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Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median (or other quantiles) of the response variable.
The theory of median-unbiased estimators was revived by George W. Brown in 1947: [8]. An estimate of a one-dimensional parameter θ will be said to be median-unbiased, if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates.
The Theil–Sen estimator is a simple robust estimation technique that chooses the slope of the fit line to be the median of the slopes of the lines through pairs of sample points. It has similar statistical efficiency properties to simple linear regression but is much less sensitive to outliers .
If the bootstrap distribution of an estimator is symmetric, then percentile confidence-interval are often used; such intervals are appropriate especially for median-unbiased estimators of minimum risk (with respect to an absolute loss function). Bias in the bootstrap distribution will lead to bias in the confidence interval.
The unconditional expectation of rainfall for an unspecified day is the average of the rainfall amounts for those 3652 days. The conditional expectation of rainfall for an otherwise unspecified day known to be (conditional on being) in the month of March, is the average of daily rainfall over all 310 days of the ten–year period that falls in ...
The conditional median occurs in the setting where we seek to estimate a random variable from a random variable , which is a noisy version of . The conditional median in this setting is given by The conditional median in this setting is given by
The median absolute deviation is a measure of statistical dispersion. Moreover, the MAD is a robust statistic , being more resilient to outliers in a data set than the standard deviation . In the standard deviation, the distances from the mean are squared, so large deviations are weighted more heavily, and thus outliers can heavily influence it.
Two basic numerical approaches to obtain the MMSE estimate depends on either finding the conditional expectation {} or finding the minima of MSE. Direct numerical evaluation of the conditional expectation is computationally expensive since it often requires multidimensional integration usually done via Monte Carlo methods .