) minimization for any \(0 with an error proportional to the noise level. On the other hand, if \(p>1\) and there is more than one underlying subspace, then with overwhelming probability the most significant subspace cannot be recovered or nearly recovered. This last result does not require spherically symmetric outliers.     

Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions

In this paper the authors show how through the use of the characteristic function of the negative logarithm of the likelihood ratio test (l.r.t.) statistic to test circular symmetry it is possible to obtain highly manageable expressions for the exact distribution of such statistic, when the number of variables, $p$ , is odd, and highly manageable and accurate approximations for an even $p$ . For the case of an even $p$ , two kinds of near-exact distributions are developed for the l.r.t. statistic which correspond, for the logarithm of the l.r.t. statistic, to a Generalized Near-Integer Gamma distribution or finite mixtures of these distributions. Numerical studies conducted in order to assess the quality of these new approximations show their impressive performance, namely when compared with the only available asymptotic distribution in the literature.     

Mod-discrete expansions

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law (without normalization) cannot be expected. The setting is one in which the simplest approximation to the $n$ -th random variable  $X_n$ is by a particular member $R_n$ of a given family of distributions, whose variance increases with  $n$ . The basic assumption is that the ratio of the characteristic function of  $X_n$ to that of  $R_n$ converges to a limit in a prescribed fashion. Our results cover and extend a number of classical examples in probability, combinatorics and number theory.     

Efficient simulation of complete and censored samples from common bivariate exponential distributions

Let $(X_{i:n},Y_{[i:n]})$ be the vector of the $i$ th $X$ -order statistic and its concomitant observed in a random sample of size $n$ where the marginal distribution of $X$ is absolutely continuous. We describe some general algorithms for simulation of complete and Type II censored samples $\{(X_{i:n}, Y_{[i:n]}), 1 \le i \le r \le n\}$ from such bivariate distributions. We study in detail several algorithms for simulating complete and censored samples from Downton, Marshall–Olkin, Gumbel (Type I) and Farlie-Gumbel-Morgenstern bivariate exponential distributions. We show that the conditioning method in conjunction with an efficient simulation of exponential order statistics that exploits the independence of spacings provides the best method with substantial savings over the basic method. Efficient simulation is essential for investigating the finite-sample distributional  properties of functions of order statistics and their concomitants.     

The distribution of points on curves over finite fields in some small rectangles

Let \(p\) be a prime. We study the distribution of points on a class of curves \(C\) over \(\mathbb{F }_p\) inside very small rectangles \(\mathcal{B }\) for which the Weil bound fails to give nontrivial information. In particular, we show that the distribution of points on \(C\) over long rectangles is Gaussian.     

Pitman closeness of k-records from two sequences to progressive Type-II censored order statistics

In this paper, we consider two independent \(k\) -record sequences with the same distribution. We determine the closeness probability of \(k\) -record values to a specific progressive Type-II censored order statistic. With this in mind, we first derive the exact expression for the Pitman closeness of records in general, and some special properties of the closeness probability are presented. Then, we apply the obtained results for the standard uniform and exponential distributions and exact expressions for the Pitman closeness are obtained. Finally, numerical results are displayed in figures.     

Objective Bayesian analysis for a capture–recapture model

In this paper, we study a special capture–recapture model, the $M_t$ model, using objective Bayesian methods. The challenge is to find a justified objective prior for an unknown population size $N$ . We develop an asymptotic objective prior for the discrete parameter $N$ and the Jeffreys’ prior for the capture probabilities $\varvec{\theta }$ . Simulation studies are conducted and the results show that the reference prior has advantages over ad-hoc non-informative priors. In the end, two real data examples are presented.     

A review on consistency and robustness properties of support vector machines for heavy-tailed distributions

Support vector machines (SVMs) belong to the class of modern statistical machine learning techniques and can be described as M-estimators with a Hilbert norm regularization term for functions. SVMs are consistent and robust for classification and regression purposes if based on a Lipschitz continuous loss and a bounded continuous kernel with a dense reproducing kernel Hilbert space. For regression, one of the conditions used is that the output variable Y has a finite first absolute moment. This assumption, however, excludes heavy-tailed distributions. Recently, the applicability of SVMs was enlarged to these distributions by considering shifted loss functions. In this review paper, we briefly describe the approach of SVMs based on shifted loss functions and list some properties of such SVMs. Then, we prove that SVMs based on a bounded continuous kernel and on a convex and Lipschitz continuous, but not necessarily differentiable, shifted loss function have a bounded Bouligand influence function for all distributions, even for heavy-tailed distributions including extreme value distributions and Cauchy distributions. SVMs are thus robust in this sense. Our result covers the important loss functions ${\epsilon}$ -insensitive for regression and pinball for quantile regression, which were not covered by earlier results on the influence function. We demonstrate the usefulness of SVMs even for heavy-tailed distributions by applying SVMs to a simulated data set with Cauchy errors and to a data set of large fire insurance claims of Copenhagen Re.     

