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A consistent estimator is an estimator whose sequence of estimates converge in probability to the quantity being estimated as the index (usually the sample size) grows without bound. In other words, increasing the sample size increases the probability of the estimator being close to the population parameter.
The James–Stein estimator may seem at first sight to be a result of some peculiarity of the problem setting. In fact, the estimator exemplifies a very wide-ranging effect; namely, the fact that the "ordinary" or least squares estimator is often inadmissible for simultaneous estimation of several parameters.
In statistics, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson, [1] is a method for estimating the total [2] and mean of a pseudo-population in a stratified sample by applying inverse probability weighting to account for the difference in the sampling distribution between the collected data and the target population.
This is one of the motivations of robust statistics – an estimator such as the sample mean is an efficient estimator of the population mean of a normal distribution, for example, but can be an inefficient estimator of a mixture distribution of two normal distributions with the same mean and different variances.
An estimator can be unbiased but not consistent. For example, for an iid sample {x 1,..., x n} one can use T n (X) = x n as the estimator of the mean E[X]. Note that here the sampling distribution of T n is the same as the underlying distribution (for any n, as it ignores all points but the last), so E[T
which is an unbiased estimator of the variance of the mean in terms of the observed sample variance and known quantities. If the autocorrelations are identically zero, this expression reduces to the well-known result for the variance of the mean for independent data. The effect of the expectation operator in these expressions is that the ...
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