The average of all the sample absolute deviations about the mean of size 3 that can be drawn from the population is 44/81, while the average of all the sample absolute deviations about the median is 4/9. Therefore, the absolute deviation is a biased estimator. However, this argument is based on the notion of mean-unbiasedness. Each measure of location has its own form of unbiasedness (see entry on biased estimator).
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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 ); for example, the sample variance is a biased estimator for the population variance. These are all illustrated below.
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Given strong evidence that CIRP holds, the forward rate unbiasedness hypothesis can serve as a test to determine whether UIRP holds (in order for the forward rate and expected spot rate to be equal, both CIRP and UIRP conditions must hold). Evidence for the validity and accuracy of the unbiasedness hypothesis, particularly evidence for cointegration between the forward rate and future spot rate, is mixed as researchers have published numerous papers demonstrating both empirical support and empirical failure of the hypothesis. UIRP is found to have some empirical support in tests for correlation between expected rates of currency depreciation and the forward premium or discount. Evidence suggests that whether UIRP holds depends on the currency examined, and deviations from UIRP have been found to be less substantial when examining longer time horizons.
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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.
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The attractiveness of different estimators can be judged by looking at their properties, such as unbiasedness, mean square error, consistency, asymptotic distribution, etc. The construction and comparison of estimators are the subjects of the estimation theory. 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.
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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." 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: :p(\theta \mid D, I) \propto p(\theta \mid I) p(D \mid \theta, I) 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.
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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. Also, an estimator's being biased does not preclude the error of an estimate from being zero in a particular instance. 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, have few outliers). Yet unbiasedness is not essential.
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In ethics, Stegmaier distinguishes between moral orientation, as a self-binding commitment due to certain norms and values, and ethical orientation, as the reflection of such self-bindings and the forgoing of reciprocity and universality.Ibid., pp. 238-240. In this way, he obtains the rightful place for virtues, which are greatly appreciated by everyone, but which moral philosophers have so far less taken into account, such as open- mindedness and unbiasedness, benevolence and friendliness, tactfulness, nobility, and goodness.Ibid., pp. 240-245.
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He was elected President of the Institute of Mathematical Statistics in 1962. Anderson's 1958 textbook, An Introduction to Multivariate Analysis, educated a generation of theorists and applied statisticians; it was "the classic" in the area until the book by Mardia, Kent and Bibby . Anderson's book emphasizes hypothesis testing via likelihood ratio tests and the properties of power functions: Admissibility, unbiasedness and monotonicity.(Pages 560–561) Anderson is also known for Anderson–Darling test of whether there is evidence that a given sample of data did not arise from a given probability distribution.
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On the basis of asymptotical unbiasedness, a moderated version of the rational expectations hypothesis can be suggested in which familiarity with the theoretical parameters is not a requirement for the relevant model. An agent with access to sufficiently vast, quality information and high-level methodological skills could specify its own quasi-relevant model describing a specific macroeconomic system. By increasing the amount of information processed, this agent could further reduce its bias. If this agent were also focal, such as a central bank, then other agents would likely accept the proposed model and adjust their expectations accordingly.
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Econometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods. Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. 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 the 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.
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The concept of invariance is sometimes used on its own as a way of choosing between estimators, but this is not necessarily definitive. For example, a requirement of invariance may be incompatible with the requirement that the estimator be mean-unbiased; on the other hand, the criterion of median-unbiasedness is defined in terms of the estimator's sampling distribution and so is invariant under many transformations. One use of the concept of invariance is where a class or family of estimators is proposed and a particular formulation must be selected amongst these. One procedure is to impose relevant invariance properties and then to find the formulation within this class that has the best properties, leading to what is called the optimal invariant estimator.
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The forward premium anomaly in currency markets (also referred to as the forward premium puzzle or the Fama puzzle) refers to the well documented empirical finding that the domestic currency appreciates when domestic nominal interest rates exceed foreign interest rates. This is perceived as puzzling in the context of the hypothesis that the expected future change in the exchange rate between two countries is equal to the interest-rate differential between these two countries; this hypothesis suggests that if all currencies are equally risky, investors would demand higher interest rates on currencies expected to fall in value. See: Forward exchange rate# Unbiasedness hypothesis. Thus, appreciation of the domestic currency when domestic interest rates are greater than foreign interest rates is called an anomaly.
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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. For practical statistics problems, it is important to determine the MVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal. This has led to substantial development of statistical theory related to the problem of optimal estimation. While combining the constraint of unbiasedness with the desirability metric of least variance leads to good results in most practical settings—making MVUE a natural starting point for a broad range of analyses—a targeted specification may perform better for a given problem; thus, MVUE is not always the best stopping point.
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On 2 August 2010, a disciplinary case was brought against Venckiene by the Judicial Discipline and Ethics Commission. It was noted that "speaking with the press, also publicly criticizing the improper pre-trial investigation, and the institutions of the pre-trial investigation, stating a negative opinion about other people, and publicly accusing them, also writing complaints in her brother's name, and using disrespectful words to describe people about who she wrote documents, violated the Judicial ethical code of conduct". In 2011, the Judicial Discipline and Ethics Commission gave a warning to Venckiene, because "Venckiene in her actions and statements in the public media violated Judges etiquette code of conduct regarding respecting a person, loyalty to the country, unbiasedness, selflessness, respect and example principles". In April 2012, G.Kryzevicius, the chairman of the Supreme Court of Lithuania, called Venckiene "an abscess in judicial and political system" and "the trouble of the whole state". Protest to support Venckiene, May 26, 2012 On 23 May, General Prosector asked the Parliament to remove Venckiene's legal immunity on 5 charges.
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