Maximum likelihood estimationIn 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.
Density estimationIn statistics, probability density estimation or simply density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function. The unobservable density function is thought of as the density according to which a large population is distributed; the data are usually thought of as a random sample from that population. A variety of approaches to density estimation are used, including Parzen windows and a range of data clustering techniques, including vector quantization.
Estimation theoryEstimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.
EstimatorIn statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the sample mean is a commonly used estimator of the population mean. There are point and interval estimators. The point estimators yield single-valued results. This is in contrast to an interval estimator, where the result would be a range of plausible values.
Sufficient statisticIn statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter". In particular, a statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than the statistic, as to which of those probability distributions is the sampling distribution.
Minimum-variance unbiased estimatorIn 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.
Fisher informationIn mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Ronald Fisher (following some initial results by Francis Ysidro Edgeworth).
Likelihood functionIn statistical inference, the likelihood function quantifies the plausibility of parameter values characterizing a statistical model in light of observed data. Its most typical usage is to compare possible parameter values (under a fixed set of observations and a particular model), where higher values of likelihood are preferred because they correspond to more probable parameter values.
Efficiency (statistics)In statistics, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator needs fewer input data or observations than a less efficient one to achieve the Cramér–Rao bound. An efficient estimator is characterized by having the smallest possible variance, indicating that there is a small deviance between the estimated value and the "true" value in the L2 norm sense.
Normal distributionIn statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.
Exponential distributionIn probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
Bernoulli trialIn the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his Ars Conjectandi (1713). The mathematical formalisation of the Bernoulli trial is known as the Bernoulli process.
Exponential familyIn probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family.
Bernoulli distributionIn probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability . Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q.
Statistical parameterIn statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population exactly follows a known and defined distribution, for example the normal distribution, then a small set of parameters can be measured which completely describes the population, and can be considered to define a probability distribution for the purposes of extracting samples from this population.
ParameterA parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. Parameter has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition.
Software development effort estimationIn software development, effort estimation is the process of predicting the most realistic amount of effort (expressed in terms of person-hours or money) required to develop or maintain software based on incomplete, uncertain and noisy input. Effort estimates may be used as input to project plans, iteration plans, budgets, investment analyses, pricing processes and bidding rounds. Published surveys on estimation practice suggest that expert estimation is the dominant strategy when estimating software development effort.
Likelihood-ratio testIn statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models, specifically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods. If the constraint (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.
Observed informationIn statistics, the observed information, or observed Fisher information, is the negative of the second derivative (the Hessian matrix) of the "log-likelihood" (the logarithm of the likelihood function). It is a sample-based version of the Fisher information. Suppose we observe random variables , independent and identically distributed with density f(X; θ), where θ is a (possibly unknown) vector.
Bias of an estimatorIn 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.
