Central Limit Theorems for Reduced U‐Statistics Under Dependence and Their Usefulness

A reduced ‐statistic is a ‐statistic with its summands drawn from a restricted but balanced set of pairs. In this article, central limit theorems are derived for reduced ‐statistics under ‐mixing, which significantly extends the work of Brown & Kildea in various aspects. It will be shown and illustrated that reduced ‐statistics are quite useful in deriving test statistics in various nonparametric testing problems.     

Further Properties of Frequentist Confidence Intervals in Regression that Utilize Uncertain Prior Information

Consider a linear regression model with n‐dimensional response vector, regression parameter and independent and identically distributed errors. Suppose that the parameter of interest is where a is a specified vector. Define the parameter where c and t are specified. Also suppose that we have uncertain prior information that . Part of our evaluation of a frequentist confidence interval for is the ratio (expected length of this confidence interval)/(expected length of standard confidence interval), which we call the scaled expected length of this interval. We say that a confidence interval for utilizes this uncertain prior information if: (i) the scaled expected length of this interval is substantially less than 1 when ; (ii) the maximum value of the scaled expected length is not too much larger than 1; and (iii) this confidence interval reverts to the standard confidence interval when the data happen to strongly contradict the prior information. Kabaila and Giri (2009) present a new method for finding such a confidence interval. Let denote the least squares estimator of . Also let and . Using computations and new theoretical results, we show that the performance of this confidence interval improves as increases and decreases.     

Reviews of Books and Teaching Materials

ActivStats 2.0.: Paul Velleman. Reading, MA: Addison-Wesley, 1998, CD-ROM, $44.25, ISBN: 0-201-31068-6.

The Active Practice of Statistics.: David S. Moore. New York: W. H. Freeman & Co., 1997, xv + 352 pp., $49.95, ISBN: ISBN 0-7167-3140-1. Reviewed by Jon Maatta

Introduction to Statistical Reasoning.: Gary Smith. Boston: McGraw-Hill, 1998, xviii + 654 pp., $46.00, ISBN: 0-07-059276-4. Reviewed by Robert W. Hayden

Modern Applied Statistics with S-Plus (2nd ed.).: William N. Venables and Brian D. Ripley. New York: Springer-Verlag, 1997, xvii + 548 pp., $59.95, ISBN: 0-387-98214-0. Reviewed by Karen Kafadar James R. Koehler

A Course in Mathematical Statistics (2nd ed.): George C. Roussas. San Diego: Academic Press, 1997, xx + 572 pp., $59.95, ISBN: 0125993153. Reviewed by David W. Macky     

Implicit Estimation for the Stochastic Volatility Model

Efficient, accurate, and fast Markov Chain Monte Carlo estimation methods based on the Implicit approach are proposed. In this article, we introduced the notion of Implicit method for the estimation of parameters in Stochastic Volatility models.

Implicit estimation offers a substantial computational advantage for learning from observations without prior knowledge and thus provides a good alternative to classical inference in Bayesian method when priors are missing.

Both Implicit and Bayesian approach are illustrated using simulated data and are applied to analyze daily stock returns data on CAC40 index.     

Wavelet Thresholding Estimation in a Poissonian Interactions Model with Application to Genomic Data

This paper deals with the study of dependencies between two given events modelled by point processes. In particular, we focus on the context of DNA to detect favoured or avoided distances between two given motifs along a genome suggesting possible interactions at a molecular level. For this, we naturally introduce a so‐called reproduction function h that allows to quantify the favoured positions of the motifs and that is considered as the intensity of a Poisson process. Our first interest is the estimation of this function h assumed to be well localized. The estimator based on random thresholds achieves an oracle inequality. Then, minimax properties of on Besov balls are established. Some simulations are provided, proving the good practical behaviour of our procedure. Finally, our method is applied to the analysis of the dependence between promoter sites and genes along the genome of the Escherichia coli bacterium.     

Correspondence Analysis in Statistical Package Programs

BMDP, Version 7.0: Available from BMDP, Statistical Software, Inc., 12121 Wilshire Blvd., Suite 300, Los Angeles, CA 90025; phone: 310-207-8800; fax: 310-207-8844. $695.

NCSS, Version 5.3: Available from NCSS, Attn: Dr. Jerry Hintze, 329 N. 1000 East, Kaysville, UT 84037; phone: 801-546-0445; fax: 801-546-3907. $295.

SAS, Version 6.07: Available from SAS Institute, Inc., SASA Campus Drive, Cary, NC 27513; phone: 919-677-8200; fax: 919-677-8123. $2525 (first year); $1175 (renewal).

SPSS, Version 6.0: Available from SPSS Inc., 444 N. Michigan Ave., Chicago, IL 60611; phone: 800-543-5831; fax: 800-841-0064. $1190.     

