2009

Optimal-order bounds on the rate of convergence to normality in the multivariate delta method

Pinelis, Iosif, Molzon, Raymond

Understand

Uniform and nonuniform Berry--Esseen (BE) bounds of optimal orders on the closeness to normality for general abstract nonlinear statistics are given, which are then used to obtain optimal bounds on the rate of convergence in the delta method for vector statistics.

Specific applications to Pearson's, non-central Student's and Hotelling's statistics, sphericity test statistics, a regularized canonical correlation, and maximum likelihood estimators (MLEs) are given; all these uniform and nonuniform BE bounds appear to be the first known results of these kinds, except for uniform BE bounds for MLEs.
When applied to the well-studied case of the central Student statistic, our general results compare well with known ones in that case, obtained previously by specialized methods.
The proofs use a Stein-type method developed by Chen and Shao, a Cram\'er-type of tilt transform, exponential and Rosenthal-type inequalities for sums of random vectors established by Pinelis, Sakhanenko, and Utev, as well as a number of other, quite recent results motivated by this study.

Built on

[author] Hoeffding, WassilyW. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30. MR0144363 (26 ##1908)
1908
Earlier work this paper cites.
[author] Pinelis, IosifI. (2012). Rosenthal-type inequalities for martingales in 2-smooth Banach spaces. http://arxiv.org/abs/1212.1912 , to appear in Theory of Probability and Applications
Original
1912
Earlier work this paper cites.
[author] Fisher, R. A.R. A. (1921). On the probable error of a coefficient of correlation deduced from a small sample. Metron 1 3–32
1921
Earlier work this paper cites.
[author] David, F. N.F. N. (1938). Tables of the ordinates and probability integral of the distribution of the correlation coefficient in small samples. Cambridge University Press, New York
1938
Earlier work this paper cites.
[author] Gayen, A. K.A. K. (1951). The frequency distribution of the product-moment correlation coefficient in random samples of any size drawn from non-normal universes. Biometrika 38 219–247. 0042097 (13,53a)
1951
Earlier work this paper cites.
[author] Hotelling, HaroldH. (1953). New light on the correlation coefficient and its transforms. J. Roy. Statist. Soc. Ser. B. 15 193–225; discussion, 225–232. 0060794 (15,728d)
1953
Earlier work this paper cites.
[author] Noether, Gottfried E.G. E. (1955). On a theorem of Pitman. Ann. Math. Statist. 26 64–68. MR0070122 (16,1133g)
1955
Earlier work this paper cites.
[author] Ito, KoichiK. (1956). Asymptotic formulae for the distribution of Hotelling’s generalized T 0 sp ⁡ 2 T_{0}\operatorname{sp}{2} statistic. Ann. Math. Statist. 27 1091–1105. MR0084969 (18,958b)
1956
Earlier work this paper cites.
[author] Ito, KoichiK. (1960). Asymptotic formulae for the distribution of Hotelling’s generalized T 0 sp ⁡ 2 T_{0}\operatorname{sp}{2} statistic. II. Ann. Math. Statist. 31 1148–1153. MR0116401 (22 ##7189)
1960
Earlier work this paper cites.
[author] Bennett, GeorgeG. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33–45
1962
Earlier work this paper cites.
[author] Rosenthal, Haskell P.H. P. (1970). On the subspaces of L p L^{p} ( p > 2 ) (p>2) spanned by sequences of independent random variables. Israel J. Math. 8 273–303. MR0271721 (42 ##6602)
1970
Earlier work this paper cites.
[author] John, S.S. (1971). Some optimal multivariate tests. Biometrika 58 123–127. 0275568 (43 ##1321)
1971
Earlier work this paper cites.
[author] Pfanzagl, J.J. (1971). The Berry-Esseen bound for minimum contrast estimates. Metrika 17 82–91. 0295467 (45 ##4533)
1971
Earlier work this paper cites.
[author] Siotani, MinoruM. (1971). An asymptotic expansion of the non-null distribution of Hotelling’s generalized T 0 sp ⁡ 2 T_{0}\operatorname{sp}{2} -statistic. Ann. Math. Statist. 42 560–571. MR0286202 (44 ##3416)
1971
Earlier work this paper cites.
[author] Muirhead, R. J.R. J. (1972). The asymptotic noncentral distribution of Hotelling’s generalized T sp ⁡ 2 0 T\operatorname{sp}{2}_{0} . Ann. Math. Statist. 43 1671–1677. MR0346999 (49 ##11719)
1972
Earlier work this paper cites.
[author] Pfanzagl, J.J. (1972/73). The accuracy of the normal approximation for estimates of vector parameters. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 171–198. 0329093 (48 ##7435)
1972
Earlier work this paper cites.
[author] Nagao, HisaoH. (1973). On some test criteria for covariance matrix. Ann. Statist. 1 700–709. 0339405 (49 ##4164)
1973
Earlier work this paper cites.
