Comparison of two measures of association in two-way contingency tables

Stuart's (1953) measure of association in contingency tables, t_C, based on Kendall's (1962) t, is compared with Goodman and Kruskal's (1954, 1959, 1963, 1972) measure G. First, it is proved that |G| ≥ |t_C|; and then it is shown that the upper bound for the asymptotic variance of G is not necessarily always smaller than the upper bound for the asymptotic variance of t_C. It is proved, however, that the upper bound for the coefficient of variation of G cannot be larger in absolute value than the upper bound for the coefficient of variation of t_C. The asymptotic variance of t_C is also derived and hence we obtain an upper bound for this asymptotic variance which is sharper than Stuart's (1953) upper bound.     

Estimating the Upper Support Point in Deconvolution

Abstract.  We consider estimation of the upper boundary point F ⁻¹ (1) of a distribution function F with finite upper boundary or 'frontier' in deconvolution problems, primarily focusing on deconvolution models where the noise density is decreasing on the positive halfline. Our estimates are based on the (non-parametric) maximum likelihood estimator (MLE) of F . We show that (1) is asymptotically never too small. If the convolution kernel has bounded support the estimator (1) can generally be expected to be consistent. In this case, we establish a relation between the extreme value index of F and the rate of convergence of (1) to the upper support point for the 'boxcar' deconvolution model. If the convolution density has unbounded support, (1) can be expected to overestimate the upper support point. We define consistent estimators , for appropriately chosen vanishing sequences ( β _n ) and study these in a particular case.     

The Choice of Statistic on Which to Base Tight Upper Confidence Limits

We consider the problem of finding an upper 1 –α confidence limit (α < ½) for a scalar parameter of interest θ in the presence of a nuisance parameter vector ψ when the data are discrete. Using a statistic T as a starting point, Kabaila & Lloyd (1997) define what they call the tight upper limit with respect to T . This tight upper limit possesses certain attractive properties. However, these properties provide very little guidance on the choice of T itself. The practical recommendation made by Kabaila & Lloyd (1997) is that T be an approximate upper 1 –α confidence limit for θ rather than, say, an approximately median unbiased estimator of θ. We derive a large sample approximation which provides strong theoretical support for this recommendation.     

On Bounds for Probability Generating Functions

Brook (1966) gave an upper bound for the moment generating function (m.g.f.) of a positive random variable (r.v.) in terms of its moments, and used this to obtain an upper bound for the probability generating function (p.g.f.) and hence the extinction probability of a simple branching process. Agresti (1974) rederived this bound of the p.g.f. and used it to obtain a lower bound of the expectation of extinction time of a branching process. In both of these applications the random variable is integer valued, and for this class we improve on Brook's bound by deriving the best upper bound of the p.g.f. Our method, which is a variant of Brook's (1966) is used later to obtain the lower bound of the p.g.f. when the third moment is also known.     

Boxplot-Based Outlier Detection for the Location-Scale Family

Boxplots are among the most widely used exploratory data analysis (EDA) tools in statistical practice. Typical applications of boxplots include eliciting information about the underlying distribution (shape, location, etc.) as well as identifying possible outliers. This article focuses on a modification using a type of lower and upper fences similar in concept to those used in a traditional boxplot; however, instead of constructing the upper and lower fences using the upper and lower quartiles, respectively, and a multiple of the interquartile range (IQR), multiples of the upper and the lower semi-interquartile ranges (SIQR), respectively, measured from the sample median, are used. Any observation beyond the proposed fences is labeled a potential outlier. An exact expression for the probability that at least one sample observation is wrongly classified as an outlier, the so-called “some-outside rate per sample” (Hoaglin et al. (1986)), is derived for the family of location-scale distributions and is used in the determination of the fence constants. Tables for the fence constants are provided for a number of well-known location-scale distributions along with some illustrations with data; the performance of the outlier detection rule is explored in a simulation study.     

Bounds on expected generalized order statistics

We establish the upper nonpositive and all the lower bounds on the expectations of generalized order statistics based on a given distribution function with the finite mean and central absolute moment of a fixed order. We also describe the distributions for which the bounds are attained. The methods of deriving the lower nonpositive (upper nonnegative) and lower nonnegative (upper nonpositive) bounds are totally different. The first one, the greatest convex minorant method is the combination of the Moriguti and well-known Hölder inequalities and the latter one is based on the maximization of some norm on the properly chosen convex set. The paper completes the results of Cramer et al. [Evaluations of expected generalized order statistics in various scale units. Appl Math. 2002;29:285–295].     

