In statistics, the **bias** (or **bias function**) of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called **unbiased**. In statistics, "bias" is an **objective** property of an estimator. Bias can also be measured with respect to the median, rather than the mean (expected value), in which case one distinguishes *median*-unbiased from the usual *mean*-unbiasedness property. Bias is a distinct concept from consistency. Consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased; see bias versus consistency for more.

- Definition
- Examples
- Sample variance
- Estimating a Poisson probability
- Maximum of a discrete uniform distribution
- Median-unbiased estimators
- Bias with respect to other loss functions
- Effect of transformations
- Bias, variance and mean squared error
- Example: Estimation of population variance
- Bayesian view
- See also
- Notes
- References
- External links

All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators (with generally small bias) are frequently used. When a biased estimator is used, bounds of the bias are calculated. A biased estimator may be used for various reasons: because an unbiased estimator does not exist without further assumptions about a population; because an estimator is difficult to compute (as in unbiased estimation of standard deviation); because an estimator is median-unbiased but not mean-unbiased (or the reverse); because a biased estimator gives a lower value of some loss function (particularly mean squared error) compared with unbiased estimators (notably in shrinkage estimators); or because in some cases being unbiased is too strong a condition, and the only unbiased estimators are not useful.

Further, mean-unbiasedness is not preserved under non-linear transformations, though median-unbiasedness is (see § Effect of transformations); for example, the sample variance is a biased estimator for the population variance. These are all illustrated below.

Suppose we have a statistical model, parameterized by a real number *θ*, giving rise to a probability distribution for observed data, , and a statistic which serves as an estimator of *θ* based on any observed data . That is, we assume that our data follow some unknown distribution (where *θ* is a fixed, unknown constant that is part of this distribution), and then we construct some estimator that maps observed data to values that we hope are close to *θ*. The **bias** of relative to is defined as^{ [1] }

where denotes expected value over the distribution (i.e., averaging over all possible observations ). The second equation follows since *θ* is measurable with respect to the conditional distribution .

An estimator is said to be **unbiased** if its bias is equal to zero for all values of parameter *θ*, or equivalently, if the expected value of the estimator matches that of the parameter.^{ [2] }

In a simulation experiment concerning the properties of an estimator, the bias of the estimator may be assessed using the mean signed difference.

The sample variance of a random variable demonstrates two aspects of estimator bias: firstly, the naive estimator is biased, which can be corrected by a scale factor; second, the unbiased estimator is not optimal in terms of mean squared error (MSE), which can be minimized by using a different scale factor, resulting in a biased estimator with lower MSE than the unbiased estimator. Concretely, the naive estimator sums the squared deviations and divides by *n,* which is biased. Dividing instead by *n* − 1 yields an unbiased estimator. Conversely, MSE can be minimized by dividing by a different number (depending on distribution), but this results in a biased estimator. This number is always larger than *n* − 1, so this is known as a shrinkage estimator, as it "shrinks" the unbiased estimator towards zero; for the normal distribution the optimal value is *n* + 1.

Suppose *X*_{1}, ..., *X*_{n} are independent and identically distributed (i.i.d.) random variables with expectation *μ* and variance *σ*^{2}. If the sample mean and uncorrected sample variance are defined as

then *S*^{2} is a biased estimator of *σ*^{2}, because

To continue, we note that by subtracting from both sides of , we get

Meaning, (by cross-multiplication) . Then, the previous becomes:

This can be seen by noting the following formula, which follows from the Bienaymé formula, for the term in the inequality for the expectation of the uncorrected sample variance above: .

In other words, the expected value of the uncorrected sample variance does not equal the population variance *σ*^{2}, unless multiplied by a normalization factor. The sample mean, on the other hand, is an unbiased^{ [3] } estimator of the population mean *μ*.^{ [2] }

Note that the usual definition of sample variance is , and this is an unbiased estimator of the population variance.

Algebraically speaking, is unbiased because:

where the transition to the second line uses the result derived above for the biased estimator. Thus , and therefore is an unbiased estimator of the population variance, *σ*^{2}. The ratio between the biased (uncorrected) and unbiased estimates of the variance is known as Bessel's correction.

