Limiting properties of some measures of information

In this paper we investigate the limiting behaviour of the measures of information due to Csiszár, Rényi and Fisher. Conditions for convergence of measures of information and for convergence of Radon-Nikodym derivatives are obtained. Our results extend the results of Kullback (1959,Information Theory and Statistics, Wiley, New York) and Kirmani (1971,Ann. Inst. Statist. Math.,23, 157–162).     

A note on cuts for contingency tables

In this note, we propose a general method to find cuts for a contingency table. Useful cuts are, in many cases, statistics S-sufficient for the nuisance parameter and S-ancillary for the parameter of interest. In general, cuts facilitate a strong form of parameter separation known to be useful for conditional inference [E.L. Lehmann, Testing Statistical Hypotheses, 2nd ed., Springer, New York, 1997, pp. 546–548]. Cuts also achieve significant dimension reduction, hence, increase computational efficiency. This is particularly true for the inference about cross-tabulated data, usually with a large number of parameters. Depending on the parameter of interest, we propose a flexible transformation to reparameterize the discrete multivariate response distribution. Inference on cell probabilities or odds ratios will require different parameterizations. The reparameterized distribution is not sum-symmetric. Thus, the finding in this paper expands the results in Barndorff-Nielsen [O.E. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory, John Wiley, New York, 1978, pp. 202–206].     

Music by Design: A Sesquicentennial Statistics Songbook

Abstract

Fox, J. (1997), Applied Regression Analysis, Linear Models, and Related Methods, Thousand Oaks, CA: Sage Publications.

French, S., and Smith, J. Q. (eds.) (1997), The Practice of Bayesian Analysis, New York: Wiley.

Hettmansperger, T. P., and McKean, J. W. (1998), Robust Nonparametric Statistical Methods, London: Arnold.

Jacoby, W. G. (1998), Statistical Graphics For Visualizing Multivariate Data, Thousand Oaks, CA: Sage Publications.

Johnson, N. L., and Balakrishnan, N. (1997), Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz, New York: Wiley.

Korner, T. W. (1995), The Pleasures of Counting, Cambridge, U.K.: Cambridge University Press.

McLachlan, G. J., and Krishman, T. (1996), The EM Algorithm and Extensions, New York: Wiley.

Ruelle, D. (1991), Chance and Chaos, Princeton, NJ: Princeton University Press.

Ryan, T. P. (1997), Modern Regression Methods, New York: Wiley.

Bernstein, D. (1997), Statistical Applications of Baseball, Alexandria, VA: SPSTATE.     

Nonparametric tests for homogeneity of scale against ordered alternatives

Summary In this paper the nonparametric several sample scale problem is considered and some tests are proposed for the hypothesis of homogeneity versus ordered alternatives. These tests are based on statistics that are weighted linear combinations of Sugiura (1965,Osaka J. Math.,2, 385–426) type statistics proposed for testing homogeneity of scale against the omnibus alternative. For each class of test statistics suggested, the member with maximum Pitman efficiency is identified. The optimal statistics are compared with their parametric and nonparametric competitors.     

The Tricomi problem for second order linear equations of mixed type with parabolic degeneracy

In Bers, 1958, Mathematical Aspects of Subsonic and Transonic Gas Dynamics (New York: Wiley); Bitsadze, 1988, Some Classes of Partial Differential Equations (New York: Gordon and Breach); Rassias, 1990, Lecture Notes on Mixed Type Partial Differential Equations (Singapore: World Scientific); Salakhitdinov and Islomov, 1987, The Tricomi problem for the general linear equation of mixed type with a nonsmooth line of degeneracy. Soviet Math. Dokl., 34, 133–136; Smirnov, 1978, Equations of Mixed type (Providence, RI: American Mathematical Society), the authors posed and discussed the Tricomi problem of second order equations of mixed type with parabolic degeneracy, which possesses important application to gas dynamics. The present article deals with the Tricomi problem for general second order equations of mixed type with parabolic degeneracy. Firstly the formulation of the problem for the equations is given, next the representations and estimates of solutions for the above problem are obtained, finally the existence of solutions for the problem is proved by the successive iteration and the method of parameter extension. In this article, we use the complex method, namely the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used (see Wen, 2002, Linear and Quasilinear Equations of Hyperbolic and Mixed Types (London: Taylor and Francis)).     