[author] Jain, Naresh C.N. C. and Marcus, Michael B.M. B. (1975). Integrability of infinite sums of independent vector-valued random variables. Trans. Amer. Math. Soc. 212 1–36. MR0385995 (52 ##6854)
1975
Earlier work this paper cites.
[author] Petrov, V. V.V. V. (1975). Sums of independent random variables. Springer-Verlag, New York. Translated from the Russian by A. A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. MR0388499 (52 ##9335)
1975
Earlier work this paper cites.
[author] Hoffmann-Jørgensen, J.J. and Pisier, G.G. (1976). The law of large numbers and the central limit theorem in Banach spaces. Ann. Probab. 4 587–599. MR0423451 (54 ##11429)
1976
Earlier work this paper cites.
[author] Bhattacharya, R. N.R. N. and Ghosh, J. K.J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434–451. 0471142 (57 ##10880)
1978
Earlier work this paper cites.
[author] Paditz, LudwigL. (1978). Abschätzungen der Konvergenzgeschwindigkeit zur Normalverteilung unter Voraussetzung einseitiger Momente. Math. Nachr. 82 131–156. 0494427 (58 ##13292)
1978
Earlier work this paper cites.
[author] de Acosta, AlejandroA. and Samur, Jorge D.J. D. (1979). Infinitely divisible probability measures and the converse Kolmogorov inequality in Banach spaces. Studia Math. 66 143–160. 565155 (81d:60004)
1979
Earlier work this paper cites.
[author] Kato, TosioT. (1995). Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin Reprint of the 1980 edition. 1335452 (96a:47025)
1980
Earlier work this paper cites.
[author] Pinelis, I. F.I. F. (1980). Estimates of moments of infinite-dimensional martingales. Math. Notes 27 459–462
1980
Earlier work this paper cites.
[author] Michel, R.R. (1981). On the constant in the nonuniform version of the Berry-Esseen theorem. Z. Wahrsch. Verw. Gebiete 55 109–117. MR606009 (82c:60042)
1981
Earlier work this paper cites.
[author] Dowson, D. C.D. C. and Landau, B. V.B. V. (1982). The Fréchet distance between multivariate normal distributions. J. Multivariate Anal. 12 450–455. 10.1016/0047-259X(82)90077-X 666017 (84a:62078)
1982
Earlier work this paper cites.
[author] Muirhead, Robb J.R. J. (1982). Aspects of multivariate statistical theory. John Wiley & Sons, Inc., New York Wiley Series in Probability and Mathematical Statistics. 652932 (84c:62073)
1982
Earlier work this paper cites.
[author] Olkin, I.I. and Pukelsheim, F.F. (1982). The distance between two random vectors with given dispersion matrices. Linear Algebra Appl. 48 257–263. 10.1016/0024-3795(82)90112-4 683223 (84f:62062)
1982
Earlier work this paper cites.
[author] Givens, Clark R.C. R. and Shortt, Rae MichaelR. M. (1984). A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31 231–240. 10.1307/mmj/1029003026 752258 (85m:60009)
1984
Earlier work this paper cites.
[author] Slavova, V. V.V. V. (1985). On the Berry-Esseen bound for Student’s statistic. In Stability problems for stochastic models (Uzhgorod, 1984). Lecture Notes in Math. 1155 355–390. Springer, Berlin. 825335 (87i:60029)
1984
Earlier work this paper cites.
[author] Horn, Roger A.R. A. and Johnson, Charles R.C. R. (1985). Matrix analysis. Cambridge University Press, Cambridge. MR832183 (87e:15001)
1985
Earlier work this paper cites.
[author] Götze, F.F. (1986). On the rate of convergence in the central limit theorem in Banach spaces. Ann. Probab. 14 922–942. MR841594 (87i:60013)
1986
Earlier work this paper cites.
[author] Pinelis, I. F.I. F. (1986). Probability inequalities for sums of independent random variables with values in a Banach space. Math. Notes 39 241–244
1986
Earlier work this paper cites.
[author] Pinelis, I. F.I. F. and Sakhanenko, A. I.A. I. (1986). Remarks on inequalities for large deviation probabilities. Theory Probab. Appl. 30 143–148
1986
Earlier work this paper cites.
[author] Hall, PeterP. (1987). Edgeworth expansion for Student’s t t statistic under minimal moment conditions. Ann. Probab. 15 920–931. MR893906 (88j:62039)
1987
Earlier work this paper cites.
[author] Zalesskii, B. A.B. A. (1988). On the accuracy of normal approximation in Banach spaces. Theory Probab. Appl. 33 239–247. MR954572 (89h:60014)
1988
Earlier work this paper cites.
[author] Koroljuk, V. S.V. S. and Borovskich, Yu. V.Y. V. (1994). Theory of U U -statistics. Mathematics and its Applications 273. Kluwer Academic Publishers Group, Dordrecht. Translated from the 1989 Russian original by P. V. Malyshev and D. V. Malyshev and revised by the authors. MR1472486 (98e:60033)
1989
Earlier work this paper cites.
[author] Pinelis, I. F.I. F. and Utev, S. A.S. A. (1989). Exact exponential bounds for sums of independent random variables. Theory Probab. Appl. 34 340–346. MR1005745 (91a:60053)
1989
Earlier work this paper cites.