The reason that an uncorrected sample variance, *S*^{2}, is biased stems from the fact that the sample mean is an ordinary least squares (OLS) estimator for *μ*: is the number that makes the sum as small as possible. That is, when any other number is plugged into this sum, the sum can only increase. In particular, the choice gives,

and then

The above discussion can be understood in geometric terms: the vector can be decomposed into the "mean part" and "variance part" by projecting to the direction of and to that direction's orthogonal complement hyperplane. One gets for the part along and for the complementary part. Since this is an orthogonal decomposition, Pythagorean theorem says , and taking expectations we get , as above (but times ). If the distribution of is rotationally symmetric, as in the case when are sampled from a Gaussian, then on average, the dimension along contributes to equally as the directions perpendicular to , so that and . This is in fact true in general, as explained above.

A far more extreme case of a biased estimator being better than any unbiased estimator arises from the Poisson distribution.^{ [4] }^{ [5] } Suppose that *X* has a Poisson distribution with expectation *λ*. Suppose it is desired to estimate

with a sample of size 1. (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.)

Since the expectation of an unbiased estimator *δ*(*X*) is equal to the estimand, i.e.

the only function of the data constituting an unbiased estimator is

To see this, note that when decomposing e^{−λ} from the above expression for expectation, the sum that is left is a Taylor series expansion of e^{−λ} as well, yielding e^{−λ}e^{−λ} = e^{−2λ} (see Characterizations of the exponential function).

If the observed value of *X* is 100, then the estimate is 1, although the true value of the quantity being estimated is 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 must be positive.

The (biased) maximum likelihood estimator

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

is smaller; compare the unbiased estimator's MSE of

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

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* given *n* is only (*n* + 1)/2; we can be certain only that *n* is at least *X* and is probably more. In this case, the natural unbiased estimator is 2*X* − 1.

The theory of median-unbiased estimators was revived by George W. Brown in 1947:^{ [6] }

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 requirement seems for most purposes to accomplish as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation.

Further properties of median-unbiased estimators have been noted by Lehmann, Birnbaum, van der Vaart and Pfanzagl.^{[ citation needed ]} In particular, median-unbiased estimators exist in cases where mean-unbiased and maximum-likelihood estimators do not exist. They are invariant under one-to-one transformations.

There are methods of construction median-unbiased estimators for probability distributions that have monotone likelihood-functions, such as one-parameter exponential families, to ensure that they are optimal (in a sense analogous to minimum-variance property considered for mean-unbiased estimators).^{ [7] }^{ [8] } One such procedure is an analogue of the Rao–Blackwell procedure for mean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao–Blackwell procedure for mean-unbiased estimation but for a larger class of loss-functions.^{ [8] }

Any minimum-variance *mean*-unbiased estimator minimizes the risk (expected loss) with respect to the squared-error loss function (among mean-unbiased estimators), as observed by Gauss.^{ [9] } A minimum-average absolute deviation * median *-unbiased estimator minimizes the risk with respect to the absolute loss function (among median-unbiased estimators), as observed by Laplace.^{ [9] }^{ [10] } Other loss functions are used in statistics, particularly in robust statistics.^{ [9] }^{ [11] }

As stated above, for univariate parameters, median-unbiased estimators remain median-unbiased under transformations that preserve order (or reverse order).

Note that, when a transformation is applied to a mean-unbiased estimator, the result need not be a mean-unbiased estimator of its corresponding population statistic. By Jensen's inequality, a convex function as transformation will introduce positive bias, while a concave function will introduce negative bias, and a function of mixed convexity may introduce bias in either direction, depending on the specific function and distribution. That is, for a non-linear function *f* and a mean-unbiased estimator *U* of a parameter *p*, the composite estimator *f*(*U*) need not be a mean-unbiased estimator of *f*(*p*). For example, the square root of the unbiased estimator of the population variance is *not* a mean-unbiased estimator of the population standard deviation: the square root of the unbiased sample variance, the corrected sample standard deviation, is biased. The bias depends both on the sampling distribution of the estimator and on the transform, and can be quite involved to calculate – see unbiased estimation of standard deviation for a discussion in this case.

