# Least empirical risk procedures in statistical inference

Applicationes Mathematicae (1993)

- Volume: 22, Issue: 1, page 55-67
- ISSN: 1233-7234

## Access Full Article

top## Abstract

top## How to cite

topNiemiro, Wojciech. "Least empirical risk procedures in statistical inference." Applicationes Mathematicae 22.1 (1993): 55-67. <http://eudml.org/doc/219083>.

@article{Niemiro1993,

abstract = {We consider the empirical risk function $Q_n(α)=\{1\over n\} \sum _\{i=1\}^n \cdot f(α,Z_i)$ (for iid $Z_i$’s) under the assumption that f(α,z) is convex with respect to α. Asymptotics of the minimum of $Q_n(α)$ is investigated. Tests for linear hypotheses are derived. Our results generalize some of those concerning LAD estimators and related tests.},

author = {Niemiro, Wojciech},

journal = {Applicationes Mathematicae},

keywords = {least distances; convex minimization; tests of significance; least absolute deviations; asymptotics; asymptotic representation; minimum of the empirical risk function; tests of significance for linear hypotheses; least distance; discriminant analysis},

language = {eng},

number = {1},

pages = {55-67},

title = {Least empirical risk procedures in statistical inference},

url = {http://eudml.org/doc/219083},

volume = {22},

year = {1993},

}

TY - JOUR

AU - Niemiro, Wojciech

TI - Least empirical risk procedures in statistical inference

JO - Applicationes Mathematicae

PY - 1993

VL - 22

IS - 1

SP - 55

EP - 67

AB - We consider the empirical risk function $Q_n(α)={1\over n} \sum _{i=1}^n \cdot f(α,Z_i)$ (for iid $Z_i$’s) under the assumption that f(α,z) is convex with respect to α. Asymptotics of the minimum of $Q_n(α)$ is investigated. Tests for linear hypotheses are derived. Our results generalize some of those concerning LAD estimators and related tests.

LA - eng

KW - least distances; convex minimization; tests of significance; least absolute deviations; asymptotics; asymptotic representation; minimum of the empirical risk function; tests of significance for linear hypotheses; least distance; discriminant analysis

UR - http://eudml.org/doc/219083

ER -

## References

top- K. Adamczyk (1993), Asymptotic properties of ANOVA test under general loss functions, Mat. Stos., to appear. Zbl0796.62059
- Z. D. Bai, C. R. Rao and Y. Q. Yin (1990), Least absolute deviations analysis of variance, Sankhyā A 52, 166-177. Zbl0727.62073
- Z. D. Bai, C. R. Rao and Y. H. Wu (1992), M-estimation of multivariate linear regression parameters under a convex discrepancy function, Statist. Sinica 2 (1), 237-254. Zbl0820.62048
- G. Basset and R. Koenker (1978), Asymptotic theory of least absolute error regression, J. Amer. Statist. Assoc. 73, 618-622. Zbl0391.62046
- P. Bloomfield and W. L. Steiger (1983), Least Absolute Deviations, Theory, Applications, Algorithms, Birkhäuser, Boston. Zbl0536.62049
- L. Bobrowski, H. Wasyluk and W. Niemiro (1987), Some technique of linear discrimination with application to analysis of thyroid diseases diagnosis, Biocybernetics Biomed. Engrg. 7, 23-32.
- P. A. Devijver and J. Kittler (1982), Pattern Recognition: A Statistical Approach, Prentice-Hall, London. Zbl0542.68071
- J. K. Ghosh (1971), A new proof of the Bahadur representation of quantiles and an application, Ann. Math. Statist. 42, 1957-1961. Zbl0235.62006
- S. J. Haberman (1989), Concavity and estimation, Ann. Statist. 17, 1631-1661. Zbl0699.62027
- J. B. S. Haldane (1948), Note on the median of a multivariate distribution, Biometrika 25, 414-415. Zbl0032.03601
- D. J. Hand (1981), Discrimination and Classification, Wiley, New York. Zbl0587.62119
- J. W. McKean and R. M. Schrader (1987), Least absolute errors analysis of variance, in: Statistical Data Analysis Based on ${L}_{1}$-norm and Related Methods, Y. Dodge (ed.), North-Holland.
- P. Milasevic and G. R. Ducharme (1987), Uniqueness of the spatial median, Ann. Statist. 15, 1332-1333. Zbl0631.62058
- W. Niemiro (1989), L^1-optimal statistical discrimination procedures and their asymptotic properties, Mat. Stos. 31, 57-89 (in Polish). Zbl0698.62060
- W. Niemiro (1992), Asymptotics for M-estimators defined by convex minimization, Ann. Statist., to appear. Zbl0786.62040
- D. Pollard (1991), Asymptotics for least absolute deviation regression estimators, Econometric Theory 7, 186-199.
- C. R. Rao (1988), Methodology based on the ${L}_{1}$-norm in statistical inference, Sankhyā A 50, 289-313. Zbl0677.62058
- R. T. Rockafellar (1970), Convex Analysis, Princeton University Press. Zbl0193.18401
- A. H. Welsh (1987), Kernel estimates of the sparsity function, in: Statistical Data Analysis Based on ${L}_{1}$-norm and Related Methods, Y. Dodge (ed.), North-Holland.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.