Asymptotic normality of numbers of observations in random regions determined by order statistics

In this paper, we study the joint limiting behaviour of numbers of observations that fall into regions determined by order statistics and Borel sets. We show that suitably centred and normed versions of these numbers are asymptotically multivariate normal under some conditions. We consider two cases: one where the population distribution function is discontinuous and the other where it is continuous and the order statistics are extreme. Finally, we compare results obtained for the two cases with their analogues for absolutely continuous distribution function and central-order statistics.     

New ASA Fellows—1967

This paper presents at an elementary level a unified presentation of concepts related to sufficiency and minimal sufficiency. Extensively discussed are techniques for showing in a particular statistical model that a given statistic is not sufficient or that a given sufficient statistic is not minimal. The applicability of these techniques is illustrated in three examples.     

Empirical Bayes testing for equivalence

For a fixed point θ₀ and a positive value c₀, this paper studies the problem of testing the hypotheses H₀:|θ−θ₀|≤c₀ against H₁:|θ−θ₀|>c₀ for the normal mean parameter θ using the empirical Bayes approach. With the accumulated past data, a monotone empirical Bayes test is constructed by mimicking the behavior of a monotone Bayes test. Such an empirical Bayes test is shown to be asymptotically optimal and its regret converges to zero at a rate (lnn)^2.5/n where n is the number of past data available, when the current testing problem is considered. A simulation study is also given, and the results show that the proposed empirical Bayes procedure has good performance for small to moderately large sample sizes. Our proposed method can be applied for testing close to a control problem or testing the therapeutic equivalence of one standard treatment compared to another in clinical trials.     

An Empirical Study of Equivalence Judgments vs. Ratio Judgments in Decision Analysis*

Two commonly used elicitation modes on strength of preference, equivalence and ratio judgments, were compared in an experiment. The result from the experiment showed that ratio judgments were less effective than equivalence judgments. Based on an iterative design for eliciting multiattribute preference structures, equivalence judgments outperformed ratio judgments in estimating single‐attribute measurable value functions, while being nearly more effective than ratio judgments in assessing multiattribute preference structures. The implications of the results from the experiment are that multiattribute decision‐making techniques should take advantage of the decision maker's inclination of making effective equivalence trade‐off judgments, and that useful techniques should be devised to incorporate different commonly used techniques, such as multiattribute utility theory and the Analytic Hierarchy Process, to elicit and consolidate equivalence trade‐off judgments.     

Business Cycle Accounting

We propose a simple method to help researchers develop quantitative models of economic fluctuations. The method rests on the insight that many models are equivalent to a prototype growth model with time‐varying wedges that resemble productivity, labor and investment taxes, and government consumption. Wedges that correspond to these variables—efficiency, labor, investment, and government consumption wedges—are measured and then fed back into the model so as to assess the fraction of various fluctuations they account for. Applying this method to U.S. data for the Great Depression and the 1982 recession reveals that the efficiency and labor wedges together account for essentially all of the fluctuations; the investment wedge plays a decidedly tertiary role, and the government consumption wedge plays none. Analyses of the entire postwar period and alternative model specifications support these results. Models with frictions manifested primarily as investment wedges are thus not promising for the study of U.S. business cycles.     

Limit laws for the cumulative number of ties for the maximum in a random sequence

Let {_Xn,n?1}$\{X_n,n?1\}$ be a sequence of independent identically distributed random variables, taking nonnegative integer values. An observation _Xn$X_n$ is a tie for the maximum if _Xn=max{_X1,…,_Xn-1}$X_n=\max \{X_1,\dots ,X_{n-1}\}$. In this paper, we obtain weak and strong laws of large numbers and central limit theorems for the cumulative number of ties for the maximum among the first n$n$ observations.     

中国政府债务对居民消费影响的实证研究

在政府债务对居民消费的影响方面,“凯恩斯主义”和“李嘉图主义”提出了对立的观点。本文运用中国宏观经济数据,分别检验了“凯恩斯主义”和“李嘉图主义”的政府债务观,识别出在中国政府债务影响居民消费的基本事实。我们的计量研究表明:在中国“凯恩斯主义”的政府债务观与现实更相关,政府债务的增加能够促进居民消费增长。居民视其持有的政府债券为净财富,他们基本上没有感受到政府债务增加对未来税收负担的影响。     

Sample size calculation for testing non-inferiority and equivalence under Poisson distribution

When counting the number of chemical parts in air pollution studies or when comparing the occurrence of congenital malformations between a uranium mining town and a control population, we often assume Poisson distribution for the number of these rare events. Some discussions on sample size calculation under Poisson model appear elsewhere, but all these focus on the case of testing equality rather than testing equivalence. We discuss sample size and power calculation on the basis of exact distribution under Poisson models for testing non-inferiority and equivalence with respect to the mean incidence rate ratio. On the basis of large sample theory, we further develop an approximate sample size calculation formula using the normal approximation of a proposed test statistic for testing non-inferiority and an approximate power calculation formula for testing equivalence. We find that using these approximation formulae tends to produce an underestimate of the minimum required sample size calculated from using the exact test procedure. On the other hand, we find that the power corresponding to the approximate sample sizes can be actually accurate (with respect to Type I error and power) when we apply the asymptotic test procedure based on the normal distribution. We tabulate in a variety of situations the minimum mean incidence needed in the standard (or the control) population, that can easily be employed to calculate the minimum required sample size from each comparison group for testing non-inferiority and equivalence between two Poisson populations.     

On quantile processes for m-dependent Rv's

We obtain strong and weak approximations for quantile processes for m-dependent random variables. We apply our results in the two-sample to obtain condidence bands for quantile plots and prove some CHERNOFF-SAVAGE theorems for m-dependent random variables     

On the efficiency of two-stage and three-stage least squares estimators

This paper utilizes the results of Kruskal (1968), Zyskind (1967), and more recently Milliken and Albohali (1984) to derive a simple necessary and sufficient condition for 3SLS to be equivalent to 2SLS. This condition depends upon the inverse of the variance:covariance matrix of the disturbances, and the set of second stage regressors of each structural equation. More importantly, this condition should prove useful for econometric students and provides an easy method for checking sufficiency.