Bias of an estimator

In statistics, the difference between an estimator's expected value and the true value of the parameter being estimated is called the bias. An estimator or decision rule having nonzero bias is said to be biased.

Although the term bias sounds pejorative, it is not necessarily used in that way in statistics. Biased estimators may have desirable properties. Not only do they sometimes have a smaller mean squared error than any unbiased estimator, but in some cases the only unbiased estimators are not even within the convex hull of the parameter space, so their use is absurd.

Definition

Suppose we are trying to estimate the parameter Failed to parse (Missing texvc executable; please see math/README to configure.): \theta \

using an estimator Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{\theta}
(that is, some function of the observed data).  Then the bias of Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{\theta}
is defined to be

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(\widehat{\theta})-\theta.\,

In words, this would be "the expected value of the estimator Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{\theta}

minus the true value  Failed to parse (Missing texvc executable; please see math/README to configure.):  \theta \ 

." This may be rewritten as

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(\widehat{\theta}-\theta).\,

which would read "the expected value of the difference between the estimator and the true value" (the expected value of Failed to parse (Missing texvc executable; please see math/README to configure.): \theta \

is precisely Failed to parse (Missing texvc executable; please see math/README to configure.):  \theta \ 
).

Examples

Estimating variance

Suppose X₁, ..., X_n are independent and identically distributed normal random variables with expectation μ and variance σ². Let

Failed to parse (Missing texvc executable; please see math/README to configure.): \overline{X}=(X_1+\cdots+X_n)/n

be the "sample average", and let

Failed to parse (Missing texvc executable; please see math/README to configure.): S^2=\frac{1}{n}\sum_{i=1}^n(X_i-\overline{X}\,)^2

be a "sample variance". We also know that the variance σ² is defined by:

Failed to parse (Missing texvc executable; please see math/README to configure.): {}\sigma^2 = \frac 1N \sum_{i=1}^N \left(x_i - \overline{x} \right)^ 2 \,

where N is the population size and x_i represents the member of the whole population.

Then S² is a "biased estimator" of σ² because

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(S^2)=\frac{N}{n-1}\sigma^2\neq\sigma^2.

In other words, the sample variance does not equal the population variance, unless multiplied by the normalization factor.

Common sense would suggest to apply the population formula to the sample as well. The reason that it is biased is that the sample mean is generally somewhat closer to the observations in the sample than the population mean is, to these observations. This is so because the sample mean is, by definition, in the middle of the sample, while the population mean may even lie outside the sample. So the deviations to the sample mean will often be smaller than the deviations to the population mean, and so, if the same formula is applied to both, then this variance estimate will on average be somewhat smaller in the sample than in the population.

Note that when a transformation is applied to an unbiased estimator, the result is not necessarily itself an unbiased estimate of its corresponding population statistic. That is, for a non-linear function f and an unbiased estimator U of a parameter p, f(U) is usually not an unbiased estimator of f(p). For example the square root of the unbiased estimator of the population variance is not an unbiased estimator of the population standard deviation.

Bias, however, is not the only consideration when choosing a statistic. Bias refers to the central tendency of the sampling distribution of a statistic, but the variance of the sampling distribution can also be an important consideration. Specifically, statistics with smaller sampling variances will yield greater statistical power. For example, while S² above is more biased than the traditional sample calculation

Failed to parse (Missing texvc executable; please see math/README to configure.): S_\mathrm{unbiased}^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\overline{X}\,)^2,

S² has a lower estimation variability than S²_unbiased because the denominator dividing the sum of squares is larger in the calculation of S², resulting in a smaller scale of final values, and therefore lower estimation variability, than that of S²_unbiased. Practically, this demonstrates that for some applications (where the amount of bias can be equated between groups/conditions) it is possible that a biased estimator can prove to be a more powerful, and therefore useful, statistic. The use of n − 1 rather than n is sometimes called Bessel's correction.

Estimating a Poisson probability

A far more extreme case of a biased estimator being better than any unbiased estimator is well-known: Suppose X has a Poisson distribution with expectation λ. It is desired to estimate

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{P}(X=0)^2=e^{-2\lambda}.\quad

(For example, when incoming calls at a telephone switchboard are modeled as a Poisson process, and λ is the average number of calls per minute, then e^−2λ is the probability that no calls arrive in the next two minutes.)

The only function of the data constituting an unbiased estimator is

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(X)=(-1)^X.\quad

If the observed value of X is 100, then the estimate is 1, although the true value of the quantity being estimated is obviously very likely to be near 0, which is the opposite extreme. And if X is observed to be 101, then the estimate is even more absurd: it is −1, although the quantity being estimated obviously must be positive.

The (biased) maximum likelihood estimator

Failed to parse (Missing texvc executable; please see math/README to configure.): e^{-2X}\quad

is far better than this unbiased estimator. Not only is its value always positive, but it is also more accurate in the sense that its mean squared error (MSE)

Failed to parse (Missing texvc executable; please see math/README to configure.): e^{-4\lambda}-2e^{\lambda(1/e^2-3)}+e^{\lambda(1/e^4-1)}

is smaller; compare the unbiased estimator's MSE of

Failed to parse (Missing texvc executable; please see math/README to configure.): 1-e^{-4\lambda}

The MSEs are a functions of the true value λ. The bias of the maximum-likelihood estimator is:

Failed to parse (Missing texvc executable; please see math/README to configure.): e^{-2\lambda}-e^{\lambda(1/e^2-1)}

.

Maximum of a discrete uniform distribution

The bias of maximum-likelihood estimators can be substantial. Consider a case where n tickets numbered from 1 through to n are placed in a box and one is selected at random, giving a value X. If n is unknown, then the maximum-likelihood estimator of n is X, even though the expectation of X is only (n + 1)/2; we can only be certain that n is at least X and is probably more. In this case, the natural unbiased estimator is 2X − 1.

See also

• Omitted-variable bias
• Consistency (statistics)

External links

• An Illuminating Counterexamplede:Verzerrung (Statistik)