Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-07-04T02:07:11.752Z Has data issue: false hasContentIssue false

6 - Stochastic Convergence and Probability Inequalities

Published online by Cambridge University Press:  05 June 2012

Pranab K. Sen
Affiliation:
University of North Carolina, Chapel Hill
Julio M. Singer
Affiliation:
Universidade de São Paulo
Antonio C. Pedroso de Lima
Affiliation:
Universidade de São Paulo
Get access

Summary

Introduction

Unbiasedness, efficiency, sufficiency, and ancillarity, as outlined in Chapters 2 and 3, are essentially finite-sample concepts, but consistency refers to indefinitely increasing samples sizes, and thus has an asymptotic nature. In general, finite-sample optimality properties of estimators and tests hold basically for a small class of probability laws, mostly related to the exponential family of distributions; consistency, however, holds under much less restricted setups as we will see. Moreover, even when finite-sample optimal statistical procedures exist, they may not lead to closed-form expressions and/or be subject to computational burden. These problems are not as bothersome when we adopt an asymptotic point of view and use the corresponding results to obtain good approximations of such procedures for large (although finite) samples. This is accomplished with the incorporation of probability inequalities, limit theorems, and other tools that will be developed in this and subsequent chapters.

In this context, a minimal requirement for a good statistical decision rule is its increasing reliability with increasing sample sizes (consistency). For an estimator, consistency relates to an increasing closeness to its population counterpart as the sample sizes become larger. In view of its stochastic nature, this closeness needs to incorporate its fluctuation around the parameter it estimates and thus requires an appropriate adaptation of the definitions usually considered in nonstochastic setups. Generally, a distance function or norm of this stochastic fluctuation is incorporated in the formulation of this closeness, and consistency refers to the convergence of this norm to 0 in some well-defined manner.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×