Profile upper Confidence Limits from Discrete Data

When the data are discrete, standard approximate confidence limits often have coverage well below nominal for some parameter values. While ad hoc adjustments may largely solve this problem for particular cases, Kabaila & Lloyd (1997) gave a more systematic method of adjustment which leads to tight upper limits, which have coverage which is never below nominal and are as small as possible within a particular class. However, their computation for all but the simplest models is infeasible. This paper suggests modifying tight upper limits by an initial replacement of the unknown nuisance parameter vector by its profile maximum likelihood estimator. While the resulting limits no longer possess the optimal properties of tight limits exactly, the paper presents both numerical and theoretical evidence that the resulting coverage function is close to optimal. Moreover these profile upper limits are much (possibly many orders of magnitude) easier to compute than tight upper limits.     

UPPER BOUNDS ON THE MINIMUM COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER MODEL SELECTION

We consider a linear regression model, with the parameter of interest a specified linear combination of the components of the regression parameter vector. We suppose that, as a first step, a data-based model selection (e.g. by preliminary hypothesis tests or minimizing the Akaike information criterion – AIC) is used to select a model. It is common statistical practice to then construct a confidence interval for the parameter of interest, based on the assumption that the selected model had been given to us  a priori . This assumption is false, and it can lead to a confidence interval with poor coverage properties. We provide an easily computed finite-sample upper bound (calculated by repeated numerical evaluation of a double integral) to the minimum coverage probability of this confidence interval. This bound applies for model selection by any of the following methods: minimum AIC, minimum Bayesian information criterion (BIC), maximum adjusted  R ², minimum Mallows'   C _P   and  t -tests. The importance of this upper bound is that it delineates general categories of design matrices and model selection procedures for which this confidence interval has poor coverage properties. This upper bound is shown to be a finite-sample analogue of an earlier large-sample upper bound due to Kabaila and Leeb.     

Confidence intervals for the mean and a percentile based on zero-inflated lognormal data

The problems of estimating the mean and an upper percentile of a lognormal population with nonnegative values are considered. For estimating the mean of a such population based on data that include zeros, a simple confidence interval (CI) that is obtained by modifying Tian's [Inferences on the mean of zero-inflated lognormal data: the generalized variable approach. Stat Med. 2005;24:3223—3232] generalized CI, is proposed. A fiducial upper confidence limit (UCL) and a closed-form approximate UCL for an upper percentile are developed. Our simulation studies indicate that the proposed methods are very satisfactory in terms of coverage probability and precision, and better than existing methods for maintaining balanced tail error rates. The proposed CI and the UCL are simple and easy to calculate. All the methods considered are illustrated using samples of data involving airborne chlorine concentrations and data on diagnostic test costs.     

A note on determining the number of outliers in an exponential sample by least squares procedure

In this paper, we suggest a least squares procedure for the determination of the number of upper outliers in an exponential sample by minimizing sample mean squared error. Moreover, the method can reduce the masking or “swamping” effects. In addition, we have also found that the least squares procedure is easy and simple to compute than test test procedure T _k suggested by Zhang (1998) for determining the number of upper outliers, since Zhang (1998) need to use the complicated null distribution of T _k . Moreover, we give three practical examples and a simulated example to illustrate the procedures. Further, simulation studies are given to show the advantages of the proposed method. Finally, the proposed least squares procedure can also determine the number of upper outliers in other continuous univariate distributions (for example, Pareto, Gumbel, Weibull, etc.). Received: May 10, 1999; revised version: June 5, 2000     

Estimation of multiple correlation coefficient when observations are missing on one of the variables

This paper proposes a control chart with variable sampling intervals (VSI) to detect increases in the expected value of the number of defects in a random sample of constant size n the upper one-sided c-VSI chart

The performance of this chart is evaluated by means of the average time to signal (ATS).The comparisons made between the standard FSI (fixed sampling intervals) and the VSI upper one-sided c - charts indicate that using variable sampling intervals can substantially reduce the average time to signal. Using stochastic ordering we prove that this reduction always occurs.

Special attention is given to the choice of the proposed control chart parameters and to the chart graphical display.     

我国35个大中城市房价-持久收入比研究

房价收入比是反映居民购房可支付能力的重要指标。为了更真实准确反映我国居民长期承受的购房负担,本文基于持久收入假说,利用我国35个大中城市数据,建立状态空间模型,估计出持久收入意义下的房价收入比,通过情景分析确定其合理上限,并分析主要特征。研究表明：①我国房价-持久收入比的合理上限为7.6,2002-2013年35个大中城市的房价-持久收入比均值是9.2,其中28个城市已超过该上限,比值最高的北京已达到14.9;②城市越发达,则房价-持久收入比越高,居民需要承受的购房压力越大,且在时间趋势上,一线与二三线城市间的差距呈现扩大趋势;③在地理分布上,我国东部、中部、东北地区、西部大中城市的房价-持久收入比呈现从高到低的排列顺序;④相比于传统方法中根据可支配收入测算的房价收入比,房价-持久收入比与其约有10%的差异。当前我国政府应采取有效措施继续限制房价,减轻居民购房负担。     

TIGHT UPPER CONFIDENCE LIMITS FROM DISCRETE DATA

Consider the problem of finding an upper 1 –α confidence limit for a scalar parameter of interest ø in the presence of a nuisance parameter vector θ when the data are discrete. Approximate upper limits T may be found by approximating the relevant unknown finite sample distribution by its limiting distribution. Such approximate upper limits typically have coverage probabilities below, sometimes far below, 1 –α for certain values of (θ, ø). This paper remedies that defect by shifting the possible values t of T so that they are as small as possible subject both to the minimum coverage probability being greater than or equal to 1 –α, and to the shifted values being in the same order as the unshifted ts. The resulting upper limits are called ‘tight’. Under very weak and easily checked regularity conditions, a formula is developed for the tight upper limits.     

