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It is an easy calculation to check that the median of the exponential distribution has value λ log 2, where log stands for natural logarithm, and so one can use Med (x) / log 2 as a simple robust estimate of λ that is consistent with the assumed model. 15. The methods of robustly fitting Weibull and gamma distributions are much more complicated than the above use of the adjusted median for the exponential distribution. JWBK076-01 JWBK076-Maronna February 16, 2006 18:7 Char Count= 0 14 INTRODUCTION We present one important application of robust fitting a gamma distribution due to Marazzi, Paccaud, Ruffieux and Beguin (1998).
Xn ; μ) = f 0 (xi − μ) i=1 The maximum likelihood estimate (MLE) of μ is the value μ—depending on x1 , . . , xn — that maximizes L(x1 , . . , xn ; μ): μ = μ(x1 , . . , xn ) = arg max L(x1 , . . 11) where “arg max” stands for “the value maximizing”. 8). g. contaminated normal). 12) i=1 where ρ = − log f 0 . 14) If F0 = N(0, 1), then and apart from a constant, ρ(x) = x 2 /2. 12) is equivalent to n μ = arg min μ (xi − μ)2 . 12) is equivalent to n μ = arg min μ |xi − μ|. 17) are the sample mean and median, respectively.
For a given distribution F, define μ0 = μ0 (F) as the solution of E F ψ(x − μ0 ) = 0. , the population mean. , a population median. 6). 21) (we say that μ is “consistent for μ0 ”), and the distribution of μ is approximately N μ0 , v n with v = E F ψ(x − μ0 )2 (E F ψ (x − μ0 ))2 . , v= E F0 ψ(x)2 E F0 ψ (x) 2 . 24) If the distribution of an estimate μ is approximately N(μ0 , v/n) for large n, we say that μ is asymptotically normal, with asymptotic value μ0 and asymptotic variance v. 25) where v0 is the asymptotic variance of the MLE, and measures how near μ is to the optimum.