Bootstrap, wild bootstrap, and asymptotic normality

SummaryWe show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend onn. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap.

[1]  R. Gill.  Non- and semi-parametric maximum likelihood estimators and the Von Mises method , 1986 .

[2]  Peter F. de Jong,et al.  A central limit theorem for generalized quadratic forms , 1987 .

[3]  Enno Mammen.  Bootstrap and wild bootstrap for high — dimensional linear random design models , 1992 .

[4]  R. Beran.  Estimated Sampling Distributions: The Bootstrap and Competitors , 1982 .

[5]  E. Mammen.  Bootstrap and Wild Bootstrap for High Dimensional Linear Models , 1993 .

[6]  B. Gnedenko,et al.  Limit Distributions for Sums of Independent Random Variables , 1955 .

[7]  B. Gnedenko,et al.  Limit distributions for sums of shrunken random variables , 1954 .

[8]  Higher — order accuracy of bootstrap for smooth functionals , 1992 .

[9]  E. Giné,et al.  Necessary Conditions for the Bootstrap of the Mean , 1989 .

[10]  D. Aldous.  The Central Limit Theorem for Real and Banach Valued Random Variables , 1981 .

[11]  C. Esseen.  On the Kolmogorov-Rogozin inequality for the concentration function , 1966 .

[12]  Rudolf Beran.  Discussion: Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis , 1986 .

[13]  P. Hall.  Central limit theorem for integrated square error of multivariate nonparametric density estimators , 1984 .

[14]  L. L. Cam,et al.  Asymptotic Methods In Statistical Decision Theory , 1986 .

[15]  Regina Y. Liu.  Bootstrap Procedures under some Non-I.I.D. Models , 1988 .

[16]  Keith Knight,et al.  On the Bootstrap of the Sample Mean in the Infinite Variance Case , 1989 .

[17]  K. Athreya.  BOOTSTRAP OF THE MEAN IN THE INFINITE VARIANCE CASE , 1987 .

[18]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[19]  M. Csörgo,et al.  Bootstrapping Empirical Functions , 1989 .

[20]  Changbao Wu,et al.  Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis , 1986 .

[21]  P. Hall.  Asymptotic Properties of the Bootstrap for Heavy-Tailed Distributions , 1990 .