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James–Stein Estimator
The James–Stein estimator is a biased estimator of the mean, \boldsymbol\theta, of (possibly) correlated Gaussian distributed random vectors Y = \ with unknown means \. It arose sequentially in two main published papers, the earlier version of the estimator was developed by Charles Stein in 1956, which reached a relatively shocking conclusion that while the then usual estimate of the mean, or the sample mean written by Stein and James as (Y_i) = , is admissible when m \leq 2, however it is inadmissible when m \geq 3 and proposed a possible improvement to the estimator that shrinks the sample means towards a more central mean vector \boldsymbol\nu (which can be chosen a priori or commonly the "average of averages" of the sample means given all samples share the same size), is commonly referred to as Stein's example or paradox. This earlier result was improved later by Willard James and Charles Stein in 1961 through simplifying the original process. It can be shown that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bias Of An Estimator
In statistics, the bias of an estimator (or bias function) 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 property of an estimator. 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. 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 a biased estimato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance Matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2 \times 2 matrix would be necessary to fully characterize the two-dimensional variation. The covariance matrix of a random vector \mathbf is typically denoted by \operatorname_ or \Sigma. Definition Throughout this article, boldfaced unsubsc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurement Error Model
In statistics, errors-in-variables models or measurement error models are regression models that account for measurement errors in the independent variables. In contrast, standard regression models assume that those regressors have been measured exactly, or observed without error; as such, those models account only for errors in the dependent variables, or responses. In the case when some regressors have been measured with errors, estimation based on the standard assumption leads to inconsistent estimates, meaning that the parameter estimates do not tend to the true values even in very large samples. For simple linear regression the effect is an underestimate of the coefficient, known as the '' attenuation bias''. In non-linear models the direction of the bias is likely to be more complicated. Motivating example Consider a simple linear regression model of the form : y_ = \alpha + \beta x_^ + \varepsilon_t\,, \quad t=1,\ldots,T, where x_^ denotes the ''true'' but uno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francis Galton
Sir Francis Galton, FRS FRAI (; 16 February 1822 – 17 January 1911), was an English Victorian era polymath: a statistician, sociologist, psychologist, anthropologist, tropical explorer, geographer, inventor, meteorologist, proto-geneticist, psychometrician and a proponent of social Darwinism, eugenics, and scientific racism. He was knighted in 1909. Galton produced over 340 papers and books. He also created the statistical concept of correlation and widely promoted regression toward the mean. He was the first to apply statistical methods to the study of human differences and inheritance of intelligence, and introduced the use of questionnaires and surveys for collecting data on human communities, which he needed for genealogical and biographical works and for his anthropometric studies. He was a pioneer of eugenics, coining the term itself in 1883, and also coined the phrase " nature versus nurture". His book ''Hereditary Genius'' (1869) was the first social sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropic Uncertainty Principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, ''x'', and momentum, ''p'', can be predicted from initial conditions. Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified. Introduced first in 1927 by the German physicist Werner ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Channel Estimation
In wireless communications, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for example, scattering, fading, and power decay with distance. The method is called Channel estimation. The CSI makes it possible to adapt transmissions to current channel conditions, which is crucial for achieving reliable communication with high data rates in multiantenna systems. CSI needs to be estimated at the receiver and usually quantized and feedback to the transmitter (although reverse-link estimation is possible in TDD systems). Therefore, the transmitter and receiver can have different CSI. The CSI at the transmitter and the CSI at the receiver are sometimes referred to as CSIT and CSIR, respectively. Different kinds of channel state information There are basically two levels of CSI, namely instantaneous CSI and statisti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communication Channel
A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used for information transfer of, for example, a digital bit stream, from one or several '' senders'' to one or several '' receivers''. A channel has a certain capacity for transmitting information, often measured by its bandwidth in Hz or its data rate in bits per second. Communicating an information signal across distance requires some form of pathway or medium. These pathways, called communication channels, use two types of media: Transmission line (e.g. twisted-pair, coaxial, and fiber-optic cable) and broadcast (e.g. microwave, satellite, radio, and infrared). In information theory, a channel refers to a theoretical ''channel model'' with certain error characteristics. In this more general view, a storage device is also a communication channel, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Telecommunication
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that feasible with the human voice, but with a similar scale of expediency; thus, slow systems (such as postal mail) are excluded from the field. The transmission media in telecommunication have evolved through numerous stages of technology, from beacons and other visual signals (such as smoke signals, semaphore telegraphs, signal flags, and optical heliographs), to electrical cable and electromagnetic radiation, including light. Such transmission paths are often divided into communication channels, which afford the advantages of multiplexing multiple concurrent communication sessions. ''Telecommunication'' is often used in its plural form. Other examples of pre-modern long-distance communication included audio messages, such as coded drumb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, maximizing a likelihood function so that, under the assumed statistical model, the Realization (probability), observed data is most probable. The point estimate, 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. If the likelihood function is Differentiable function, differentiable, the derivative test for finding maxima can be applied. In some cases, the first-order conditions of the likelihood function can be solved analytically; for instance, the ordinary least squares estimator for a linear regression model maximizes the likelihood when ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prior Probability
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable. Bayes' theorem calculates the renormalized pointwise product of the prior and the likelihood function, to produce the ''posterior probability distribution'', which is the conditional distribution of the uncertain quantity given the data. Similarly, the prior probability of a random event or an uncertain proposition is the unconditional probability that is assigned before any relevant evidence is taken into account. Priors can be created using a num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Bayes Method
Empirical Bayes methods are procedures for statistical inference in which the prior probability distribution is estimated from the data. This approach stands in contrast to standard Bayesian methods, for which the prior distribution is fixed before any data are observed. Despite this difference in perspective, empirical Bayes may be viewed as an approximation to a fully Bayesian treatment of a hierarchical model wherein the parameters at the highest level of the hierarchy are set to their most likely values, instead of being integrated out. Empirical Bayes, also known as maximum marginal likelihood, represents a convenient approach for setting hyperparameters, but has been mostly supplanted by fully Bayesian hierarchical analyses since the 2000s with the increasing availability of well-performing computation techniques. Introduction Empirical Bayes methods can be seen as an approximation to a fully Bayesian treatment of a hierarchical Bayes model. In, for example, a two-stag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Admissible Decision Rule
In statistical decision theory, an admissible decision rule is a rule for making a decision such that there is no other rule that is always "better" than it (or at least sometimes better and never worse), in the precise sense of "better" defined below. This concept is analogous to Pareto efficiency. Definition Define sets \Theta\,, \mathcal and \mathcal, where \Theta\, are the states of nature, \mathcal the possible observations, and \mathcal the actions that may be taken. An observation x \in \mathcal\,\! is distributed as F(x\mid\theta)\,\! and therefore provides evidence about the state of nature \theta\in\Theta\,\!. A decision rule is a function \delta:\rightarrow , where upon observing x\in \mathcal, we choose to take action \delta(x)\in \mathcal\,\!. Also define a loss function L: \Theta \times \mathcal \rightarrow \mathbb, which specifies the loss we would incur by taking action a \in \mathcal when the true state of nature is \theta \in \Theta. Usually we will take thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
