Statistical Science

Larry Brown’s Contributions to Parametric Inference, Decision Theory and Foundations: A Survey

James O. Berger and Anirban DasGupta

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This article gives a panoramic survey of the general area of parametric statistical inference, decision theory and foundations of statistics for the period 1965–2010 through the lens of Larry Brown’s contributions to varied aspects of this massive area. The article goes over sufficiency, shrinkage estimation, admissibility, minimaxity, complete class theorems, estimated confidence, conditional confidence procedures, Edgeworth and higher order asymptotic expansions, variational Bayes, Stein’s SURE, differential inequalities, geometrization of convergence rates, asymptotic equivalence, aspects of empirical process theory, inference after model selection, unified frequentist and Bayesian testing, and Wald’s sequential theory. A reasonably comprehensive bibliography is provided.

Article information

Statist. Sci., Volume 34, Number 4 (2019), 621-634.

First available in Project Euclid: 8 January 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Admissibility ancillary asymptotic equivalence Bayes conditional confidence differential inequality Edgeworth expansions estimated confidence minimax sequential shrinkage sufficiency


Berger, James O.; DasGupta, Anirban. Larry Brown’s Contributions to Parametric Inference, Decision Theory and Foundations: A Survey. Statist. Sci. 34 (2019), no. 4, 621--634. doi:10.1214/19-STS717.