Optimal designs for covariates’ models

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. The D-criterion is used to judge the ‘goodness’ of any design for estimating the parameters of this model. Since this criterion is based on the determinant of the information matrix M(d) of a design d, upper bounds for |M(d)| yield lower bounds for the D-efficiency of any design d in estimating the vector of parameters in the model. We consider here only classes of designs d for which the number N of observations to be taken is a multiple of V, that is, there exists R≥2 such that N=V×R.Under these conditions, we determine the maximum of |M(d)|, and conditions under which the maximum is attained. These conditions include R being even, each treatment level being observed the same number of times, that is, R times, and N being a multiple of four. For the other cases of congruence of N (modulo 4) we further determine upper bounds on |M (d)| for equireplicated designs, i.e. for designs with equal number of observations per treatment level. These upper bounds are shown to depend also on the congruence of V (modulo 4). For some triples (N,V,K), the upper bounds determined are shown to be attained.Construction methods yielding families of designs which attain the upper bounds of |M(d)| are presented, for each of the sixteen cases of congruence of N and V.We also determine the upper bound for D-optimal designs for estimating only the treatment parameters, when first order regression on one continuous covariate is present.     

Epidemics in heterogeneous communities: estimation of R0 and secure vaccination coverage

A stochastic multitype model for the spread of an infectious disease in a community of heterogeneous individuals is analysed. In particular, estimates of R ₀ (the basic reproduction number) and the critical vaccination coverage are derived, where estimation is based on final size data of an outbreak in the community. It is shown that these key parameters cannot be estimated consistently from data; only upper and lower bounds can be estimated. Confidence regions for the upper bounds are derived, thus giving conservative estimates of R ₀ and the fractions necessary to vaccinate.     

A cramer - von mises type goodness of fit test with asymmetric weight function

A goodness of fit test of the Cramer - von Mises type, which gives more weight to the upper (or to the lower) tail of the distribution, is proposed and studied. It is found the orthogonal representation of the test for the case of a simple null hypothesis. The characteristic function of the asymptotic null distribution is found and inverted to get percentage points. The asymptotic power of the test is obtained for the normal null hypothesis, against mean and variance shifts and more asymmetric alternatives.

Also the case of the exponential null hypothesis is studied. It is found that the test, which emphasizes the upper tail, has more power than those of Anderson - Darling and Cramer - von Mises, against alternatives which differ from the null hypothesis mainly in the upper tail, and less power when the main difference is in the lower tail of the distribution.     

Improved variance bounds for unbiased estimation in inverse sampling

Sathe (1977) derived sharper variance bounds for inverse sample unbiased estimator of the negative binomial parameter p. In the present writing improved upper/lower variance bounds are achieved and the relative improvement is numerically illustrated.     

Bounds on the number of constraints for balanced arrays of strength t

We describe a method, based on non-negative definiteness of moment matrices, for deriving upper bounds on the number of constraints in balanced arrays of strength t, involving two or more symbols. It is seen that the method covers, in particular, those due to Rafter and Seiden (1974) and Chopra (1982,1983).     

Bounds on the mean of the maximum of a number of normal random variables

A simple method producing lower and upper bounds on E max(X₁,...,X_n) is presented under assumption that the X_i's are independent normal random variables. Furthermore the upper bounds are determined when the X_i's are normal and positively correlated     

The limiting bound of Efron's W-formula for hypothesis testing when a nuisance parameter is present only under the alternative

When testing a hypothesis with a nuisance parameter present only under the alternative, the maximum of a test statistic over the nuisance parameter space has been proposed. Different upper bounds for the one-sided tail probabilities of the maximum tests were provided. Davies (1977. Biometrika 64, 247–254) studied the problem when the parameter space is an interval, while Efron (1997. Biometrika 84, 143–157) considered the problem with some finite points of the parameter space and obtained a W-formula. We study the limiting bound of Efron's W-formula when the number of points in the parameter space goes to infinity. The conditions under which the limiting bound of the W-formula is identical to that of Davies are given. The results are also extended to two-sided tests. Examples are used to illustrate the conditions, including case-control genetic association studies. Efficient calculations of upper bounds for the tail probability with finite points in the parameter space are described.