Estimator
This is in contrast to an interval estimator, where the result would be a range of plausible values.
However, in robust statistics, statistical theory goes on to consider the balance between having good properties, if tightly defined assumptions hold, and having worse properties that hold under wider conditions.
then the estimator is traditionally written by adding a circumflex over the symbol:
The definition places virtually no restrictions on which functions of the data can be called the "estimators".
The attractiveness of different estimators can be judged by looking at their properties, such as unbiasedness, mean square error, consistency, asymptotic distribution, etc.
The estimate in this case is a single point in the parameter space.
Firstly, in estimating the probability density functions of random variables and secondly in estimating the spectral density function of a time series.
is defined as the expected value (probability-weighted average, over all samples) of the squared errors; that is, It is used to indicate how far, on average, the collection of estimates are from the single parameter being estimated.
Suppose the parameter is the bull's-eye of a target, the estimator is the process of shooting arrows at the target, and the individual arrows are estimates (samples).
Then high MSE means the average distance of the arrows from the bull's eye is high, and low MSE means the average distance from the bull's eye is low.
For example, even if all arrows hit the same point, yet grossly miss the target, the MSE is still relatively large.
Finally, even if all arrows grossly miss the target, if they nevertheless all hit the same point, the variance is zero.
If the parameter is the bull's eye of a target and the arrows are estimates, then a relatively high absolute value for the bias means the average position of the arrows is off-target, and a relatively low absolute bias means the average position of the arrows is on target.
The ideal situation is to have an unbiased estimator with low variance, and also try to limit the number of samples where the error is extreme (that is, to have few outliers).
An alternative to the version of "unbiased" above, is "median-unbiased", where the median of the distribution of estimates agrees with the true value; thus, in the long run half the estimates will be too low and half too high.
For example, if a genetic theory states there is a type of leaf (starchy green) that occurs with probability
To find if your estimator is unbiased it is easy to follow along the equation
The sequence is strongly consistent, if it converges almost surely to the true value.
This occurs frequently in estimation of scale parameters by measures of statistical dispersion.
[5] An asymptotically normal estimator is a consistent estimator whose distribution around the true parameter θ approaches a normal distribution with standard deviation shrinking in proportion to
Note that convergence will not necessarily have occurred for any finite "n", therefore this value is only an approximation to the true variance of the estimator, while in the limit the asymptotic variance (V/n) is simply zero.
To be more specific, the distribution of the estimator tn converges weakly to a dirac delta function centered at
The central limit theorem implies asymptotic normality of the sample mean
More generally, maximum likelihood estimators are asymptotically normal under fairly weak regularity conditions — see the asymptotics section of the maximum likelihood article.
However, not all estimators are asymptotically normal; the simplest examples are found when the true value of a parameter lies on the boundary of the allowable parameter region.
This equation relates the mean squared error with the estimator bias:[4] The first term represents the mean squared error; the second term represents the square of the estimator bias; and the third term represents the variance of the estimator.
The variance of the good estimator (good efficiency) would be smaller than the variance of the bad estimator (bad efficiency).
For example: If an estimator is not efficient, the frequency vs. value graph, there will be a relatively more gentle curve.
Plotting these two curves on one graph with a shared y-axis, the difference becomes more obvious.
In some cases an unbiased efficient estimator exists, which, in addition to having the lowest variance among unbiased estimators, satisfies the Cramér–Rao bound, which is an absolute lower bound on variance for statistics of a variable.