Validity of heavy-traffic steady-state approximations in many-server queues with abandonment

We consider \(GI/Ph/n+M\) parallel-server systems with a renewal arrival process, a phase-type service time distribution, \(n\) homogenous servers, and an exponential patience time distribution with positive rate. We show that in the Halfin–Whitt regime, the sequence of stationary distributions corresponding to the normalized state processes is tight. As a consequence, we establish an interchange of heavy-traffic and steady-state limits for \(GI/Ph/n+M\) queues.     

Coupon collector’s problems with statistical applications to rankings

Some new exact distributions on coupon collector’s waiting time problems are given based on a generalized Pólya urn sampling. In particular, usual Pólya urn sampling generates an exchangeable random sequence. In this case, an alternative derivation of the distribution is also obtained from de Finetti’s theorem. In coupon collector’s waiting time problems with $m$ kinds of coupons, the observed order of $m$ kinds of coupons corresponds to a permutation of $m$ letters uniquely. Using the property of coupon collector’s problems, a statistical model on the permutation group of $m$ letters is proposed for analyzing ranked data. In the model, as the parameters mean the proportion of the $m$ kinds of coupons, the observed ranking can be intuitively understood. Some examples of statistical inference are also given.     

A U-statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens–Fisher setting

A two-sample test statistic is presented for testing the equality of mean vectors when the dimension, $p$ , exceeds the sample sizes, $n_i,\; i = 1, 2$ , and the distributions are not necessarily normal. Under mild assumptions on the traces of the covariance matrices, the statistic is shown to be asymptotically Chi-square distributed when $n_i, p \rightarrow \infty $ . However, the validity of the test statistic when $p$ is fixed but large, including $p > n_i$ , and when the distributions are multivariate normal, is shown as special cases. This two-sample Chi-square approximation helps us establish the validity of Box’s approximation for high-dimensional and non-normal data to a two-sample setup, valid even under Behrens–Fisher setting. The limiting Chi-square distribution of the statistic is obtained using the asymptotic theory of degenerate $U$ -statistics, and using a result from classical asymptotic theory, it is further extended to an approximate normal distribution. Both independent and paired-sample cases are considered.     

Square Function Characterization of Weak Hardy Spaces

We obtain a new square function characterization of the weak Hardy space \(H^{p,\infty }\) for all \(p\in (0,\infty )\) . This space consists of all tempered distributions whose smooth maximal function lies in weak \(L^p\) . Our proof is based on interpolation between \(H^p\) spaces. The main difficulty we overcome is the lack of a good dense subspace of \(H^{p,\infty }\) which forces us to work with general \(H^{p,\infty }\) distributions.     

Asymptotic Behavior of the Maximum and Minimum Singular Value of Random Vandermonde Matrices

This study examines various statistical distributions in connection with random Vandermonde matrices and their extension to \(d\) -dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to be \(O(\log ^{1/2}{N^{d}})\) and \(\Omega ((\log N^{d} /(\log \log N^d))^{1/2})\) , respectively, where \(N\) is the dimension of the matrix, generalizing the results in Tucci and Whiting (IEEE Trans Inf Theory 57(6):3938–3954, 2011). We further study the behavior of the minimum singular value of these random matrices. In particular, we prove that the minimum singular value is at most \(N\exp (-C\sqrt{N}))\) with high probability where \(C\) is a constant independent of \(N\) . Furthermore, the value of the constant \(C\) is determined explicitly. The main result is obtained in two different ways. One approach uses techniques from stochastic processes and in particular a construction related to the Brownian bridge. The other one is a more direct analytical approach involving combinatorics and complex analysis. As a consequence, we obtain a lower bound for the maximum absolute value of a random complex polynomial on the unit circle, which may be of independent mathematical interest. Lastly, for each sequence of positive integers \(\{k_p\}_{p=1}^{\infty }\) we present a generalized version of the previously discussed matrices. The classical random Vandermonde matrix corresponds to the sequence \(k_{p}=p-1\) . We find a combinatorial formula for their moments and show that the limit eigenvalue distribution converges to a probability measure supported on \([0,\infty )\) . Finally, we show that for the sequence \(k_p=2^{p}\) the limit eigenvalue distribution is the famous Marchenko–Pastur distribution.     

