On the Convergence Rates of Extreme Generalized Order Statistics

A classical result of extreme value theory yields that in case of a linear normalization three possible types of limit distributions are possible. As proved recently a similar classification of the limit distributions holds for extreme generalized order statistics which provide a general concept of ordered random variables. In this paper, we derive results for the convergence rates of the nth and (n-r+1)st generalized order statistic, respectively. It turns out that the rate is highly influenced by the choice of the normalizing sequence. Moreover, we show that a uniform bound of order 1/n holds for underlying generalized Pareto distributions, whereas for the standard normal distribution the convergence might be very slow. Similar results for ordinary order statistics are included.     

Bayes estimators whose range has convex hull of zero prior probability

It is shown that in a general statistical decision problem Bayes procedures under convex loss can possess the following property: The induced prior measure of the convex hull of the range of the corresponding Bayes rule is equal to zero. A stronger statement holds in the case of binomial distributions.     

A new approach to the credibility formula

The usual credibility formula holds whenever, (i) claim size distribution is a member of the exponential family of distributions, (ii) prior distribution conjugates with claim size distribution, and (iii) square error loss has been considered. As long as, one of these conditions is violent, the usual credibility formula no longer holds. This article, using the mean square error minimization technique, develops a simple and practical approach to the credibility theory. Namely, we approximate the Bayes estimator with respect to a general loss function and general prior distribution by a convex combination of the observation mean and mean of prior, say, approximate credibility formula. Adjustment of the approximate credibility for several situations and its form for several important losses are given.     

An invariance property of quadratic forms in random vectors with a selection distribution,with application to sample variogram and covariogram estimators

We study conditions under which an invariance property holds for the class of selection distributions. First, we consider selection distributions arising from two uncorrelated random vectors. In that setting, the invariance holds for the so-called C{\cal{C}} -class and for elliptical distributions. Second, we describe the invariance property for selection distributions arising from two correlated random vectors. The particular case of the distribution of quadratic forms and its invariance, under various selection distributions, is investigated in more details. We describe the application of our invariance results to sample variogram and covariogram estimators used in spatial statistics and provide a small simulation study for illustration. We end with a discussion about other applications, for example such as linear models and indices of temporal/spatial dependence.     

Niche width and scale in organizational competition: A computational approach

In some organizational applications, the principle of allocation (PoA) and scale advantage (SA) oppose each other. While PoA implies that organizations with wide niches get punished, SA holds that large organizations gain an advantage because of scale efficiencies. The opposition occurs because many large organizations also possess wide niches. However, analyzing these theoretical mechanisms implies a possible trade-off between niche width and size: if both PoA and SA are strong, then organizations must be either focused or large to survive, resulting in a dual market structure, as proposed by the theory of resource partitioning. This article develops a computational model used to study this trade-off, and investigates the properties of organizational populations with low/high SA and low/high PoA. The model generates three expected core “corner” solutions: (1) the dominance of large organizations in the strong SA setting; (2) the proliferation of narrow-niche organizations in the strong PoA setting; and (3) a bifurcated or dual market structure if both SA and PoA are present. The model also allows us to identify circumstances under which narrow-niche (specialists) or wide-niche (generalists) organizations thrive. We also use the model to examine the claim that concentrated resource distributions are more likely to generate partitioned or bifurcated populations. We also investigate the consequences of environments comprised of ordered versus unordered positions.     

The Law of Large Numbers for self-exciting correlated defaults

We consider a model of correlated defaults in which the default times of multiple entities depend not only on common and specific factors, but also on the extent of past defaults in the market, via the average loss process, including the average number of defaults as a special case. The paper characterizes the average loss process when the number of entities becomes large, showing that under some monotonicity conditions the limiting average loss process can be determined by a fixed point problem. We also show that the Law of Large Numbers holds under certain compatibility conditions.     

Bundling decisions in supply chains

Firms often sell products in bundles to extract consumer surplus. While most bundling decisions studied in the literature are geared to integrated firms, we examine a decentralized supply chain where the suppliers retain decision rights. Using a generic distribution of customers’ reservation price we establish equilibrium solutions for three different bundling scenarios in a supply chain, and generate interesting insights for distributions with specific forms. We find that (i) in supply chain bundling the retailer’s margin equals the margin of each independent supplier, and it equals the combined margin when the suppliers are in a coalition, (ii) when the suppliers form a coalition to bundle their products the bundling gain in the supply chain is higher and retail price is lower than when the retailer bundles the products, (iii) the supply chain has more to gain from bundling relative to an integrated firm, (iv) the first-best supply chain bundling remains viable over a larger set of parameter values than those in the case of the integrated firm, (v) supplier led bundling is preferable to separate sales over a wider range of parameter values than if the retailer led the bundling, and (vi) if the reservation prices are uniformly distributed bundling can be profitable when the variable costs are low and valuations of the products are not significantly different from one another. For normally distributed reservation prices, we show that the bundling set is larger and the bundling gain is higher than that for a uniform distribution.     

