Faculty of Economics and Business Administration Publications Database

A quantitative weak law of large numbers and its application to the delta method

Volume: 18
Number: 1
Pages: 84 - 95
Month: March
ISSN-Print: 1066-5307
Link External Source: Online Version
Year: 2009

Let Tn be a statistic of the form Tn = f(X¯n), where X¯n is the samplemean of a sequence of independent random variables and f denotes a prescribed function taking values in a separable Banach space. In order to establish asymptotic expansions for bias and variance of Tn conventional theorems typically impose restrictive boundedness conditions upon f or its derivatives; moreover, these conditions are sufficient but not necessary. It is shown that a quantitative version of the weak law of large numbers is both sufficient and necessary for the accuracy of Taylor expansions of Tn. In particular, boundedness conditions may be replaced by mild requirements upon the global growth of f.