Export citation


  • Agresti, A. and Coull, B. A. (1998). Approximate is better than “exact” for interval estimation of binomial proportions. Amer. Statist. 52 119–126.
  • Bahadur, R. R. (1954). Sufficiency and statistical decision functions. Ann. Math. Stat. 25 423–462.
  • Barankin, E. W. and Maitra, A. P. (1963). Generalization of the Fisher–Darmois–Koopman–Pitman theorem on sufficient statistics. Sankhya, Ser. A 25 217–244.
  • Barron, A. R. (1986). Entropy and the central limit theorem. Ann. Probab. 14 336–342.
  • Berger, J. O. (1976). Admissible minimax estimation of a multivariate normal mean with arbitrary quadratic loss. Ann. Statist. 4 223–226.
  • Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer Series in Statistics. Springer, New York.
  • Berger, J. O., Brown, L. D. and Wolpert, R. L. (1994). A unified conditional frequentist and Bayesian test for fixed and sequential simple hypothesis testing. Ann. Statist. 22 1787–1807.
  • Berger, J. O. and Strawderman, W. E. (1996). Choice of hierarchical priors: Admissibility in estimation of normal means. Ann. Statist. 24 931–951.
  • Berger, J., Bock, M. E., Brown, L. D., Casella, G. and Gleser, L. (1977). Minimax estimation of a normal mean vector for arbitrary quadratic loss and unknown covariance matrix. Ann. Statist. 5 763–771.
  • Berk, R., Brown, L. D. and Zhao, L. (2009). Statistical inference after model selection. J. Quant. Criminol. 26 217–236.
  • Berk, R., Brown, L., Buja, A., Zhang, K. and Zhao, L. (2013). Valid post-selection inference. Ann. Statist. 41 802–837.
  • Bickel, P. J. (1981). Minimax estimation of the mean of a normal distribution when the parameter space is restricted. Ann. Statist. 9 1301–1309.
  • Bickel, P. J. and Doksum, K. A. (2016). Mathematical Statistics—Basic Ideas and Selected Topics. Vol. 2, 2nd ed. Texts in Statistical Science Series. CRC Press, Boca Raton, FL.
  • Blackwell, D. (1951). On the translation parameter problem for discrete variables. Ann. Math. Stat. 22 393–399.
  • Blyth, C. R. and Still, H. A. (1983). Binomial confidence intervals. J. Amer. Statist. Assoc. 78 108–116.
  • Borovkov, A. A. and Sakhanenko, A. I. (1980). Estimates for averaged quadratic risk. Probab. Math. Statist. 1 185–195.
  • Brown, L. (1964). Sufficient statistics in the case of independent random variables. Ann. Inst. Statist. Math. 35 1456–1474.
  • Brown, L. D. (1966). On the admissibility of invariant estimators of one or more location parameters. Ann. Inst. Statist. Math. 37 1087–1136.
  • Brown, L. (1967). The conditional level of Student’s $t$ test. Ann. Inst. Statist. Math. 38 1068–1071.
  • Brown, L. D. (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Ann. Inst. Statist. Math. 42 855–903.
  • Brown, L. D. (1978). A contribution to Kiefer’s theory of conditional confidence procedures. Ann. Statist. 6 59–71.
  • Brown, L. D. (1979). A heuristic method for determining admissibility of estimators—with applications. Ann. Statist. 7 960–994.
  • Brown, L. D. (1981). A complete class theorem for statistical problems with finite sample spaces. Ann. Statist. 9 1289–1300.
  • Brown, L. D. (1982). A proof of the central limit theorem motivated by the Cramér–Rao inequality. In Statistics and Probability: Essays in Honor of C. R. Rao (G. Kallianpur, P. R. Krishnaiah and J. K. Ghosh, eds.) 141–148. North-Holland, Amsterdam.
  • Brown, L. D. (1983). Comments on “The Robust Bayesian Viewpoint” by Berger, J. O. In Robustness of Bayesian Analyses (J. B. Kadane, ed.) 126–133. Elsevier, Amsterdam.
  • Brown, L. D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Institute of Mathematical Statistics Lecture Notes—Monograph Series (S. S. Gupta, ed.) 9. IMS, Hayward, CA.
  • Brown, L. D. (1988). The differential inequality of a statistical estimation problem. In Statistical Decision Theory and Related Topics, IV, Vol. 1 (West Lafayette, Ind., 1986) (S. S. Gupta and J. O. Berger, eds.) 299–324. Springer, New York.
  • Brown, L. D. (1990). The 1985 Wald Memorial Lectures. An ancillarity paradox which appears in multiple linear regression. Ann. Statist. 18 471–538.
  • Brown, L. D. (1994). Minimaxity, more or less. In Statistical Decision Theory and Related Topics, V (West Lafayette, IN, 1992) (S. S. Gupta and J. O. Berger, eds.) 1–18. Springer, New York.
