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
3 - 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 consists of using a Taylor expansion to approximate a random vector of the form (Tn) by the polynomial + ‘ (Tn–) + … in Tn–. It is a simple but useful method to deduce the limit law of(Tn)–from that of Tn–. Applications include the non robustness of the chi-square test for normal variances and variance stabilizing transformations.
Basic Result
Suppose an estimator Tn for a parameter is available, but the quantity of interest isfor some known function . A natural estimator is (Tn). How do the asymptotic properties of (Tn) follow from those of Tn
A first result is an immediate consequence of the continuous-mapping theorem. If the sequence Tn converges in probability to andis continuous at then (Tn) converges in probability to
Of greater interest is a similar question concerning limit distributions. In particular, if y'n(Tn) converges weakly to a limit distribution, is the same true for y'n((Tn) Ifis differentiable, then the answer is affirmative. Informally, we have
If y'n(Tn)–T for some variable T, then we expect that y'n((Tn)–(e»)–‘ In particular, if y'n(Tn–e) is asymptotically normal then we expect that y'n((Tn)–(e») is asymptotically normal This is proved in greater generality in the following theorem.
In the preceding paragraph it is silently understood that Tn is real-valued, but we are more interested in considering statistics(Tn) that are formed out of several more basic statistics. Consider the situation that … , Tn,k) is vector-valued, and thatis a given function defined at least on a neighbourhood of Recall that is differentiable at if there exists a linear map (matrix) such that
All the expressions in this equation are vectors of length m, and IIhll is the Euclidean norm. The linear map is sometimes called a “total derivative,” as opposed to partial derivatives.
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- Asymptotic Statistics , pp. 25 - 34Publisher: Cambridge University PressPrint publication year: 1998
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