Pitman closeness of current records for location-scale families

In this paper, the largest and the smallest observations are considered, at the time when a new record of either kind (upper or lower) occurs based on a sequence of independent random variables with identical continuous distributions. These statistics are referred to as current upper and lower records, respectively, in the statistical literature. We derive expressions for the Pitman closeness of current records to a common population parameter and then apply these results to location-scale families of distributions with a special emphasis on the estimation of quantiles. In the case of symmetric distributions, we show that this criterion possesses some symmetry properties. Exact expressions are derived for the Pitman closeness probabilities in the case of Uniform (?1, 1) and exponential distributions. Moreover, for the population median, we show that the Pitman closeness probability is distribution-free.     

Optimal static-dynamic hedges for exotic options under convex risk measures

We study the problem of optimally hedging exotic derivatives positions using a combination of dynamic trading strategies in underlying stocks and static positions in vanilla options when the performance is quantified by a convex risk measure. We establish conditions for the existence of an optimal static position for general convex risk measures, and then analyze in detail the case of shortfall risk with a power loss function. Here we find conditions for uniqueness of the static hedge. We illustrate the computational challenge of computing the market-adjusted risk measure in a simple diffusion model for an option on a non-traded asset.     

A uniform asymptotic estimate for discounted aggregate claims with subexponential tails

In this paper we study the tail probability of discounted aggregate claims in a continuous-time renewal model. For the case that the common claim-size distribution is subexponential, we obtain an asymptotic formula, which holds uniformly for all time horizons within a finite interval. Then, with some additional mild assumptions on the distributions of the claim sizes and inter-arrival times, we further prove that this formula holds uniformly for all time horizons. In this way, we significantly extend a recent result of Tang [Tang, Q., 2007. Heavy tails of discounted aggregate claims in the continuous-time renewal model. J. Appl. Probab. 44 (2), 285–294].     

Local Walsh-average-based estimation and variable selection for single-index models

We propose a robust estimation procedure based on local Walsh-average regression(LWR) for single-index models. Our novel method provides a root-n consistent estimate of the single-index parameter under some mild regularity conditions; the estimate of the unknown link function converges at the usual rate for the nonparametric estimation of a univariate covariate. We theoretically demonstrate that the new estimators show significant efficiency gain across a wide spectrum of non-normal error distributions and have almost no loss of efficiency for the normal error. Even in the worst case, the asymptotic relative efficiency(ARE) has a lower bound compared with the least squares(LS) estimates; the lower bounds of the AREs are 0.864 and 0.8896 for the single-index parameter and nonparametric function, respectively. Moreover, the ARE of the proposed LWR-based approach versus the ARE of the LS-based method has an expression that is closely related to the ARE of the signed-rank Wilcoxon test as compared with the t-test. In addition, to obtain a sparse estimate of the single-index parameter, we develop a variable selection procedure by combining the estimation method with smoothly clipped absolute deviation penalty; this procedure is shown to possess the oracle property. We also propose a Bayes information criterion(BIC)-type criterion for selecting the tuning parameter and further prove its ability to consistently identify the true model. We conduct some Monte Carlo simulations and a real data analysis to illustrate the finite sample performance of the proposed methods.     

The asymptotic expansion as well as the exact moments of the stein estimator when the population means are nearly equal

Summary The purpose of this note is to derive the asymptotic distributions, means and variances of the Stein estimator, as well as that of the quadratic loss function for the vector case when the population means are nearly equal. These results are given in Section 3 and are obtained by using a method similar to the perturbation method, used by Nagao [4]. In Section 4 exact moments of the Stein estimator are also derived. Financially supported by the CSIR and the University of the OFS Research Fund.     

Conditional tail risk measures for the skewed generalised hyperbolic family

This paper deals with the estimation of loss severity distributions arising from historical data on univariate and multivariate losses. We present an innovative theoretical framework where a closed-form expression for the tail conditional expectation (TCE) is derived for the skewed generalised hyperbolic (GH) family of distributions. The skewed GH family is especially suitable for equity losses because it allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to the widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its special cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. The GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on equity losses. The TCE risk measure computed for the skewed family of GH distributions provides a conservative estimator of risk, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolios of losses, allowing the multivariate GH distribution to capture the combination of correlated risks and demonstrate how the TCE of the portfolio can be decomposed into individual components, representing individual risks in the aggregate (portfolio) loss.     

