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Notionally, theoretical adjustments might be obtainable to lead to unbiased estimates but, unlike those for the normal distribution, these would typically depend on the estimated parameters. If the requirement is simply to reduce the bias of an estimated standard deviation, rather than to eliminate it entirely, then two practical approaches are ...
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.
This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.
For a normal distribution with unknown mean and variance, the sample mean and (unbiased) sample variance are the MVUEs for the population mean and population variance. However, the sample standard deviation is not unbiased for the population standard deviation – see unbiased estimation of standard deviation.
The connection of maximum likelihood estimation to OLS arises when this distribution is modeled as a multivariate normal. Specifically, assume that the errors ε have multivariate normal distribution with mean 0 and variance matrix σ 2 I. Then the distribution of y conditionally on X is
For the normal distribution, an unbiased estimator is given by s / c 4 , where the correction factor (which depends on N) is given in terms of the Gamma function, and equals: = (). This arises because the sampling distribution of the sample standard deviation follows a (scaled) chi distribution , and the correction factor is the mean of ...
In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) [1] states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. [2]
Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model). In this case, the errors are the deviations of the observations from the population mean, while the residuals are the deviations of the observations from the sample mean.