Redescending M-estimator - Examples

Examples

1. Hampel's three-part M estimators have Ψ functions which are odd functions and defined for any x by:

\Psi(x)= \begin{cases} x, & 0\le |x| \le a \text{ (central segment)}\\ a\, \operatorname{sign}(x), & a\le |x| \le b \text{ (high and low flat segments)}\\ \frac{a(r-|x|)}{r-b}\,\operatorname{sign}(x),& b\le |x| \le r \text{ (end slopes)}\\ 0,& r\le |x| \qquad\, \text{(left and right tails)} \end{cases}

This function is plotted in the following figure for a=1.645, b=3 and r=6.5.

2. Tukey's biweight or bisquare M estimators have Ψ functions for any positive k, which defined by:

This function is plotted in the following figure for k=5.

3. Andrew's sine wave M estimator has the following Ψ function:

This function is plotted in the following figure.

Read more about this topic:  Redescending M-estimator

Famous quotes containing the word examples:

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)

    “It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
    —G.C. (Georg Christoph)