Statistical assumptionStatistics, like all mathematical disciplines, does not infer valid conclusions from nothing. Inferring interesting conclusions about real statistical populations almost always requires some background assumptions. Those assumptions must be made carefully, because incorrect assumptions can generate wildly inaccurate conclusions. Here are some examples of statistical assumptions: Independence of observations from each other (this assumption is an especially common error). Independence of observational error from potential confounding effects.
Errors-in-variables modelsIn 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.
Ancillary statisticAn ancillary statistic is a measure of a sample whose distribution (or whose pmf or pdf) does not depend on the parameters of the model. An ancillary statistic is a pivotal quantity that is also a statistic. Ancillary statistics can be used to construct prediction intervals. They are also used in connection with Basu's theorem to prove independence between statistics. This concept was first introduced by Ronald Fisher in the 1920s, but its formal definition was only provided in 1964 by Debabrata Basu.
Bonferroni correctionIn statistics, the Bonferroni correction is a method to counteract the multiple comparisons problem. The method is named for its use of the Bonferroni inequalities. An extension of the method to confidence intervals was proposed by Olive Jean Dunn. Statistical hypothesis testing is based on rejecting the null hypothesis if the likelihood of the observed data under the null hypotheses is low. If multiple hypotheses are tested, the probability of observing a rare event increases, and therefore, the likelihood of incorrectly rejecting a null hypothesis (i.
Random elementIn probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.
Noether's theoremNoether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.
