# Sufficient conditions for the strong consistency of least squares estimator with α-stable errors

João Tiago Mexia; João Lita da Silva

Discussiones Mathematicae Probability and Statistics (2007)

- Volume: 27, Issue: 1-2, page 27-45
- ISSN: 1509-9423

## Access Full Article

top## Abstract

top## How to cite

topJoão Tiago Mexia, and João Lita da Silva. "Sufficient conditions for the strong consistency of least squares estimator with α-stable errors." Discussiones Mathematicae Probability and Statistics 27.1-2 (2007): 27-45. <http://eudml.org/doc/277074>.

@article{JoãoTiagoMexia2007,

abstract = {Let $Y_\{i\} = x_\{i\}^\{T\}β + e_\{i\}$, 1 ≤ i ≤ n, n ≥ 1 be a linear regression model and suppose that the random errors e₁, e₂, ... are independent and α-stable. In this paper, we obtain sufficient conditions for the strong consistency of the least squares estimator β̃ of β under additional assumptions on the non-random sequence x₁, x₂,... of real vectors.},

author = {João Tiago Mexia, João Lita da Silva},

journal = {Discussiones Mathematicae Probability and Statistics},

keywords = {linear models; least squares estimator; strong consistency; stability; least squares estimates; regression models; Marcinkiewicz-Zygmund law; undefined errors mean values},

language = {eng},

number = {1-2},

pages = {27-45},

title = {Sufficient conditions for the strong consistency of least squares estimator with α-stable errors},

url = {http://eudml.org/doc/277074},

volume = {27},

year = {2007},

}

TY - JOUR

AU - João Tiago Mexia

AU - João Lita da Silva

TI - Sufficient conditions for the strong consistency of least squares estimator with α-stable errors

JO - Discussiones Mathematicae Probability and Statistics

PY - 2007

VL - 27

IS - 1-2

SP - 27

EP - 45

AB - Let $Y_{i} = x_{i}^{T}β + e_{i}$, 1 ≤ i ≤ n, n ≥ 1 be a linear regression model and suppose that the random errors e₁, e₂, ... are independent and α-stable. In this paper, we obtain sufficient conditions for the strong consistency of the least squares estimator β̃ of β under additional assumptions on the non-random sequence x₁, x₂,... of real vectors.

LA - eng

KW - linear models; least squares estimator; strong consistency; stability; least squares estimates; regression models; Marcinkiewicz-Zygmund law; undefined errors mean values

UR - http://eudml.org/doc/277074

ER -

## References

top- [1] B.D.O. Anderson and J.B. Moore, On martingales and least squares linear system identification, Technical report EE7522 (1975).
- [2] B.D.O. Anderson and J.B. Moore, A matrix Kronecker lemma, Linear Algebra and its Applications 15 (1976), 227-234. Zbl0356.15019
- [3] P. Billingsley, Probability and Measure, (third edition) John Wiley & Sons 1995.
- [4] Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, Springer 1997.
- [5] H. Drygas, Consistency of the least squares and Gauss-Markov estimators in regression models, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 17 (1971), 309-326. Zbl0204.52801
- [6] H. Drygas, Weak and strong consistency of the least squares estimators in regression model, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 34 (1976), 119-127. Zbl0307.62047
- [7] W. Feller, An Introduction to Probability Theory and Its Applications - Volume I, (third edition) John Wiley & Sons 1968. Zbl0155.23101
- [8] W. Feller, An Introduction to Probability Theory and Its Applications - Volume II, (second edition) John Wiley & Sons 1971. Zbl0219.60003
- [9] I.A. Ibragimov and Yu.V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen (1971). Zbl0219.60027
- [10] M. Jin, Some new results of the strong consistency of multiple regression coefficients, p. 514-519 in: 'Proceedings of the Second Asian Mathematical Conference 1995' (Tangmanee, S. & Schulz, E. eds.), World Scientific. Zbl0952.62064
- [11] M. Jin and X. Chen, Strong consistency of least squares estimate in multiple regression when the error variance is infinite, Stat. Sin. 9 (1) (1999), 289-296. Zbl0913.62024
- [12] T.L. Lai, H. Robbins and C.Z. Wei, Strong consistency of least squares estimates in multiple regression, Proc. Natl. Acad. Sci. USA 75 (7) (1978), 3034-3036. Zbl0386.62051
- [13] T.L. Lai, H. Robbins and C.Z. Wei, Strong consistency of least squares estimates in multiple regression II, J. Multivariate Anal. 9 (1979), 343-362. Zbl0416.62051
- [14] J.T. Mexia, P. Corte Real, M.L. Esquível and J. Lita da Silva, Convergência do estimador dos mínimos quadrados em modelos lineares, Estaística Jubilar, Actas do XII Congresso da Sociedade Portuguesa de Estatística, Edições SPE (2005), 455-466.
- [15]J.T. Mexia and J. Lita da Silva, Least squares estimator consistency: a geometric approach, Discussiones Mathematicae - Probability and Statistics 26 (1) (2006), 19-45. Zbl1128.62029
- [16] G. Samorodnitsky and M.S. Taqqu, Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall 1994. Zbl0925.60027
- [17] V.V. Uchaikin and V.M. Zolotarev, Chance and Stability, Stable Distributions and Their Applications, Ultrech 1999. Zbl0944.60006
- [18] V.M. Zolotarev, One-Dimensional Stable Distributions, American Mathematical Society, Providence, R.I. 1986. Zbl0589.60015

## 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.