Average worth and simultaneous estimation of the selected subset

Suppose a subset of populations is selected from the given k gamma G( _i,p ) (i = 1,2,...,k)populations, using Gupta's rule (1963, Ann. Inst. Statist. Math., 14, 199–216). The problem of estimating the average worth of the selected subset is first considered. The natural estimator is shown to be positively biased and the UMVUE is obtained using Robbins' UV method of estimation (1988, Statistical Decision Theory and Related Topics IV, Vol. 1 (eds. S. S. Gupta and J. O. Berger), 265–270, Springer, New York). A class of estimators that dominate the natural estimator for an arbitrary k is derived. Similar results are observed for the simultaneous estimation of the selected subset.     

Statistical inference based on aligned ranks for two-way MANOVA with interaction

Multiresponse experiments in two-way layouts with interactions, having equal number of observations per cell, are considered. Robust procedures based on aligned ranks for statistical inference of interactions, main effects and an overall mean response in the models are proposed. Large sample properties of the proposed tests, estimators and confidence regions as the cell size tends to infinity are investigated. For the univariate case, it is found that the asymptotic relative efficiencies (ARE's) of the proposed procedures relative to classical procedures agree with the ARE-results of the two-sample rank test relative to the t-test. In addition, robustness due to Huber (1981, Robust Statistics, Wiley, New York) can be drawn.     

Bayesian linear prediction in finite populations

In this paper, Bayesian linear prediction of the total of a finite population is considered in situations where the observation error variance is parameter dependent. Connections with least squares prediction (Royall (1976, J. Amer. Statist. Assoc., 71, 657–664)) in mixed linear models (Theil (1971, Principles of Econometrics, Wiley, New York)), are established. Extensions to the case of dynamic (state dependent) superpopulation models are also proposed.     

Score and Wald tests for the multivariate growth curve model with missing data

We present the score and Wald test analogues to Srivastava's (1985, Comm. Statist. A—Theory Methods, 14, 775–792) likelihood ratio tests for the multivariate growth curve model with missing data, and illustrate their use with data from an immunotherapy experiment (Fukushima et al. (1982, Int. J. Cancer, 29, 107–112, 113–117)).     

Empirical Bayes detection of a change in distribution

The problem of detection of a change in distribution is considered. Shiryayev (1963, Theory Probab. Appl., 8, pp. 22–46, 247–264 and 402–413; 1978, Optimal Stopping Rules, Springer, New York) solved the problem in a Bayesian framework assuming that the prior on the change point is Geometric (p). Shiryayev showed that the Bayes solution prescribes stopping as soon as the posterior probability of the change having occurred exceeds a fixed level. In this paper, a myopic policy is studied. An empirical Bayes stopping time is investigated for detecting a change in distribution when the prior is not completely known.Research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant GP 7987.     

Local Bahadur efficiency of rank tests for the independence problem

In previous papers [Approximate and local Bahadur efficiency of linear rank tests in the two-sample problem, Ann. Statist.7, 1246–1255, 1979; Local comparison of linear rank tests in the Bahadur sense, Metrika, 1979] the author developed for linear rank tests of the one-sample symmetry and the k-sample problem (k ≥ 2) a theory of local comparison, based on the concept of Bahadur efficiency. In the present article this theory is carried over to rank tests of the independence problem.     

The influence of sample extremes on the maximum of partial sums of independent random variables

Many papers exist dealing with the distribution of the maximum of partial sums of independent random variables and studying the relationship between the sums and the sample extremes (see [L. Takacs,Combinator Methods in the Theory of Stochastic Processes, Wiley, New York (1967)], [J. Calambos,The Asymptotic Theory of Extreme Order Statistics, Wiley, New York (1978)]). The present paper deals with the effect the sample extremes have on the distribution of the maximum of sums of independent random variables. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993     

The limiting normality of the test statistic for the two-sample problem induced by a convex sum distance

Critchlow (1992, J. Statist. Plann. Inference, 32, 325–346) proposed a method of a unified construction of a class of rank tests. In this paper, we introduce a convex sum distance and prove the limiting normality of the test statistics for the two-sample problem derived by his method.     

Weighted Kaplan-Meier Tests for Umbrella Alternatives

The problem of testing for umbrella alternatives in a one-way layout with right-censored survival data is considered. Testing procedures based on the two-sample weighted Kaplan-Meier statistics suggested by Pepe and Fleming (1989, Biometrics, 45, 497–507; 1991, J. Roy. Statist. Soc. Ser. B, 53, 341–352) are suggested for both cases when the peak of the umbrella is known or unknown. The asymptotic relative efficiency of the weighted Kaplan-Meier test and the weighted logrank test proposed by Chen and Wolfe (2000, Statist. Sinica, 10, 595–612) is computed for the umbrella peak-known setting where the piecewise exponential survival distributions have the proportional or crossing hazards, or the related hazards differ at early or late times. Moreover, the results of a Monte Carlo study are presented to investigate the level and power performances of the umbrella tests. Finally, application of the proposed procedures to an appropriated data set is illustrated.     

