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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.
Parameter estimation using computation algorithms to arrive at coefficients that best fit the selected ARIMA model. The most common methods use maximum likelihood estimation or non-linear least-squares estimation. Statistical model checking by testing whether the estimated model conforms to the specifications of a stationary univariate process ...
In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares problem where the coefficients are not allowed to become negative.
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:
To estimate the area under a curve the trapezoid rule is applied first to one-piece, then two, then four, and so on. One-piece. Note since it starts and ends at zero, this approximation yields zero area. Two-piece Four-piece Eight-piece. After trapezoid rule estimates are obtained, Richardson extrapolation is applied.
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 ...
Thus detrending does not solve the estimation problem. In order to still use the Box–Jenkins approach, one could difference the series and then estimate models such as ARIMA, given that many commonly used time series (e.g. in economics) appear to be stationary in first differences. Forecasts from such a model will still reflect cycles and ...