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The weighted median is shown in red and is different than the ordinary median. In statistics, a weighted median of a sample is the 50% weighted percentile. [1] [2] [3] It was first proposed by F. Y. Edgeworth in 1888. [4] [5] Like the median, it is useful as an estimator of central tendency, robust against outliers. It allows for non-uniform ...
In statistics, the Hodges–Lehmann estimator is a robust and nonparametric estimator of a population's location parameter.For populations that are symmetric about one median, such as the Gaussian or normal distribution or the Student t-distribution, the Hodges–Lehmann estimator is a consistent and median-unbiased estimate of the population median.
The Marshall-Edgeworth index, credited to Marshall (1887) and Edgeworth (1925), [11] is a weighted relative of current period to base period sets of prices. This index uses the arithmetic average of the current and based period quantities for weighting. It is considered a pseudo-superlative formula and is symmetric. [12]
For context, the best single point estimate by L-estimators is the median, with an efficiency of 64% or better (for all n), while using two points (for a large data set of over 100 points from a symmetric population), the most efficient estimate is the 27% midsummary (mean of 27th and 73rd percentiles), which has an efficiency of about 81% ...
For normally distributed random variables inverse-variance weighted averages can also be derived as the maximum likelihood estimate for the true value. Furthermore, from a Bayesian perspective the posterior distribution for the true value given normally distributed observations and a flat prior is a normal distribution with the inverse-variance weighted average as a mean and variance ().
A winsorized mean is a winsorized statistical measure of central tendency, much like the mean and median, and even more similar to the truncated mean.It involves the calculation of the mean after winsorizing — replacing given parts of a probability distribution or sample at the high and low end with the most extreme remaining values, [1] typically doing so for an equal amount of both ...
Such an estimator is not necessarily an M-estimator of ρ-type, but if ρ has a continuous first derivative with respect to , then a necessary condition for an M-estimator of ψ-type to be an M-estimator of ρ-type is (,) = (,). The previous definitions can easily be extended to finite samples.
Two very commonly used loss functions are the squared loss, () =, and the absolute loss, () = | |.The squared loss function results in an arithmetic mean-unbiased estimator, and the absolute-value loss function results in a median-unbiased estimator (in the one-dimensional case, and a geometric median-unbiased estimator for the multi-dimensional case).