  • Brown, L. D. (1998). Minimax Theory, in Encyclopedia of Biostatistics (P. Armitage and T. Colton, eds.). Wiley, New York.
  • Brown, L. D. (2000). An essay on statistical decision theory. J. Amer. Statist. Assoc. 95 1277–1281.
  • Brown, L. D., ed. (2002). An analogy between statistical equivalence and stochastic strong limit theorems. Invited address at the International Congress of Mathematicians, Beijing, August, 2002.
  • Brown, L. D., Cai, T. T. and DasGupta, A. (2001). Interval estimation for a binomial proportion. Statist. Sci. 16 101–133.
  • Brown, L. D., Cai, T. T. and DasGupta, A. (2002). Confidence intervals for a binomial proportion and asymptotic expansions. Ann. Statist. 30 160–201.
  • Brown, L. D., Cai, T. T. and DasGupta, A. (2003). Interval estimation in exponential families. Statist. Sinica 13 19–49.
  • Brown, L. D. and Cohen, A. (1981). Inadmissibility of large classes of sequential tests. Ann. Statist. 9 1239–1247.
  • Brown, L. D., Cohen, A. and Samuel-Cahn, E. (1983). A sharp necessary condition for admissibility of sequential tests—necessary and sufficient conditions for admissibility of SPRTs. Ann. Statist. 11 640–653.
  • Brown, L. D., Cohen, A. and Strawderman, W. E. (1979). Monotonicity of Bayes sequential tests. Ann. Statist. 7 1222–1230.
  • Brown, L. D. and Farrell, R. H. (1990). A lower bound for the risk in estimating the value of a probability density. J. Amer. Statist. Assoc. 85 1147–1153.
  • Brown, L. D. and Gajek, L. (1990). Information inequalities for the Bayes risk. Ann. Statist. 18 1578–1594.
  • Brown, L. D. and Hwang, J. T. (1982). A unified admissibility proof. In Statistical Decision Theory and Related Topics, III, Vol. 1 (West Lafayette, Ind., 1981) 205–230. Academic Press, New York.
  • Brown, L. D. and Liu, R. C. (1993). Bounds on the Bayes and minimax risk for signal parameter estimation. IEEE Trans. Inform. Theory 39 1386–1394.
  • Brown, L. D. and Low, M. G. (1991). Information inequality bounds on the minimax risk (with an application to nonparametric regression). Ann. Statist. 19 329–337.
  • Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398.
  • Brown, L. D. and Purves, R. (1973). Measurable selections of extrema. Ann. Statist. 1 902–912.
  • Brown, L. D. and Zhao, L. (2003). Direct asymptotic equivalence of nonparametric regression and the infinite-dimensional location problem. Preprint Univ. Pennsylvania, Philadelphia, PA.
  • Brown, L. D., Carter, A. V., Low, M. G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074–2097.
  • Brown, L., DasGupta, A., Haff, L. R. and Strawderman, W. E. (2006). The heat equation and Stein’s identity: Connections, applications. J. Statist. Plann. Inference 136 2254–2278.
  • Clevenson, M. L. and Zidek, J. V. (1975). Simultaneous estimation of the means of independent Poisson laws. J. Amer. Statist. Assoc. 70 698–705.
  • Cohen, A. (1965). Estimates of linear combinations of the parameters in the mean vector of a multivariate distribution. Ann. Inst. Statist. Math. 36 78–87.
  • DasGupta, A. (1983). Bayes and minimax estimation in one and multiparameter families. Ph.D. thesis, Indian Statistical Institute.
  • DasGupta, A. (2005). A conversation with Larry Brown. Statist. Sci. 20 193–203.
  • DasGupta, A. (2008). Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. Springer, New York.
  • DasGupta, A. and Sinha, B. K. (1986). Estimation in the multiparameter exponential family: Admissibility and inadmissibility results. Statist. Decisions 4 101–130.
  • Diaconis, P. and Stein, C. (1982). Lecture notes on statistical decision theory. Dept. Statistics, Stanford Univ., Stanford, CA.
  • Donoho, D. L., Liu, R. C. and MacGibbon, B. (1990). Minimax risk over hyperrectangles, and implications. Ann. Statist. 18 1416–1437.
  • Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508–539.
  • Dynkin, E. B. (1961). Necessary and sufficient statistics for a family of probability distributions. In Select. Transl. Math. Statist. and Probability, Vol. 1 17–40. IMS and Amer. Math. Soc., Providence, RI.
  • Eaton, M. L. (1992). A statistical diptych: Admissible inferences—recurrence of symmetric Markov chains. Ann. Statist. 20 1147–1179.
  • Fisher, R. A. (1923). On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond. Ser. A 222 309–368.
  • Ghosh, B. K. (1979). A comparison of some approximate confidence intervals for the binomial parameter. J. Amer. Statist. Assoc. 74 894–900.