While bias quantifies the *average* difference to be expected between an estimator and an underlying parameter, an estimator based on a finite sample can additionally be expected to differ from the parameter due to the randomness in the sample.

One measure which is used to try to reflect both types of difference is the mean square error,^{ [1] }

This can be shown to be equal to the square of the bias, plus the variance:^{ [1] }

When the parameter is a vector, an analogous decomposition applies:^{ [12] }

where

is the trace of the covariance matrix of the estimator.

An estimator that minimises the bias will not necessarily minimise the mean square error.

For example,^{ [13] } suppose an estimator of the form

is sought for the population variance as above, but this time to minimise the MSE:

If the variables *X*_{1} ... *X*_{n} follow a normal distribution, then *nS*^{2}/σ^{2} has a chi-squared distribution with *n* − 1 degrees of freedom, giving:

and so

With a little algebra it can be confirmed that it is *c* = 1/(*n* + 1) which minimises this combined loss function, rather than *c* = 1/(*n* − 1) which minimises just the bias term.

More generally it is only in restricted classes of problems that there will be an estimator that minimises the MSE independently of the parameter values.

However it is very common that there may be perceived to be a * bias–variance tradeoff *, such that a small increase in bias can be traded for a larger decrease in variance, resulting in a more desirable estimator overall.

Most bayesians are rather unconcerned about unbiasedness (at least in the formal sampling-theory sense above) of their estimates. For example, Gelman and coauthors (1995) write: "From a Bayesian perspective, the principle of unbiasedness is reasonable in the limit of large samples, but otherwise it is potentially misleading."^{ [14] }

Fundamentally, the difference between the Bayesian approach and the sampling-theory approach above is that in the sampling-theory approach the parameter is taken as fixed, and then probability distributions of a statistic are considered, based on the predicted sampling distribution of the data. For a Bayesian, however, it is the *data* which are known, and fixed, and it is the unknown parameter for which an attempt is made to construct a probability distribution, using Bayes' theorem:

Here the second term, the likelihood of the data given the unknown parameter value θ, depends just on the data obtained and the modelling of the data generation process. However a Bayesian calculation also includes the first term, the prior probability for θ, which takes account of everything the analyst may know or suspect about θ *before* the data comes in. This information plays no part in the sampling-theory approach; indeed any attempt to include it would be considered "bias" away from what was pointed to purely by the data. To the extent that Bayesian calculations include prior information, it is therefore essentially inevitable that their results will not be "unbiased" in sampling theory terms.

But the results of a Bayesian approach can differ from the sampling theory approach even if the Bayesian tries to adopt an "uninformative" prior.

For example, consider again the estimation of an unknown population variance σ^{2} of a Normal distribution with unknown mean, where it is desired to optimise *c* in the expected loss function

A standard choice of uninformative prior for this problem is the Jeffreys prior, , which is equivalent to adopting a rescaling-invariant flat prior for **ln(σ ^{2})**.

One consequence of adopting this prior is that *S*^{2}/σ^{2} remains a pivotal quantity, i.e. the probability distribution of *S*^{2}/σ^{2} depends only on *S*^{2}/σ^{2}, independent of the value of *S*^{2} or σ^{2}:

However, while

in contrast

— when the expectation is taken over the probability distribution of σ^{2} given *S*^{2}, as it is in the Bayesian case, rather than *S*^{2} given σ^{2}, one can no longer take σ^{4} as a constant and factor it out. The consequence of this is that, compared to the sampling-theory calculation, the Bayesian calculation puts more weight on larger values of σ^{2}, properly taking into account (as the sampling-theory calculation cannot) that under this squared-loss function the consequence of underestimating large values of σ^{2} is more costly in squared-loss terms than that of overestimating small values of σ^{2}.

The worked-out Bayesian calculation gives a scaled inverse chi-squared distribution with *n* − 1 degrees of freedom for the posterior probability distribution of σ^{2}. The expected loss is minimised when *cnS*^{2} = <σ^{2}>; this occurs when *c* = 1/(*n* − 3).

Even with an uninformative prior, therefore, a Bayesian calculation may not give the same expected-loss minimising result as the corresponding sampling-theory calculation.

