Asymptotic,Higher Order Borovkov A,Mogulskii A (1993)Large deviations and testing of truly approximating a target quantity up to the claimed A statistical hypotheses.Siberian Adv Math 2(3,4);3(1,2) degree of accuracy. Chernoff H(1952)A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations.Ann Math Stat Classical asymptotics is based on the notion of asymp- 23:493-507 totic distribution,often derived from the central limit Dembo A,ZeitouniO(1998)Large deviations techniques and appli- theorem (see Central Limit Theorems),and usually the cations,2nd edn.Springer,New York approximations are correct up to O(n-2),where n is Deuschel J-D.Stroock D(1989)Large deviations.Academic.Boston the sample size.Higher order asymptotics provides refine- Groeneboom P.Oosterhoff J (1981)Bahadur efficiency and small ments based on asymptotic expansions of the distribution sample efficiency.Int Stat Rev 49:127-141 Hodges J,Lehmann EL(1956)The efficiency of some nonparametric or density function of an estimator of a parameter.They are competitors of the f-test.Ann Math Stat 26:324-335 rooted in the Edgeworth theory,which is itselfa refinement Kakizawa Y(2005)Bahadur exact slopes of some tests for spectral of the central limit theorem.The theory of higher order densities.J Nonparametric Stat 17:745-764 asymptotic is very much related with the corresponding Kallenberg WCM(1983)Intermediate efficiency,theory and exam to Approximations to distributions treated as well in this ples.Ann Stat 11:170-182 Kallenberg WCM(1982)Chernoff efficiency and deficiency.Ann Encyclopedia. Stat10:583-594 When higher order asymptotic is correct up to Nikitin Y (1995)Asymptotic efficiency of nonparametric tests. o(n),it is second order asymptotic.When further Cambridge University Press,Cambridge terms are picked up,so that the asymptotic is correct up Otsu T(2010)On Bahadur efficiency of empirical likelihood.JEcon to o(n),it is third order asymptotic.In his pioneer- 157:248-256 Pitman EJG(1949)Lecture notes on nonparametric statistical infer- ing papers,C.R.Rao coined the term second order effi- ence.Columbia University,Mimeographed ciency for a concept that would now is called third order Puhalskii A,Spokoiny V(1998)On large-deviation efficiency in efficiency.The new terminology is essentially owing to statistical inference.Bernoulli 4:203-272 Pfanzagl and Takeuchi. Serfling R(1980)Approximation theorems of mathematical statis- tics.Wiley,New York Serfling R(2010)Asymptotic relative efficiency in estimation.In: About the Author Lovric M (ed)International encyclopedia of statistical sciences. Professor Abril is co-editor of the Revista de la Sociedad Springer Argentina de Estadistica (Journal of the Argentinean Sta- Taniguchi M(2001)On large deviation asymptotics of some tests in tistical Society). time series.J Stat Plann Inf 97:191-200 Van der Vaart AW (1998)Asymptotic statistics.Cambridge Univer- sity Press,Cambridge Cross References Wieand HS(1976)A condition under which the Pitman and Bahadur Approximations to Distributions approaches to efficiency coincide.Ann Statist 4:1003-1011 Edgeworth Expansion References and Further Reading Abril JC(1985)Asymptotic expansions for time series problems Asymptotic,Higher Order with applications to moving average models.PhD thesis.The London School of Economics and Political Science,University of London,England JUAN CARLOS ABRIL Barndorff-Nielsen O,Cox DR(1979)Edgeworth and saddle-point President of the Argentinean Statistical Society,Professor approximations with statistical applications.J R Stat Soc B Universidad Nacional de Tucuman and Consejo Nacional 41:279-312 de Investigaciones Cientificas y Tecnicas,San Miguel de Daniels HE(1954)Saddlepoint approximations in statistics.Ann Math Stat 25:631-650 Tucuman,Argentina Durbin J(1980)Approximations for the densities of sufficient esti- mates.Biometrika 67:311-333 Higher order asymptotic deals with two sorts of closely Feller W(1971)An introduction to probability theory and its appli- related things.First,there are questions of approxima- cations,vol 2,2nd edn.Wiley,New York Ghosh JK(1994)Higher order Asymptotic.NSF-CBMS Regional tion.One is concerned with expansions or inequalities Conference Series in Probability and Statistics,4.Hayward and for a distribution function.Second,there are inferential Alexandria:Institute of Mathematical Statistics and American issues.These involve,among other things,the applica- Statistical Association tion of the ideas of the study of higher order efficiency, Pfanzagl J(1979)Asymptotic expansions in parametric statistical admissibility and minimaxity.In the matter of expansions, theory.In:Krishnaiah PR(ed)Developments in statistics,vol.3. Academic,New York,pp 1-97 it is as important to have usable,explicit formulas as a rig- Rao CR (1961)Asymptotic efficiency and limiting information.In orous proof that the expansions are valid in the sense of