A Note on Edgeworth Expansion
Reza Habibi *
Iran Banking Institute, Central Bank of Iran, Tehran, Iran.
*Author to whom correspondence should be addressed.
Abstract
The Edgeworth expansion plays important role in approximating the distribution function, specially the tail probabilities of a complicated statistic. For example, sometimes, the test statistic, in hand, is too complicated and deriving its quantiles is too hard. However, these quantiles are necessary for decision making in hypothesis testing. This problem is seen frequently in change point analysis. Thus, in these fields, the Edgeworth expansion is valuable mean. The traditional Edgeworth expansion is derived using the approximation of characteristic function by Taylor expansion. In the current note, an alternative method is proposed to derive this expansion. This paper is concerned with application of Euler-Lagrange equation in Edgeworth expansion. The method is proposed and error analysis shows that the method is accurate. The application of bootstrap method is observed. Finally, a conclusion section is proposed.
Keywords: Bootstrap, calculus of variations, dynamic programming, Edgeworth expansion, error analysis, Euler-Lagrange equation