Similar

[author] Zalesskii, B. A.B. A. (1990). The accuracy of Gaussian approximation in Banach spaces. Theory Probab. Appl. 34 747–748. MR1036728 (91e:60021)
1990
Cited alongside, same era.
[author] Götze, F.F. (1991). On the rate of convergence in the multivariate CLT. Ann. Probab. 19 724–739. MR1106283 (92g:60028)
1991
Cited alongside, same era.
[author] Bentkus, VidmantasV. and Götze, FriedrichF. (1993). On smoothness conditions and convergence rates in the CLT in Banach spaces. Probab. Theory Related Fields 96 137–151. MR1227029 (94f:60010)
1993
Cited alongside, same era.
[author] Pinelis, IosifI. (1994). Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22 1679–1706. MR1331198 (96b:60010)
1994
Cited alongside, same era.
[author] Kano, YutakaY. (1995). An asymptotic expansion of the distribution of Hotelling’s T 2 T^{2} -statistic under general distributions. Amer. J. Math. Management Sci. 15 317–341. 1397511 (97h:62044)
1995
Cited alongside, same era.
[author] Pinelis, I.I. (1995). Optimum bounds on moments of sums of independent random vectors. Siberian Adv. Math. 5 141–150. MR1387858 (97c:60043)
1995
Cited alongside, same era.
[author] Bentkus, V.V., Bloznelis, M.M. and Götze, F.F. (1996). A Berry-Esséen bound for Student’s statistic in the non-i.i.d. case. J. Theoret. Probab. 9 765–796. MR1400598 (97e:60036)
1996
Cited alongside, same era.
[author] Bentkus, V.V. and Götze, F.F. (1996). The Berry-Esseen bound for Student’s statistic. Ann. Probab. 24 491–503. MR1387647 (97f:62021)
1996
Cited alongside, same era.
[author] Fujikoshi, YasunoriY. (1997). An asymptotic expansion for the distribution of Hotelling’s T 2 T^{2} -statistic under nonnormality. J. Multivariate Anal. 61 187–193. 1452272 (98h:62091)
1997
Cited alongside, same era.
[author] Giné, EvaristE., Götze, FriedrichF. and Mason, David M.D. M. (1997). When is the Student t t -statistic asymptotically standard normal? Ann. Probab. 25 1514–1531. MR1457629 (98j:60033)
1997
Cited alongside, same era.
[author] Heyde, Christopher C.C. C. (1997). Quasi-likelihood and its application. Springer Series in Statistics. Springer-Verlag, New York A general approach to optimal parameter estimation. 10.1007/b98823 1461808 (99f:62003)
1997
Cited alongside, same era.
[author] Shao, Qi-ManQ.-M. (1997). Self-normalized large deviations. Ann. Probab. 25 285–328. MR1428510 (98b:60056)
1997
Cited alongside, same era.