Partial penalized empirical likelihood ratio test under sparse case

A consistent test via the partial penalized empirical likelihood approach for the parametric hypothesis testing under the sparse case, called the partial penalized empirical likelihood ratio (PPELR) test, is proposed in this paper. Our results are demonstrated for the mean vector in multivariate analysis and regression coefficients in linear models, respectively. And we establish its asymptotic distributions under the null hypothesis and the local alternatives of order n^?1/2 under regularity conditions. Meanwhile, the oracle property of the partial penalized empirical likelihood estimator also holds. The proposed PPELR test statistic performs as well as the ordinary empirical likelihood ratio test statistic and outperforms the full penalized empirical likelihood ratio test statistic in term of size and power when the null parameter is zero. Moreover, the proposed method obtains the variable selection as well as the p-values of testing. Numerical simulations and an analysis of Prostate Cancer data confirm our theoretical findings and demonstrate the promising performance of the proposed method in hypothesis testing and variable selection.     

Optimizing delivery lead time/inventory placement in a two-stage production/distribution system

In this paper we study a system composed of a supplier and buyer(s). We assume that the buyer faces random demand with a known distribution function. The supplier faces a known production lead time. The main objective of this study is to determine the optimal delivery lead time and the resulting location of the system inventory. In a system with a single-supplier and a single-buyer it is shown that system inventory should not be split between a buyer and supplier. Based on system parameters of shortage and holding costs, production lead times, and standard deviations of demand distributions, conditions indicating when the supplier or buyer(s) should keep the system inventory are derived. The impact of changes to these parameters on the location of system inventory is examined. For the case with multiple buyers, it is found that the supplier holds inventory for the buyers with the smallest standard deviations, while the buyers with the largest standard deviations hold their own inventory.     

A Problem in Reliability Optimization

We consider the system efficiency measure as the average asymptotic gain earned by the system for one time unit. Such defined efficiency is a continuous function of the load that the system is to support. Each increment of the load causes an increment of the gain. On the other hand, it also causes a loss of system reliability, giving in this way a loss. This loss follows from the fact that a less reliable system is in the off state more frequently than in the lower-load case. The goal is to find the load that maximizes the efficiency.The lifetime distribution of the system is defined as exponential. As models of the relationship between the load and the gain as well as the relationship between the load and the failure rate, power and exponential functions have been taken. The analytical conditions for the existence of maximum positive gain and the analysis of the behaviour of the gain as the load increases are presented.     

Likelihood Inference under Generalized Hybrid Censoring Scheme with Comp eting Risks

Statistical inference is developed for the analysis of generalized type-II hybrid censoring data under exponential competing risks model. In order to solve the problem that approximate methods make unsatisfactory performances in the case of small sample size, we establish the exact conditional distributions of estimators for parameters by conditional moment generating function(CMGF). Furthermore, confidence intervals(CIs) are constructed by exact distributions, approximate distributions as well as bootstrap method respectively, and their performances are evaluated by Monte Carlo simulations. And finally, a real data set is analyzed to illustrate all the methods developed here.     

Local asymptotic normality and asymptotical minimax efficiency of the MLE under random censorship

Here we study the problems of local asymptotic normality of the parametric family of distributions and asymptotic minimax efficient estimators when the observations are subject to right censoring. Local asymptotic normality will be established under some mild regularity conditions. A lower bound for local asymptotic minimax risk is given with respect to a bowl-shaped loss function, and furthermore a necessary and sufficient condition is given in order to achieve this lower bound. Finally, we show that this lower bound can be attained by the maximum likelihood estimator in the censored case and hence it is local asymptotic minimax efficient.     

Asymptotic properties of sample quantiles of discrete distributions

The asymptotic distribution of sample quantiles in the classical definition is well-known to be normal for absolutely continuous distributions. However, this is no longer true for discrete distributions or samples with ties. We show that the definition of sample quantiles based on mid-distribution functions resolves this issue and provides a unified framework for asymptotic properties of sample quantiles from absolutely continuous and from discrete distributions. We demonstrate that the same asymptotic normal distribution result as for the classical sample quantiles holds at differentiable points, whereas a more general form arises for distributions whose cumulative distribution function has only one-sided differentiability. For discrete distributions with finite support, the same type of asymptotics holds and the sample quantiles based on mid-distribution functions either follow a normal or a two-piece normal distribution. We also calculate the exact distribution of these sample quantiles for the binomial and Poisson distributions. We illustrate the asymptotic results with simulations.     

The loss of stability of symmetrical equilibrium positions

Systems of differential equations possessing a finite (or compact) symmetry group and depending on one parameter are considered. The nature of the loss of stability of equilibrium positions is investigated in cases when, owing to symmetry, the linearized problem has multiple eigenvalues. Conditions are presented that determine whether the loss of stability when the parameter is varied is soft or hard, for double eigenvalues λ - zero or pure imaginary. Cases of triple zero eigenvalues λ corresponding to tetrahedral (or cubic) symmetry, are considered.