Sample extremes: an elementary introduction

Sample extremes: an elementary introduction This note is of a didactic nature. We will derive the limit distributions for maxima of i.i.d. random variables and give sufficient conditions for their domains of attraction. [2] These results can be found in the works of GNEDENKO and VON MISES[4]. The present exposition contains new proofs which are intended to be sufficiently simple to be used in an elementary course of probability theory. 2 The limit distributions We will be concerned with the following problem. Suppose X I , X,, ... are independent real-valued random variables with common distribution function (df) F . We define for n = 1, 2, .. Y, = max(X,, X2, ...,X,). We remark that one can interpret all the results that follow as results for minima by noting that min (XI, ..., X,)= --ax ( - X I , ..., -X,,). It follows from the independence of the X i that P(Y, < x } = P ( X , < x , ..., x, < x } = F ( x ) . a,,> 0 and b,(n = 1,2, ...) such that the sequence P(u,- '( Y, We ask for conditions which enables one to choose sequences of real constants http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Statistica Neerlandica Wiley

Sample extremes: an elementary introduction

Statistica Neerlandica, Volume 30 (4) – Dec 1, 1976

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Publisher
Wiley
Copyright
Copyright © 1976 Wiley Subscription Services, Inc., A Wiley Company
ISSN
0039-0402
eISSN
1467-9574
DOI
10.1111/j.1467-9574.1976.tb00275.x
Publisher site
See Article on Publisher Site

Abstract

This note is of a didactic nature. We will derive the limit distributions for maxima of i.i.d. random variables and give sufficient conditions for their domains of attraction. [2] These results can be found in the works of GNEDENKO and VON MISES[4]. The present exposition contains new proofs which are intended to be sufficiently simple to be used in an elementary course of probability theory. 2 The limit distributions We will be concerned with the following problem. Suppose X I , X,, ... are independent real-valued random variables with common distribution function (df) F . We define for n = 1, 2, .. Y, = max(X,, X2, ...,X,). We remark that one can interpret all the results that follow as results for minima by noting that min (XI, ..., X,)= --ax ( - X I , ..., -X,,). It follows from the independence of the X i that P(Y, < x } = P ( X , < x , ..., x, < x } = F ( x ) . a,,> 0 and b,(n = 1,2, ...) such that the sequence P(u,- '( Y, We ask for conditions which enables one to choose sequences of real constants

Journal

Statistica NeerlandicaWiley

Published: Dec 1, 1976

References

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