Proceedings of Fourth Berkeley Symposium on MathematicalAsymptotic, Higher Order A  A Borovkov A, Mogulskii A () Large deviations and testing of statistical hypotheses. Siberian Adv Math (, ); (, ) Chernoff H () A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations. Ann Math Stat :– Dembo A, Zeitouni O () Large deviations techniques and appli￾cations, nd edn. Springer, New York Deuschel J-D, Stroock D () Large deviations. Academic, Boston Groeneboom P, Oosterhoff J () Bahadur efficiency and small sample efficiency. Int Stat Rev :– Hodges J, Lehmann EL () The efficiency of some nonparametric competitors of the t-test. Ann Math Stat :– Kakizawa Y () Bahadur exact slopes of some tests for spectral densities. J Nonparametric Stat :– Kallenberg WCM () Intermediate efficiency, theory and exam￾ples. Ann Stat :– Kallenberg WCM () Chernoff efficiency and deficiency. Ann Stat :– Nikitin Y () Asymptotic efficiency of nonparametric tests. Cambridge University Press, Cambridge Otsu T () On Bahadur efficiency of empirical likelihood. J Econ :– Pitman EJG () Lecture notes on nonparametric statistical infer￾ence. Columbia University, Mimeographed Puhalskii A, Spokoiny V () On large-deviation efficiency in statistical inference. Bernoulli :– Serfling R () Approximation theorems of mathematical statis￾tics. Wiley, New York Serfling R () Asymptotic relative efficiency in estimation. In: Lovric M (ed) International encyclopedia of statistical sciences. Springer Taniguchi M () On large deviation asymptotics of some tests in time series. J Stat Plann Inf :– Van der Vaart AW () Asymptotic statistics. Cambridge Univer￾sity Press, Cambridge Wieand HS () A condition under which the Pitman and Bahadur approaches to efficiency coincide. Ann Statist :– Asymptotic, Higher Order Juan Carlos Abril President of the Argentinean Statistical Society, Professor Universidad Nacional de Tucumán and Consejo Nacional de Investigaciones Cientícas y Técnicas, San Miguel de Tucumán, Argentina Higher order asymptotic deals with two sorts of closely related things. First, there are questions of approxima￾tion. One is concerned with expansions or inequalities for a distribution function. Second, there are inferential issues. ese involve, among other things, the applica￾tion of the ideas of the study of higher order e›ciency, admissibility and minimaxity. In the matter of expansions, it is as important to have usable, explicit formulas as a rig￾orous proof that the expansions are valid in the sense of truly approximating a target quantity up to the claimed degree of accuracy. Classical asymptotics is based on the notion of asymp￾totic distribution, oŸen derived from the central limit theorem (see 7Central Limit eorems), and usually the approximations are correct up to O(n −/ ), where n is the sample size. Higher order asymptotics provides rene￾ments based on asymptotic expansions of the distribution or density function of an estimator of a parameter.ey are rooted in the Edgeworth theory, which is itself a renement of the central limit theorem. e theory of higher order asymptotic is very much related with the corresponding to Approximations to distributions treated as well in this Encyclopedia. When higher order asymptotic is correct up to o(n −/ ), it is second order asymptotic. When further terms are picked up, so that the asymptotic is correct up to o(n − ), it is third order asymptotic. In his pioneer￾ing papers, C. R. Rao coined the term second order e›- ciency for a concept that would now is called third order e›ciency. e new terminology is essentially owing to Pfanzagl and Takeuchi. About the Author Professor Abril is co-editor of the Revista de la Sociedad Argentina de Estadística (Journal of the Argentinean Sta￾tistical Society). Cross References 7Approximations to Distributions 7Edgeworth Expansion References and Further Reading Abril JC () Asymptotic expansions for time series problems with applications to moving average models. PhD thesis. The London School of Economics and Political Science, University of London, England Barndorff-Nielsen O, Cox DR () Edgeworth and saddle-point approximations with statistical applications. J R Stat Soc B :– Daniels HE () Saddlepoint approximations in statistics. Ann Math Stat :– Durbin J () Approximations for the densities of sufficient esti￾mates. Biometrika :– Feller W () An introduction to probability theory and its appli￾cations, vol , nd edn. Wiley, New York Ghosh JK () Higher order Asymptotic. NSF-CBMS Regional Conference Series in Probability and Statistics, . Hayward and Alexandria: Institute of Mathematical Statistics and American Statistical Association Pfanzagl J () Asymptotic expansions in parametric statistical theory. In: Krishnaiah PR (ed) Developments in statistics, vol. . Academic, New York, pp – Rao CR () Asymptotic efficiency and limiting information. In Proceedings of Fourth Berkeley Symposium on Mathematical