On Maximally Inflected Hyperbolic Curves

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert’s method we show that for any integers \(d\) and \(r\) such that \(4\le r \le 2d^2-2d\) , there is a non-singular hyperbolic curve of degree \(2d\) in \({\mathbb R}^2\) with exactly \(r\) line segments in the boundary of its convex hull. We also give a complete classification of possible distributions of inflection points among the ovals of a maximally inflected non-singular hyperbolic curve of degree \(6\) .     

Recent results in the theory and applications of CARMA processes

Just as ARMA processes play a central role in the representation of stationary time series with discrete time parameter, \((Y_n)_{n\in \mathbb {Z}}\) , CARMA processes play an analogous role in the representation of stationary time series with continuous time parameter, \((Y(t))_{t\in \mathbb {R}}\) . Lévy-driven CARMA processes permit the modelling of heavy-tailed and asymmetric time series and incorporate both distributional and sample-path information. In this article we provide a review of the basic theory and applications, emphasizing developments which have occurred since the earlier review in Brockwell (2001a, In D. N. Shanbhag and C. R. Rao (Eds.), Handbook of Statistics 19; Stochastic Processes: Theory and Methods (pp. 249–276), Amsterdam: Elsevier).     

Poisson and compound Poisson approximations in conventional and nonconventional setups

It was shown in Kifer (Israel J Math, 2013) that for any subshift of finite type considered with a Gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically Poisson distributed. Here we not only extend this result to all \(\psi \) -mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to Poisson or to compound Poisson distributions. It turns out that for all nonperiodic points a limiting distribution is always Poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is Bernoulli in which case the limiting distribution always exists. Thus we describe, essentially completely, limiting distributions of multiple recurrence numbers in this setup. As a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed. Most of the results are new also for the widely studied single recurrencies case (see, for instance, Haydn and Vaienti Discret Contin Dyn Syst A 10:589–616, 2004; Probab Theory Relat Fields 144:517–542, 2009; Abadi and Saussol Stoch Process Appl 121:314–323, 2011; Abadi and Vergne Nonlinearity 21:2871–2885, 2008), as well.     

On convolvability conditions for distributions

We investigate several conditions of the convolvability and ${\mathcal{S}'}$ -convolvability of distributions and we show their equivalence by characterizing the partial summability of distribution kernels by multiplicative properties. More generally, partial summability to the power p and the partial vanishing at infinity of kernels are characterized by multiplicative properties. As an application we state several sufficient equivalent conditions ensuring the validity of the equation, $$ (\partial_jS)\ast T=S\ast (\partial_j T).$$ Furthermore, it is shown that the Chevalley condition for the convolvability of two distributions ${S,T\in\mathcal{D}'}$ , i.e., $$ (\varphi\ast S)(\psi\ast\check T)\in L^1\quad\text{for all }\varphi,\psi\in\mathcal D,$$ is equivalent with $$S(x-y)T(y-z)\in\mathcal D'_{xz}\hat\otimes L^1_y.$$      

Closure of some heavy-tailed distribution classes under random convolution*

In this paper, we consider the closure property of a random convolution $ \sum\nolimits_{n = 0}^\infty {{p_n}{F^*}^n} $ , where F is a heavy-tailed distribution on [0, ??), and p _n (n?=?0, 1, . . . ) are the local probabilities of a nonnegative integer-valued random variable. We obtain conditions under which the fact that distribution F belongs to the dominatedly varying-tailed class, long-tailed class, or to the intersection of these classes implies that $ \sum\nolimits_{n = 0}^\infty {{p_n}{F^*}^n} $ is in the same class.     

Discontinuous Galerkin method for hyperbolic equations involving \delta -singularities: negative-order norm error estimates and applications

In this paper, we develop and analyze discontinuous Galerkin (DG) methods to solve hyperbolic equations involving $\delta $ -singularities. Negative-order norm error estimates for the accuracy of DG approximations to $\delta $ -singularities are investigated. We first consider linear hyperbolic conservation laws in one space dimension with singular initial data. We prove that, by using piecewise $k$ th degree polynomials, at time $t$ , the error in the $H^{-(k+2)}$ norm over the whole domain is $(k+1/2)$ th order, and the error in the $H^{-(k+1)}(\mathbb R \backslash \mathcal R _t)$ norm is $(2k+1)$ th order, where $\mathcal R _t$ is the pollution region due to the initial singularity with the width of order $\mathcal O (h^{1/2} \log (1/h))$ and $h$ is the maximum cell length. As an application of the negative-order norm error estimates, we convolve the numerical solution with a suitable kernel which is a linear combination of B-splines, to obtain $L^2$ error estimate of $(2k+1)$ th order for the post-processed solution. Moreover, we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying Duhamel’s principle. Numerical examples including an acoustic equation and the nonlinear rendez-vous algorithms are given to demonstrate the good performance of DG methods for solving hyperbolic equations involving $\delta $ -singularities.