Then

[author] Jing, Bing-YiB.-Y., Shao, Qi-ManQ.-M. and Zhou, WangW. (2008). Towards a universal self-normalized moderate deviation. Trans. Amer. Math. Soc. 360 4263–4285. 2395172 (2009d:60065)
2008
Later among the works it cites.
[author] Kakizawa, YoshihideY. and Iwashita, ToshiyaT. (2008). Hotelling’s one-sample and two-sample T 2 T^{2} tests and the multivariate Behrens-Fisher problem under nonnormality. J. Statist. Plann. Inference 138 3379–3404. 2450082 (2009m:62166)
2008
Later among the works it cites.
[author] Kosorok, Michael R.M. R. (2008). Introduction to empirical processes and semiparametric inference. Springer Series in Statistics. Springer, New York. 2724368 (2012b:62005)
2008
Later among the works it cites.
[author] Pang, Tian-xiaoT.-x., Zhang, Li-xinL.-x. and Wang, Jian-fengJ.-f. (2008). Precise asymptotics in the self-normalized law of the iterated logarithm. J. Math. Anal. Appl. 340 1249–1262. 2390926 (2009c:60070)
2008
Later among the works it cites.
[author] Gilliam, D. S.D. S., Hohage, T.T., Ji, X.X. and Ruymgaart, F.F. (2009). The Fréchet derivative of an analytic function of a bounded operator with some applications. Int. J. Math. Math. Sci. Art. ID 239025, 17. 2496753 (2010a:47030)
2009
Closest in time.
[author] Pinelis, IosifI. (2009). Optimal two-value zero-mean disintegration of zero-mean random variables. Electron. J. Probab. 14 no. 26, 663–727. MR2486818
2009
Closest in time.
[author] Tyurin, IlyaI. (2009). New estimates of the convergence rate in the Lyapunov theorem (preprint). http://arxiv.org/abs/0912.0726
Original
2009
Closest in time.
[author] Wang, QiyingQ. and Hall, PeterP. (2009). Relative errors in central limit theorems for Student’s t t statistic, with applications. Statist. Sinica 19 343–354. 2487894 (2010b:60076)
2009
Closest in time.
[author] Bhattacharya, RabiR. and Holmes, SusanS. (2010). An exposition of Götze’s estimation of the rate of convergence in the multivariate central limit theorem (preprint). http://arxiv.org/abs/1003.4254
Original
2010
Closest in time.
[author] Chen, Song XiS. X., Zhang, Li-XinL.-X. and Zhong, Ping-ShouP.-S. (2010). Tests for high-dimensional covariance matrices. J. Amer. Statist. Assoc. 105 810–819. 2724863 (2011h:62167)
2010
Closest in time.
[author] Fisher, Thomas J.T. J., Sun, XiaoqianX. and Gallagher, Colin M.C. M. (2010). A new test for sphericity of the covariance matrix for high dimensional data. J. Multivariate Anal. 101 2554–2570. 2719881 (2012d:62169)
2010
Closest in time.
[author] Small, Christopher G.C. G. (2010). Expansions and asymptotics for statistics. Monographs on Statistics and Applied Probability 115. CRC Press, Boca Raton, FL. 2681183 (2011m:62005)
2010
Closest in time.
[author] Chen, Louis H. Y.L. H. Y. and Fang, XiaoX. (2011). Multivariate normal approximation by Stein’s method: the concentration inequality approach (preprint). http://arxiv.org/abs/1111.4073
Original
2011
Closest in time.
[author] Gaines, G.G., Kaphle, K.K. and Ruymgaart, F.F. (2011). Application of a delta-method for random operators to testing equality of two covariance operators. Math. Methods Statist. 20 232–245. 2908760
2011
Closest in time.
[author] Pinelis, IosifI. (2011). On the Berry-Esseen bound for the Student statistic (preprint). http://arxiv.org/abs/1101.3286
Original
2011
Closest in time.
[author] Pinelis, IosifI. (2011). Monotone tail and moment ratio properties of Student’s family of distributions (preprint). http://arxiv.org/abs/1101.3289
Original
2011
Closest in time.
[author] Pinelis, IosifI. (2011). Exact lower bounds on the exponential moments of truncated random variables. J. Appl. Probab. 48 547–560. 2840315 (2012f:60083)
2011
Closest in time.
[author] Pinelis, IosifI. (2011). Exact bounds on the truncated-tilted mean, with applications (preprint). http://arxiv.org/abs/1103.3683
Original
2011
Closest in time.