Empirical Likelihood for Compound Poisson Processes

Let {N(t), t > 0} be a Poisson process with rate λ > 0, independent of the independent and identically distributed random variables with mean μ and variance . The stochastic process is then called a compound Poisson process and has a wide range of applications in, for example, physics, mining, finance and risk management. Among these applications, the average number of objects, which is defined to be λμ, is an important quantity. Although many papers have been devoted to the estimation of λμ in the literature, in this paper, we use the well‐known empirical likelihood method to construct confidence intervals. The simulation results show that the empirical likelihood method often outperforms the normal approximation and Edgeworth expansion approaches in terms of coverage probabilities. A real data set concerning coal‐mining disasters is analyzed using these methods.     

On the Evolution and Asymptotic Analysis of Open Markov Populations: Application to Consumption Credit

In this paper, we study, by means of randomized sampling, the long-run stability of some open Markov population fed with time-dependent Poisson inputs. We show that state probabilities within transient states converge—even when the overall expected population dimension increases without bound—under general conditions on the transition matrix and input intensities.

Following the convergence results, we obtain ML estimators for a particular sequence of input intensities, where the sequence of new arrivals is modeled by a sigmoidal function. These estimators allow for the forecast, by confidence intervals, of the evolution of the relative population structure in the transient states.

Applying these results to the study of a consumption credit portfolio, we estimate the implicit default rate.     

Approximate confidence bounds for ratios of variance components in two-way unbalanced crossed models with interactions

Consider the general unbalanced two-factor crossed components-of-variance model with interaction given by Y_ijk: = μ+A_i: +B_j: + C_ij: +E_ijk: (i = 1,2, … a; j = 1,…,b; k = 1,…,.n_ij:=0) A_i:,B_j:, C_ij: and E_ijk: are independent unobservable random variables. Also A_i:sim; N(0,σ² _A),B_j: ～ N(0,σ² _B), C_ij:～N(0,s² _C:) and E_ijk:～N(0,s² _E:). In this paper approximate confidence bounds are obtained for ρ_A: = ρ² _A/² and ρ_B: = ρ² _B:/ρ² (where σ² = σ² _A:+ σ² _B+σ² _Cσ² _E) for special cases of the above model. The balanced incomplete block model is studied as a special case.     

Book Reviews

Books reviewed:
David Griffiths, W. Douglas Stirling, and K. Laurence Weldon, Understanding Data: Principles and Practice of Statistics
Ingwer Borg and Patrick Groenen, Modern Multidimensional Scaling: Theory and Applications
Jeffrey H. Dorfman, Bayesian Economics Through Numerical Methods: A Guide to Econometrics and Decision-making with Prior Information
Marek Musiela and Marek Rutkowski, Martingale Methods in Financial Modelling: Theory and Applications
Aad W. van der Vaart and Jon A. Wellner, Weak Convergence and Empirical Processes     

Statistical Analysis Of Cometary Data

In this study an attempt is made to assess statistically the validity of two theories as to the origin of comets. This subject still leads to great controversy amongst astronomers but recently two main schools of thought have developed.

These are that comets are of

(i) planetary origin,

(ii) interstellar origin.

Many theories have been expanded within each school of thought but at the present time one theory in each is generally accepted. This paper sets out to identify the statistical implications of each theory and evaluate each theory in terms of their implications.     

Is there an alternative to the european farm typology?

The 1978 European Community Typology for Agricultural Holdings is described in this paper and contrasted with a data based, polythetic-multivariate classification based on cluster analysis.

The requirement to reduce the size of the variable set employed in an optimisation-partition method of clustering suggested the value of principal components and factor analysis for the identification of major ‘source’ dimensions against which to measure farm differences and similarities.

The Euclidean cluster analysis incorporating the reduced dimensions quickly converged to a stable solution and was little influenced by the initial number or nature of ‘seeding’ partitions of the data.

The assignment of non-sampled observations from the population to cluster classes was completed using classification functions.

The final scheme, based on a sample of over 2,000 observations, was found to be both capable of interpretation and meaningful in terms of agricultural structure and practice and much superior in its explanatory power when compared with a version of the principal activity typology.     

Simulation of weibull and gamma autoregressive stationary process

This paper presents two simple non-Gaussian first-order autoregressive markovian processes which are easy to simulate via a computer. The autoregressive Gamma process {X_n:} is constructed according to the stochastic difference equation X_n:=V_n:X_n?1+?_n:, where {?_n:} is an i.i.d. Exponential sequence and {V_n:} is i.i.d. with Power-function distribution defined on the interval [0,1). The autoregressive Weibull process {X_n:} is constructed from the probabilistic model X_n:= k.min (X_n?1:, Y_n:) where {Y_n:} is an i.i.d. Weibull sequence and k > 1.     

A threshold based approach to merge data in financial risk management

According to the last proposals by the Basel Committee, banks are allowed to use statistical approaches for the computation of their capital charge covering financial risks such as credit risk, market risk and operational risk.

It is widely recognized that internal loss data alone do not suffice to provide accurate capital charge in financial risk management, especially for high-severity and low-frequency events. Financial institutions typically use external loss data to augment the available evidence and, therefore, provide more accurate risk estimates. Rigorous statistical treatments are required to make internal and external data comparable and to ensure that merging the two databases leads to unbiased estimates.