  • Halmos, P. R. and Savage, L. J. (1949). Application of the Radon–Nikodym theorem to the theory of sufficient statistics. Ann. Math. Stat. 20 225–241.
  • Hsuan, F. C. (1979). A stepwise Bayesian procedure. Ann. Statist. 7 860–868.
  • Hwang, J. T. and Brown, L. D. (1991). Estimated confidence under the validity constraint. Ann. Statist. 19 1964–1977.
  • James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 361–379. Univ. California Press, Berkeley, CA.
  • Johnson, B. McK. (1971). On the admissible estimators for certain fixed sample binomial problems. Ann. Inst. Statist. Math. 42 1579–1587.
  • Johnson, O. (2004). Information Theory and the Central Limit Theorem. Imperial College Press, London.
  • Johnstone, I. (1984). Admissibility, difference equations and recurrence in estimating a Poisson mean. Ann. Statist. 12 1173–1198.
  • Johnstone, I. and Lalley, S. (1984). On independent statistical decision problems and products of diffusions. Z. Wahrsch. Verw. Gebiete 68 29–47.
  • Kagan, A. M., Linnik, Y. V. and Rao, C. R. (1973). Characterization Problems in Mathematical Statistics. Wiley, New York.
  • Karlin, S. (1958). Admissibility for estimation with quadratic loss. Ann. Inst. Statist. Math. 29 406–436.
  • Kiefer, J. (1977). Conditional confidence statements and confidence estimators. J. Amer. Statist. Assoc. 72 789–827.
  • Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent $\mathrm{RV}$’s and the sample $\mathrm{DF}$. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent RV’s, and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 33–58.
  • Korostelev, A. and Korosteleva, O. (2011). Mathematical Statistics. Graduate Studies in Mathematics, Asymptotic Minimax Theory 119. Amer. Math. Soc., Providence, RI.
  • Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Series in Statistics. Springer, New York.
  • Leeb, H. and Pötscher, B. M. (2008). Can one estimate the unconditional distribution of post-model-selection estimators? Econometric Theory 24 338–376.
  • Lehmann, E. L. (1959). Testing Statistical Hypotheses. Wiley, New York.
  • Lehmann, E. L. and Romano, J. P. (2008). Testing Statistical Hypotheses. Springer, New York.
  • Lehmann, E. L. and Stein, C. M. (1953). The admissibility of certain invariant statistical tests involving a translation parameter. Ann. Math. Stat. 24 473–479.
  • Levit, B. Y. (1987). On bounds for the minimax risk. In Probability Theory and Mathematical Statistics, Vol. II (Vilnius, 1985) 203–216. VNU Sci. Press, Utrecht.
  • Liu, R. C. and Brown, L. D. (1993). Nonexistence of informative unbiased estimators in singular problems. Ann. Statist. 21 1–13.
  • Meeden, G., Ghosh, M., Srinivasan, C. and Vardeman, S. (1989). The admissibility of the Kaplan–Meier and other maximum likelihood estimators in the presence of censoring. Ann. Statist. 17 1509–1531.
  • Morris, C. N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10 65–80.
  • Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430.
  • Olkin, I. and Sobel, M. (1979). Admissible and minimax estimation for the multinomial distribution and for $k$ independent binomial distributions. Ann. Statist. 7 284–290.
  • Pötscher, B. M. (1991). Effects of model selection on inference. Econometric Theory 7 163–185.
  • Rukhin, A. L. (1995). Admissibility: Survey of a concept in progress. Int. Stat. Rev. 63 95–115.
  • Sinha, B. K. and Gupta, A. D. (1984). Admissibility of generalized Bayes and Pitman estimates in the nonregular family. Comm. Statist. Theory Methods 13 1709–1721.
  • Skibinsky, M. and Rukhin, A. L. (1989). Admissible estimators of binomial probability and the inverse Bayes rule map. Ann. Inst. Statist. Math. 41 699–716.
  • Sobel, M. (1953). An essentially complete class of decision functions for certain standard sequential problems. Ann. Math. Stat. 24 319–337.
  • Stein, C. (1959). The admissibility of Pitman’s estimator of a single location parameter. Ann. Inst. Statist. Math. 30 970–979.
  • Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135–1151.
  • Tsybakov, A. B. (2004). Introduction à L’estimation Non-paramétrique. Mathématiques & Applications (Berlin) [Mathematics & Applications] 41. Springer, Berlin.
  • Wald, A. (1945). Sequential tests of statistical hypotheses. Ann. Math. Stat. 16 117–186.