Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- 1 Introduction
- 2 Stochastic Convergence
- 3 Delta Method
- 4 Moment Estimators
- 5 M–and Z-Estimators
- 6 Contiguity
- 7 Local Asymptotic Normality
- 8 Efficiency of Estimators
- 9 Limits of Experiments
- 10 Bayes Procedures
- 11 Projections
- 12 U -Statistics
- 13 Rank, Sign, and Permutation Statistics
- 14 Relative Efficiency of Tests
- 15 Efficiency of Tests
- 16 Likelihood Ratio Tests
- 17 Chi-Square Tests
- 18 Stochastic Convergence in Metric Spaces
- 19 Empirical Processes
- 20 Functional Delta Method
- 21 Quantiles and Order Statistics
- 22 L-Statistics
- 23 Bootstrap
- 24 Nonparametric Density Estimation
- 25 Semiparametric Models
- References
- Index
20 - Functional Delta Method
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- 1 Introduction
- 2 Stochastic Convergence
- 3 Delta Method
- 4 Moment Estimators
- 5 M–and Z-Estimators
- 6 Contiguity
- 7 Local Asymptotic Normality
- 8 Efficiency of Estimators
- 9 Limits of Experiments
- 10 Bayes Procedures
- 11 Projections
- 12 U -Statistics
- 13 Rank, Sign, and Permutation Statistics
- 14 Relative Efficiency of Tests
- 15 Efficiency of Tests
- 16 Likelihood Ratio Tests
- 17 Chi-Square Tests
- 18 Stochastic Convergence in Metric Spaces
- 19 Empirical Processes
- 20 Functional Delta Method
- 21 Quantiles and Order Statistics
- 22 L-Statistics
- 23 Bootstrap
- 24 Nonparametric Density Estimation
- 25 Semiparametric Models
- References
- Index
Summary
The delta method was introduced in Chapter 3 as an easy way to tum the weak convergence of a sequence of random vectors into the weak convergence of transformations of the type. It is useful to apply a similar technique in combination with the more powerful convergence of stochastic processes. In this chapter we consider the delta method at two levels. The first section is of a heuristic character and limited to the case that Tn is the empirical distribution. The second section establishes the delta method rigorously and in general, completely parallel to the delta method for, for Hadamard differentiable maps between normed spaces.
von Mises Calculus
Let be the empirical distribution of a random sample X1, … , Xn from a distribution P. Many statistics can be written in the form whereis a function that maps every distribution of interest into some space, which for simplicity is taken equal to the real line. Because the observations can be regained from completely (unless there are ties), any statistic can be expressed in the empirical distribution. The special structure assumed here is that the statistic can be written as a fixed functionof , independent of n, a strong assumption.
Because converges to P astends to infinity, we may hope to find the asymptotic behavior of through a differential analysis ofin a neighborhood of P. A first-order analysis would have the form
where is a “derivative” and the remainder is hopefully negligible. The simplest approach towards defining a derivative is to consider the function for a fixed perturbation H and as a function of the real-valued argument t. Iftakes its values in JR, then this function is just a function from the reals to the reals.
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- Asymptotic Statistics , pp. 291 - 303Publisher: Cambridge University PressPrint publication year: 1998
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