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Cornish-Fisher Expansions

  • Rabi Bhattacharya
  • Manfred Denker
Chapter
Part of the DMV Seminar book series (OWS, volume 14)

Abstract

Consider an estimator θ̂ n of a real valued parameter θ lying in an open set Θ, and let \( {{s} \left/ {{\sqrt {n} }} \right.} \) be an estimate of its standard error. Suppose under P θ (i.e., when θ is the true parameter value) the distribution function G n (x;θ) of \( {{{\sqrt {n} ({{\hat{\theta }}_n} - \theta )}} \left/ {s} \right.} \) has the asymptotic expansion
$$ {G_n}(x,\theta ) = \Phi (x) + \sum\limits_{{r = 1}}^{{s - 2}} {{n^{{ - \frac{r}{2} }}}} {q_r}(x;\theta )\phi (x) + o({n^{{{{{ - (s - 2)}} \left/ {2} \right.}}}}) = {\Psi_{{s,n}}}(x,\theta ) + o({n^{{{{{ - (s - 2)}} \left/ {2} \right.}}}}) $$
(3.1)
uniformly in x1 and θ in any compact K ⊂ Θ. Here q r (x;θ) are polynomials in x whose coefficients are smooth functions of θ (say, (s-2)-times continuously differentiable in θ). The usual large sample confidence interval for θ having nominal coverage (or nominal confidence level) 1 - α is [\( \left[ {{{{{{\hat{\theta }}_n} - {z_{{1 - \frac{\alpha }{2}}}}s}} \left/ {{\sqrt {n} }} \right.},{{{{{\hat{\theta }}_n} + {z_{{1 - \frac{alpha}{2}}}}s}} \left/ {{\sqrt {n} }} \right.}} \right] \)], where z p is the unique number satisfying
$$ \Phi ({z_p}) \equiv \int\limits_{{ - \infty }}^{{{z_p}}} {\phi (x)dx = p(0 < p < 1)} $$
(3.2)
.

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Copyright information

© Birkhäuser Verlag Basel 1990

Authors and Affiliations

  • Rabi Bhattacharya
    • 1
  • Manfred Denker
    • 2
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Institut für MathematischeStochastikGöttingenGermany

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