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In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only ...
Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a n converge to 0 monotonically, but this condition is not necessary for convergence.
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet , and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
Many significance tests have an estimation counterpart; [26] in almost every case, the test result (or its p-value) can be simply substituted with the effect size and a precision estimate. For example, instead of using Student's t-test , the analyst can compare two independent groups by calculating the mean difference and its 95% confidence ...
For this reason, it is common to use statistical software designed to handle to the approach – virtually all modern statistical packages feature this capability. The main approaches to fitting Box–Jenkins models are nonlinear least squares and maximum likelihood estimation. Maximum likelihood estimation is generally the preferred technique.
There are two main uses of the term calibration in statistics that denote special types of statistical inference problems. Calibration can mean a reverse process to regression, where instead of a future dependent variable being predicted from known explanatory variables, a known observation of the dependent variables is used to predict a corresponding explanatory variable; [1]
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This test is an approximate one and assumes that the time-series is Gaussian. In the above, z 1−α/2 is the quantile of the normal distribution; SE is the standard error, which can be computed by Bartlett's formula for MA(ℓ) processes: =