Unbiased estimators in the sense of Lehmann and their discrimination rates (II): Multi-parameter cases

Summary The notion ofdiscrimination rate of any unbiased estimator in the sense of Lehmann is, as defined by the author (1982,Ann. Inst. Statist. Math.,34, A, 19–37), extended to multi-parameter cases. This research was supported in part by a Grant-in-Aid for Scientific Research of the Japanese Ministry of Education, Science and Culture. The Institute of Statistical Mathematics     

Symmetrized approximate score rank tests for the two-sample case

Rank test statistics for the two-sample problem are based on the sum of the rank scores from either sample. However, a critical difference can occur when approximate scores are used since the sum of the rank scores from sample 1 is not equal to minus the sum of the rank scores from sample 2. By centering and scaling as described in Hajek and Sidak (1967, Theory of Rank Tests, Academic Press, New York) for the uncensored data case the statistics computed from each sample become identical. However such symmetrized approximate scores rank statistics have not been proposed in the censored data case. We propose a statistic that treats the two approximate scores rank statistics in a symmetric manner. Under equal censoring distributions the symmetric rank tests are efficient when the score function corresponds to the underlying model distribution. For unequal censoring distributions we derive a useable expression for the asymptotic variance of our symmetric rank statistics.     

Octahedral designs

An oriented octahedral design of order v, or OCT(v), is a decomposition of all oriented triples on v points into oriented octahedra. Hanani [H. Hanani, Decomposition of hypergraphs into octahedra, Second International Conference on Combinatorial Mathematics (New York, 1978), Annals of the New York Academy of Sciences, 319, New York Academy of Science, New York, 1979, pp. 260–264.] settled the existence of these designs in the unoriented case. We show that an OCT(v) exists if and only if v≡1, 2, 6 (mod 8) (the admissible numbers), and moreover the constructed OCT(v) are unsplit, i.e. their octahedra cannot be paired into mirror images. We show that an OCT(v) with a subdesign OCT(U) exists if and only if v and u are admissible and v≥u+4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:319–327, 2010     

On a Multiparameter Version of Tukey's Linear Sensitivity Measure and its Properties

A multiparameter version of Tukey's (1965, Proc. Nat. Acad. Sci. U.S.A., 53, 127–134) linear sensitivity measure, as a measure of informativeness in the joint distribution of a given set of random variables, is proposed. The proposed sensitivity measure, under some conditions, is a matrix which is non-negative definite, weakly additive, monotone and convex. Its relation to Fisher information matrix and the best linear unbiased estimator (BLUE) are investigated. The results are applied to the location-scale model and it is observed that the dispersion matrix of the BLUE of the vector location-scale parameter is the inverse of the sensitivity measure. A similar property was established by Nagaraja (1994, Ann. Inst. Statist. Math., 46, 757–768) for the single parameter case when applied to the location and scale models. Two illustrative examples are included.     

Rank estimates in a class of semiparametric two-sample models

We consider a two-sample semiparametric model involving a real parameter and a nuisance parameter F which is a distribution function. This model includes the proportional hazard, proportional odds, linear transformation and Harrington-Fleming models (1982, Biometrika, 69, 533–546). We propose two types of estimates based on ranks. The first is a rank approximation to Huber's M-estimates (1981, Robust Statistics, Wiley) and the second is a Hodges-Lehmann type rank inversion estimate (1963, Ann. Math. Statist., 34, 598–611). We obtain asymptotic normality and efficiency results. The estimates are consistent and asymptotically normal generally but fully efficient only for special cases.Research partially supported by National Science Foundation Grant DMS-86-02083 and National Institute of General Medical Sciences Grant SSS-Y1RO1GM35416-01     

A comparison of higher-order local powers of a class of one-way MANOVA tests under general distributions

The purpose of this paper is to investigate the effect of nonnormality upon the nonnull distributions of some MANOVA test statistics under normality. It is shown that whatever the underlying distributions, the difference of the local powers up to order N^-1 (N is the total number of observations) after either Bartlett's type adjustment or Cornish-Fisher's type adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, second ed., 1984 and third ed., 2003, Wiley, New York] under normality. The performance of higher-order results in finite samples is examined using simulation studies.