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In probability theory and statistics, **skewness** is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.

In probability theory and statistics, **variance** is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .

The **weighted arithmetic mean** is similar to an ordinary arithmetic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

In statistics, **maximum likelihood estimation** (**MLE**) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

In probability theory, the **law of large numbers** (**LLN**) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and will tend to become closer to the expected value as more trials are performed.

In statistics, the **mean squared error** (**MSE**) or **mean squared deviation** (**MSD**) of an estimator measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive is because of randomness or because the estimator does not account for information that could produce a more accurate estimate.

In statistics and optimization, **errors** and **residuals** are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "theoretical value". The **error** of an observed value is the deviation of the observed value from the (unobservable) *true* value of a quantity of interest, and the **residual** of an observed value is the difference between the observed value and the *estimated* value of the quantity of interest. The distinction is most important in regression analysis, where the concepts are sometimes called the **regression errors** and **regression residuals** and where they lead to the concept of studentized residuals.

In estimation theory and statistics, the **Cramér–Rao bound** (**CRB**) expresses a lower bound on the variance of unbiased estimators of a deterministic parameter, stating that the variance of any such estimator is at least as high as the inverse of the Fisher information. The result is named in honor of Harald Cramér and C. R. Rao, but has independently also been derived by Maurice Fréchet, Georges Darmois, as well as Alexander Aitken and Harold Silverstone.

In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. **Estimation of covariance matrices** then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in **R**^{p×p}; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has normal distribution, the sample covariance matrix has Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data require deeper considerations. Another issue is the robustness to outliers, to which sample covariance matrices are highly sensitive.

**Directional statistics** is the subdiscipline of statistics that deals with directions, axes or rotations in **R**^{n}. More generally, directional statistics deals with observations on compact Riemannian manifolds including the Stiefel manifold.

In statistics, a **consistent estimator** or **asymptotically consistent estimator** is an estimator—a rule for computing estimates of a parameter *θ*_{0}—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to *θ*_{0}. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to *θ*_{0} converges to one.

In statistics a **minimum-variance unbiased estimator (MVUE)** or **uniformly minimum-variance unbiased estimator (UMVUE)** is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

In probability theory and directional statistics, the **von Mises distribution** is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle on a circle is a wrapped normally distributed random variable with an unwrapped variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation. The von Mises distribution is the maximum entropy distribution for circular data when the real and imaginary parts of the first circular moment are specified. The von Mises distribution is a special case of the von Mises–Fisher distribution on the *N*-dimensional sphere.

In statistics, a **pivotal quantity** or **pivot** is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters. A pivot quantity need not be a statistic—the function and its *value* can depend on the parameters of the model, but its *distribution* must not. If it is a statistic, then it is known as an *ancillary statistic.*

In estimation theory and decision theory, a **Bayes estimator** or a **Bayes action** is an estimator or decision rule that minimizes the posterior expected value of a loss function. Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation.

In statistics and in particular statistical theory, **unbiased estimation of a standard deviation** is the calculation from a statistical sample of an estimated value of the standard deviation of a population of values, in such a way that the expected value of the calculation equals the true value. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the use of significance tests and confidence intervals, or by using Bayesian analysis.

In statistics, **Bessel's correction** is the use of *n* − 1 instead of *n* in the formula for the sample variance and sample standard deviation, where *n* is the number of observations in a sample. This method corrects the bias in the estimation of the population variance. It also partially corrects the bias in the estimation of the population standard deviation. However, the correction often increases the mean squared error in these estimations. This technique is named after Friedrich Bessel.

In statistics, **Stein's unbiased risk estimate (SURE)** is an unbiased estimator of the mean-squared error of "a nearly arbitrary, nonlinear biased estimator." In other words, it provides an indication of the accuracy of a given estimator. This is important since the true mean-squared error of an estimator is a function of the unknown parameter to be estimated, and thus cannot be determined exactly.

In probability theory and directional statistics, a **wrapped normal distribution** is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion and is a solution to the heat equation for periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.

In the comparison of various statistical procedures, **efficiency** is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator, experiment, or test needs fewer observations than a less efficient one to achieve a given performance. This article primarily deals with efficiency of estimators.

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