[author] Pinelis, IosifI. (2011). Improved nonuniform Berry–Esseen-type bounds (preprint). http://arxiv.org/abs/1109.0741
Original
2011
Closest in time.
[author] Pinelis, IosifI. (2011). Optimal re-centering bounds, with applications to Rosenthal-type concentration of measure inequalities (preprint). http://arxiv.org/abs/1111.2622
Original
2011
Closest in time.
[author] Shevtsova, IrinaI. (2011). On the absolute constants in the Berry-Esseen type inequalities for identically distributed summands (preprint). http://arxiv.org/abs/1111.6554
Original
2011
Closest in time.
[author] Pinelis, IosifI. (2012). An exact bound on the truncated-tilted mean for symmetric distributions (preprint). http://arxiv.org/abs/1205.5234
Original
2012
Closest in time.
[author] Gamboa, FabriceF., Janon, AlexandreA., Klein, ThierryT., Lagnoux-Renaudie, AgnesA. and Prieur, ClémentineC. (2013). Statistical inference for Sobol pick freeze Monte Carlo method (preprint). http://arxiv.org/pdf/1303.6447.pdf
Original
2013
Closest in time.
[author] Horgan, DonaghD. and Murphy, Colin C.C. C. (2013). On the Convergence of the Chi Square and Noncentral Chi Square Distributions to the Normal Distribution. IEEE Communications Letters 17 2233–2236
2013
Closest in time.
[author] Kotevski, ZoranZ. and Mitrevski, PeceP. (2013). Hybrid fluid modeling approach for performance analysis of P2P live video streaming systems. Peer-to-Peer Networking and Applications 7 410–426
2013
Closest in time.
[author] Küchler, UweU. and Tappe, StefanS. (2013). Tempered stable distributions and processes. Stochastic Process. Appl. 123 4256–4293. 10.1016/j.spa.2013.06.012 3096354
2013
Closest in time.
[author] Pinelis, IosifI. (2013). An optimal bound on the quantiles of a certain kind of distributions (preprint). http://arxiv.org/pdf/1301.0294.pdf
Original
2013
Closest in time.
[author] Pinelis, IosifI. (2013). Exact Rosenthal-type inequalities for p = 3 p=3 , and related results. Statist. Probab. Lett. 83 2634–2637. 3118206
2013
Closest in time.
[author] Rippl, ThomasT., Munk, AxelA. and Sturm, AnjaA. (2013). Limit laws of the empirical Wasserstein distance: Gaussian distributions (preprint). http://arxiv.org/pdf/1507.04090v1.pdf
Original
2013
Closest in time.
[author] Li, KeK. (2014). Second-order asymptotics for quantum hypothesis testing. Ann. Statist. 42 171–189. 10.1214/13-AOS1185 3178460
2014
Closest in time.
[author] Pinelis, IosifI. (2014). On the Bennett-Hoeffding inequality. Ann. Inst. H. Poincaré Probab. Statist. 50 15–27
2014
Closest in time.
[author] Wasserman, L.L., Kolar, M.M. and Rinaldo, A.A. (2014). Berry-Esseen bounds for estimating undirected graphs. Electron. J. Stat. 8 1188–1224
2014
Closest in time.
[author] Zeifman, AlexanderA., Korolev, VictorV., Satin, YacovY., Korotysheva, AnnaA. and Bening, VladimirV. (2014). Perturbation bounds and truncations for a class of Markovian queues. Queueing Syst. 76 205–221. 10.1007/s11134-013-9388-0 3158075
Original
2014
Closest in time.
[author] Anastasiou, AndreasA. and Ley, ChristopheC. (2015). New simpler bounds to assess the asymptotic normality of the maximum likelihood estimator. http://arxiv.org/abs/1508.04948
Original
2015
Closest in time.
[author] Anastasiou, AndreasA. and Reinert, GesineG. (2015). Bounds for the normal approximation of the maximum likelihood estimator. http://arxiv.org/abs/1411.2391
Original
2015
Closest in time.
[author] MolavianJazi, E.E. (2015). Private communication
2015
Closest in time.
[author] Pinelis, IosifI. (2016). Optimal-order bounds on the rate of convergence to normality for maximum likelihood estimators. http://arxiv.org/abs/1601.02177
Original
2016
Closest in time.
[author] Wang, QiyingQ. and Jing, Bing-YiB.-Y. (1999). An exponential nonuniform Berry-Esseen bound for self-normalized sums. Ann. Probab. 27 2068–2088. MR1742902 (2001c:60045)
2088
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…