By Christopher A. Field, Elvezio Ronchetti

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4. The value of log(F/l - F) is plotted against t for the 5% contaminated normal and the Cauchy. 01% level. Similar results hold for the Cauchy. These results imply that the approximations accurately reflect the distributional behaviour of the estimates and provide a very useful tool for determining small sample properties of interest. 5) for Cauchy Some calculations with the mean suggest that any accuracy obtained by including the first neglected term is of the order of round off errors and so the first term in the expansion is all that is needed for good accuracy.

4) holds for arbitrary α. In order to approximate htθtfl(t0)y it is necessary to express Tn as an approximate mean and evaluate its cumulants. The development follows that of Field (1982) and Tingley (1987, p. 49-52). The first part is very similar to expansions used to demonstrate properties of maximum likelihood estimates. The result will be stated in terms of do, the true value of θ and then will be modified for the case of the conjugate density Λto which is centered at to. 5 hold. 1)) = 1 - 0(l/>/n) where dQ may depend on k.

This implies that the fourth moment is 0(n~ 2 ). We can now find a constant Ci, so that for n sufficiently large,i^dΓn—flol4) < C\n~2. Letting A be the region \t — θ0 \ < d, then Cm-*> f(t-θo)4fn(t)dt+ I JAC JA >δ4 (t-θoγfn(t)dt ί fn(t)dt = 64P(\Tn - 0O| > «2) JA This implies P(\Tn - 0 O | > δ2) = 0(n~ 2 ). 3 implies that 1 - C2/n < fn(t)/gn(t) < 1 + C2/n for \t - θo\ < δ2 where gn(t) = (n/2'κ)ιl2c-n{t)A(i)lσ(t) and C2 does not depend, on n or t. Let {Bn} be a sequence of Borel sets such that P(Tn £ Bn) converges to a positive limit.