By Christopher F. Baum
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Additional resources for A review of Stata 8.1 and its time series
60). 4 Quantiles For α ∈ (0, 1) and F a continuous and increasing distribution function, the α-quantile of F is the unique value q(α) such that F(q(α)) = α. If F is discontinuous, such a value might not exist. , sgn lim (F(t) − α) = sgn t↑q(α) lim (F(t) − α) , t↓q(α) where “↑” and “↓” denote the limits from the left and from the right, respectively. It is easy to show that such a value always exists. It is unique if F is increasing. Otherwise, it is not necessarily unique, and hence we may speak of an α-quantile.
2 A ψ-function will denote a function ψ which is the derivative of a ρ-function, which implies in particular that 1 ψ is odd and ψ(x) ≥ 0 for x ≥ 0. 3 Trimmed means Another approach to robust estimation of location would be to discard a proportion of the largest and smallest values. 2). The reader may think that we are again suppressing observations. Note, however, that no subjective choice has been made: the result is actually a function of all observations (even of those that have not been included in the sum).
Maronna, R. Douglas Martin and V´ıctor J. Yohai C 2006 John Wiley & Sons, Ltd ISBN: 0-470-01092-4 JWBK076-02 JWBK076-Maronna February 16, 2006 18:7 Char Count= 0 18 LOCATION AND SCALE way to measure the approximation is with mean squared error (MSE): MSE(μ) = E(μ − μ)2 (other measures will be developed later). The MSE can be decomposed as MSE(μ) = Var(μ) + Bias(μ)2 , with Bias(μ) = Eμ − μ where “E” stands for the expectation. Note that if μ is the sample mean and c is any constant, then μ(x1 + c, .