Abstract
Quite many authors have dealt with the estimation of the parameters of normal distribution on the basis of non-homogeneous sets: Hald A. 1949 [1], Arango-Castillo L. and Takahara G. 2018 [2]. All the robust methods are based on the assumption that the results affected by gross errors can be found to the left and/or to the right of censoring, or truncated, points. However, as a rule, the (intrinsic) distribution of observations is complex (mixed) consisting of two or more distributions. Then the existing methods, such as ML, Huber’s, etc., yield enlarged estimates for the normal-distribution variance. By studying better estimates the present author has invented new method, called PEROBLS D, based on the Tukeyan mixed-distribution model in which both the contamination rate (percentage) and the parameters of both distributions, forming the mixed one, are estimated, and for the parameters of the basic normal distribution better estimates are obtained than by the existing methods.
Highlights
Introduction and History of Robust EstimationThe history of this problem is older than 300 years
Quite many authors have dealt with the estimation of the parameters of normal distribution on the basis of non-homogeneous sets: Hald A. 1949 [1], Arango-Castillo L. and Takahara G. 2018 [2]
The parameter estimates in the PEROBLS D method are close to the exact ones, whereas in the case of application of the Maximum Likelihood (ML) and Huber-mad methods the variance of the basic distribution is overestimated
Summary
The history of this problem is older than 300 years. Galileo as long ago as in 1632 used the least absolute sum in order to reduce the effect of observational errors to the estimate of the measured quantity [3], whereas Rudjer Boscovich, is the first who, as early as in 1757, rejected clearly outlying observations [4], done by Daniel Bernouli 1777 [4]. G. Perović used since long ago, see “Anonymous” 1821 [4]. The mixed distribution models have been considered since long ago: Glaisher 1872/1873, Stone 1873, Edgeworth 1883, Newcomb 1886, Jeffreys 1932/ 1939 [4]. Tukey in 1960 [5] defined a mixed model as a mixture of two normal distributions of a basic Φ ( x −θ ) σ and of a contaminating Φ ( x −θ ) 3σ distribution:
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