The domain of partial attraction of a Poisson law

Groshev gave a characterization of the union of domains of partial attraction of all Poisson laws in 1941. His classical condition is expressed by the underlying distribution function and disguises the role of the mean of the attracting distribution. In the present paper we start out from results of the recent probabilistic approach and derive characterizations for any fixed >0 in terms of the underlying quantile function. The approach identifies the portion of the sample that contributes the limiting Poisson behavior of the sum, delineates the effect of extreme values, and leads to necessary and sufficient conditions all involving . It turns out that the limiting Poisson distributions arise in two qualitatively different ways depending upon whether >1 or <1. A concrete construction, illustrating all the results, also shows that in the boundary case when =1 both possibilities may occur.     

AN EFFICIENT SEQUENTIAL DESIGN FOR SENSITIVITY EXPERIMENTS

In sensitivity experiments, the response is binary and each experimental unit has a critical stimulus level that cannot be observed directly. It is often of interest to estimate extreme quantiles of the distribution of these critical stimulus levels over the tested products. For this purpose a new sequential scheme is proposed with some commonly used models. By using the bootstrap repeated-sampling principle, reasonable prior distributions based on a historic data set are specified. Then, a Bayesian strategy for the sequential procedure is provided and the estimator is given. Further, a high order approximation for such an estimator is explored and its consistency is proven. A simulation study shows that the proposed method gives superior performances over the existing methods.     

Asymptotics of the regression quantile basic solution under misspecification

We consider the asymptotic distribution of covariate values in the quantile regression basic solution under weak assumptions. A diagnostic procedure for assessing homogeneity of the conditional densities is also proposed. The research for this paper was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Nonparametric inference on the difference of location parameters of correlated variables from fragmentary samples

Summary In this paper, two types of robust estimators and approximate confidence intervals for the difference of location parameters of correlated random variables are proposed and investigated when some observations are missing. It is shown that the suggested estimators are consistent and asymptotically normally distributed. In addition, the proposed approximate confidence intervals are also shown to enjoy some nice asymptotic properties.     

Limit theorems for the median deviation

Summary In this paper, we obtain a strong law and central limit theorem for the median deviation under only very mild smoothness conditions on the underlying distribution. Under an additional condition implied by symmetry, we derive a weak Bahadur representation for the median deviation and establish the asymptotic equivalence of the median deviation and the semi-interquartile range.     

On Quantiles and the Central Limit Question for Strongly Mixing Sequences

Three classes of strictly stationary, strongly mixing random sequences are constructed, in order to provide further information on the borderline of the central limit theorem for strictly stationary, strongly mixing random sequences. In these constructions, a key role is played by quantiles, as in a related construction of Doukhan et al.⁽¹¹⁾     

Subsample and half-sample methods

Hartigan's subsample and half-sample methods are both shown to be inefficient methods of estimating the sampling distributions. In the sample mean case the bootstrap is known to correct for skewness. But irrespective of the population, the estimates based on the subsample method, have skewness factor zero. This problem persists even if we take only samples of size less than or equal to half of the original sample. For linear statistics it is possible to correct this by considering estimates based on subsamples of size n, when the sample size is n. In the sample mean case can be taken as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeimaiaab6% cacaqG1aGaaeikaiaabgdacaqGGaGaeyOeI0IaaeiiaiaabgdacaqG% VaWaaOaaaeaacaqG1aaaleqaaOGaaeykaaaa!3E8A!\[{\text{0}}{\text{.5(1 }} - {\text{ 1/}}\sqrt {\text{5}} {\text{)}}\]. In spite of these negative results, the half-sample method is useful in estimating the variance of sample quantiles. It is shown that this method gives as good an estimate as that given by the bootstrap method. A major advantage of the half-sample method is that it is shown to be robust in estimating the mean square error of estimators of parameters of a linear regression model when the errors are heterogeneous. Bootstrap is known to give inconsistent results in this case; although, it is more efficient in the case of homogeneous errors.Research supported in part by NSA Grant MDA904-90-H-1001 and by NASA Grant NAGW-1917.     

A note on bootstrapping the variance of sample quantile

Summary Letm _{n, p} denote thep-th quantile based onn observations and let λ _p denote the population quantile. In this paper consistency of the bootstrap estimate of variance of is established.     

Tests of significance using selected sample quantiles

Large sample tests of significance for the location parameter, the scale parameter, and quantiles for a location-scale family of distributions based on a few optimally chosen sample quantiles are considered.     

Inference for intermediate Haezendonck–Goovaerts risk measure

Recently Haezendonck–Goovaerts (H–G) risk measure has received much attention in actuarial science. Nonparametric inference has been studied by Ahn and Shyamalkumar (2014) and Peng et al. (2015) when the risk measure is defined at a fixed level. In risk management, the level is usually set to be quite near one by regulators. Therefore, especially when the sample size is not large enough, it is useful to treat the level as a function of the sample size, which diverges to one as the sample size goes to infinity. In this paper, we extend the results in Peng et al. (2015) from a fixed level to an intermediate level. Although the proposed maximum empirical likelihood estimator for the H–G risk measure has a different limit for a fixed level and an intermediate level, the proposed empirical likelihood method indeed gives a unified interval estimation for both cases. A simulation study is conducted to examine the finite sample performance of the proposed method.