Huber-Sense Robust M-Estimation of a Scale Parameter, with Application to the Exponential Distribution.
FLORIDA STATE UNIV TALLAHASSEE DEPT OF STATISTICS
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A theory of robust M-estimation of a location parameter was developed by Huber 1964 and applied to estimation of the mean of a normal distribution. This theory is applicable to the problem of robust estimation of a scale parameter, since nonnegative data X having scale parameter theta may be transformed by y log x into data Y having location parameter log theta. Equivalently, in the present article the authors reformulate Hubers location parameter results in the scale parameter content--that is transform the theorems instead of the data--and we the authors apply the results in connection with the problem of robust estimation of the parameter theta of the exponential distribution, 1 - exp -xtheta, x 0. Whereas the maximum likelihood estimator of theta is the sample mean, the robust M-estimator, which is the solution of a minimax problem based on the asymptotic variance criterion, turns out to be a type of Winsorized mean. Numerical illustration is provided using a data set of Proschan 1963 consisting of the time intervals between successive failures of the air conditioning systems of a fleet of jet airplanes.
- Statistics and Probability