The goal of this paper is to propose a correct statistical treatment to make the external and internal data comparable and, therefore, mergeable. Such methodology augments internal losses with relevant, rather than redundant, external loss data.     

Reviews of Books and Teaching Materials

Workshop Statistics: Discovery with Data and Minitab. Allan J. ROSSMAN and Beth L. CHANCE. New York: Springer-Verlag, 1998, xxviii + 486 pp. $39.95 (P).

Data Analysis with Microsoft Excel. Kenneth N. BERK and Patrick CAREY. Pacific Grove, CA: Duxbury, 1998, xxi + 503 pp. $29.95 (P), ISBN: 0-534-52929-1.

Statistical Laboratory Exercises Using Excel: A Guide to Understanding Data. Tania PRVAN and Peter PETOCZ. Brisbane: Jacaranda Wiley, 1998, vi + 57 pp., $15.95 (P + 1 disc), ISBN: 0-471-34050-2. Reviewed by Richard Cleary

Data, Statistics, and Decision Models with Excel. Donald L. HARNETT and James E. HORRELL. New York: Wiley, 1998, xviii + 605, $93.95, ISBN: 0-471-13398-1. Reviewed by Sue B. Schou

Maple V© Student Version: Release 5. Waterloo Maple, Inc., New York: Springer-Verlag, 1998, $99.

Statistics with Stata 5. Lawrence C. HAMILTON. Pacific Grove, CA: Duxbury, 1998, x + 325 pp. $34.95, ISBN: 0-534-26559-6. Reviewed by Richard Goldstein     

Book Reviews

Statistics for Engineering Problem Solving.: Stephen B. Vardeman. Boston, MA: PWS Publishing, 1994. 840 pp. $59.95. Reviewed by Soren Bisgaard

A Data-Based Approach to Statistics.: Ronald L. Iman. Belmont, CA: Wadsworth Publishing, 1994. xiv + 898 pp. $51.95. Reviewed by Katherine Halvorsen

Concepts in Probability and Stochastic Modeling.: James J. Higgins and Sallie Keller-McNulty. Belmont, CA: Duxbury Press, 1995. xi + 420 pp. $67.43. Reviewed by David M. Nickerson

Statistics (6th ed.).: James T. McClave and Frank H. Dietrich II. New York: Dellen/Macmillan College Publishing, 1994. xix + 967 pp. $66.00. Reviewed by Katherine Halvorsen

Data Analysis for Monitoring Human Rights.: Herbert Spirer and Louise Spirer. Washington, DC: American Association for the Advancement of Science, 1994. xiv + 126 pp. $15.00. Reviewed by Diane G. Saphire     

Combining evidence from clinical trials in conditional or accelerated approval

Conditional (European Medicines Agency) or accelerated (U.S. Food and Drug Administration) approval of drugs allows earlier access to promising new treatments that address unmet medical needs. Certain post-marketing requirements must typically be met in order to obtain full approval, such as conducting a new post-market clinical trial. We study the applicability of the recently developed harmonic mean ${\chi }^2$-test to this conditional or accelerated approval framework. The proposed approach can be used both to support the design of the post-market trial and the analysis of the combined evidence provided by both trials. Other methods considered are the two-trials rule, Fisher's criterion and Stouffer's method. In contrast to some of the traditional methods, the harmonic mean ${\chi }^2$-test always requires a post-market clinical trial. If the $p$-value from the pre-market clinical trial is $\ll 0.025$, a smaller sample size for the post-market clinical trial is needed than with the two-trials rule. For illustration, we apply the harmonic mean ${\chi }^2$-test to a drug which received conditional (and later full) market licensing by the EMA. A simulation study is conducted to study the operating characteristics of the harmonic mean ${\chi }^2$-test and two-trials rule in more detail. We finally investigate the applicability of these two methods to compute the power at interim of an ongoing post-market trial. These results are expected to aid in the design and assessment of the required post-market studies in terms of the level of evidence required for full approval.     

Meta-Analysis Software: A Comparative Review

DSTAT, Version 1.10: Available from Lawrence Erlbaum Associates, Inc., 10 Industrial Ave., Mahwah, NJ 07430-2262; phone: 800-926-6579

TRUE EPISTAT, Version 4.0: Available from Epistat Services, 2011 Cap Rock Circle, Richardson, TX 75080; phone: 214-680-1376; fax: 214-680-1303.

FAST*PRO, Version 1.0: Available from Academic Press, Inc., 955 Massachusetts Avenue, Cambridge, MA 02139; phone: 800-321-5068; fax: 800-336-7377.

Meta-analysts conduct studies in which the responses are analytic summary measurements, such as risk differences, effect sizes, p values, or z statistics, obtained from a series of independent studies. The motivation for conducting a meta-analysis is to integrate research findings over studies in order to summarize the evidence about treatment efficacy or risk factors. This article presents a comparative review of three meta-analytic software packages: DSTAT, TRUE EPISTAT, and FAST*PRO.