The unbiased estimation of the fraction of variance explained by a model
Fig 14
Illustrative schematic of confidence interval estimation.
Given an observed estimate x* (green dashed vertical) from the distribution of the estimator X with CDF T(x) (solid black curve) associated with the parameter being estimated μ (black dashed vertical), the upper limit of the α-level confidence interval is the μU (purple vertical dashed) corresponding to the cumulative distribution of XU, U(x) (solid purple curve) that would generate values less than x* with probability α/2 (purple horizontal dashed). Thus U(x) is defined by U(x*) = α/2. Under the assumption the family of CDFs of X are stochastically increasing in μ, the event that T(x) ≥ α/2 corresponds to the event that μ < μU, thus the upper limit of the confidence interval contains the true value of μ. In graphical terms, if the black horizontal dashed line is above the purple, then it is guaranteed that the purple vertical dashed is to the right of the black. Thus these two events have the same probability: Pr(μ ≤ μU) = Pr(α/2 ≤ T(X)) = 1 − α/2. Here we have used generic symbols for illustrative purposes, but for reference to the proof (see Methods, “Proof of α-level confidence intervals”), the notation used here correspond as follows: ,
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