Eta-squared is a biased estimator of the variance explained by the model in the population ( it estimates only the effect size in the sample ).

In fact, the distribution of the sample mean will be equal to the distribution of the samples themselves ; i. e., the sample mean of a large sample is no better ( or worse ) an estimator of x < sub > 0 </ sub > than any single observation from the sample.

One simple method is to take the median value of the sample as an estimator of x < sub > 0 </ sub > and half the sample interquartile range as an estimator of γ.

However, because of the fat tails of the Cauchy distribution, the efficiency of the estimator decreases if more than 24 % of the sample is used.

Also, while the maximum likelihood estimator is asymptotically efficient, it is relatively inefficient for small samples.

The truncated sample mean using the middle 24 % order statistics is about 88 % as asymptotically efficient an estimator of x < sub > 0 </ sub > as the maximum likelihood estimate.

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result ( the estimate ) are distinguished.

This is in contrast to an interval estimator, where the result would be a range of plausible values ( or vectors or functions ).

An " estimator " or " point estimate " is a statistic ( that is, a function of the data ) that is used to infer the value of an unknown parameter in a statistical model.

If the parameter is denoted θ then the estimator is typically written by adding a " hat " over the symbol:.

Being a function of the data, the estimator is itself a random variable ; a particular realization of this random variable is called the " estimate ".

In the context of decision theory, an estimator is a type of decision rule, and its performance may be evaluated through the use of loss functions.

When the word " estimator " is used without a qualifier, it usually refers to point estimation.

Then an " estimator " is a function that maps the sample space to a set of sample estimates.

An estimator of is usually denoted by the symbol.

It is often convenient to express the theory using the algebra of random variables: thus if X is used to denote a random variable corresponding to the observed data, the estimator ( itself treated as a random variable ) is symbolised as a function of that random variable,.

For a given sample, the " error " of the estimator is defined as

Often, people refer to a " biased estimate " or an " unbiased estimate ," but they really are talking about an " estimate from a biased estimator ," or an " estimate from an unbiased estimator.

In fact, even if all estimates have astronomical absolute values for their errors, if the expected value of the error is zero, the estimator is unbiased.

The ideal situation, of course, 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, have few outliers ).

In particular, for an unbiased estimator, the variance equals the MSE.

An estimator is unbiased if its expected value is the true value of the parameter ; It is consistent if it converges to the true value as sample size gets larger, and it is efficient if the estimator has lower standard error than other unbiased estimators for a given sample size.

Ordinary least squares ( OLS ) is often used for estimation since it provides the BLUE or " best linear unbiased estimator " ( where " best " means most efficient, unbiased estimator ) given the Gauss-Markov assumptions.

The Gauss-Markov theorem shows that the OLS estimator is the best ( minimum variance ), unbiased estimator assuming the model is linear, the expected value of the error term is zero, errors are homoskedastic and not autocorrelated, and there is no perfect multicollinearity.

The OLS estimator remains unbiased, however.

where is the unique symmetric unbiased estimator of the third cumulant and is the symmetric unbiased estimator of the second cumulant.

While the first one may be seen as the variance of the sample considered as a population, the second one is the unbiased estimator of the population variance, meaning that its expected value E is equal to the true variance of the sampled random variable ; the use of the term n − 1 is called Bessel's correction.

Often, if just a little bias is permitted, then an estimator can be found with lower MSE and / or fewer outlier sample estimates.

* In statistics, a statistic is called complete if it does not allow an unbiased estimator of zero.

Least squares corresponds to the maximum likelihood criterion if the experimental errors have a normal distribution and can also be derived as a method of moments estimator.

If the errors are correlated, the resulting estimator is BLUE if the weight matrix is equal to the inverse of the variance-covariance matrix of the observations.

Even among unbiased estimators, if the distribution is not Gaussian the best ( minimum mean square error ) estimator of the variance may not be

Formally, if T is a complete sufficient statistic for θ and E ( g ( T )) = τ ( θ ) then g ( T ) is the minimum-variance unbiased estimator ( MVUE ) of τ ( θ ).

The Rao – Blackwell theorem states that if g ( X ) is any kind of estimator of a parameter θ, then the conditional expectation of g ( X ) given T ( X ), where T is a sufficient statistic, is typically a better estimator of θ, and is never worse.

The improved estimator is unbiased if and only if the original estimator is unbiased, as may be seen at once by using the law of total expectation.

Note that the error, e, depends not only on the estimator ( the estimation formula or procedure ), but on the sample.

Note that the sampling deviation, d, depends not only on the estimator, but on the sample.

The theorem states that any estimator which is unbiased for a given unknown quantity and which is based on only a complete, sufficient statistic ( and on no other data-derived values ) is the unique best unbiased estimator of that quantity.

The theorem seems very weak: it says only that the Rao – Blackwell estimator is no worse than the original estimator.

) More commonly, however, the expected value ( mean or average ) of the sampled values is chosen ; this is a Bayes estimator that takes advantage of the additional data about the entire distribution that is available from Bayesian sampling, whereas a maximization algorithm such as expectation maximization ( EM ) is capable of only returning a single point from the distribution.

Generally, instrumental variables estimators only have desirable asymptotic, not finite sample, properties, and inference is based on asymptotic approximations to the sampling distribution of the estimator.

* The orthogonality principle: An estimator is MMSE if and only if

It is only on average that the new estimator is better.

The reason that this estimator is useful is that the inverse regression line is actually unaffected by the Malmquist Bias, so long as the selection effects are only based on magnitude.