### Table 3.1: Table of weight functions for M-estimation.

1996

Cited by 33

### Table 1. Error percentage and m-estimates obtained in our experiments

2005

"... In PAGE 7: ... Ensemble predictions are provided to the right of the labeled data in each of the figures. In Table1 , we provide an estimate of the true error for our experiments. Because this error is based on a point-to-point comparison between our labeled test set and the predictions upon the test set, and because we know regions are salient rather than points, we could potentially lower the error by utilizing image processing techniques such as erosion and dilation.... ..."

Cited by 3

### Table 1: A few commonly used M-estimators

"... In PAGE 34: ... This avoids having to search through the complete parameter space. Table1 lists a few commonly used in uence functions. They are graphically depicted in Fig.... ..."

### Table 1: M-estimator and M-smoother (weighted M-estimator) cost functions and outputs for various lter families.

"... In PAGE 3: ... The cost function thus obtained is (u) = log(K2+u2), where the parameter K controls the robustness of the estimator; a more detailed descrip- tion is given in Section 2. The rst three rows of Table1 show the cost functions for the sample mean, median and myriad estimators. From a signal processing point of view, these plain location M-estimators, using i.... In PAGE 3: ... By extending this principle to the more general M-estimator of (1), we obtain a weighted M-estimator: ^ 4 = arg min N X i=1 (pwi (xi ? )): (2) For reasons to be described shortly, these estimators are also referred to as M-smoothers. The last three rows of Table1 show the di erent M-smoothers obtained using the cost functions for the linear, median and myriad smoother families, respectively. The notations wi xi (for the weighted median) and wi xi (for the weighted myriad), shown in the last... In PAGE 10: ...1 M- lters Before we can extend the general class of M-smoothers, it is useful to examine the special cases of the weighted mean and median smoothers. Referring to Table1 , the output of a weighted mean smoother, with input vector x and a weight vector w of non-negative weights, is given by ^ (w; x) = mean (wi xi)jN i=1 = N X i=1 wixi = N X i=1 wi: (11) Notice from Table 1 that ^ minimizes the cost function N X i=1 wi(xi ? )2. When the weights are allowed to be both positive and negative, fwi 2 RgN i=1, a natural extension of (11) would be to write ~ (w; x) 4 = N X i=1 wixi = N X i=1 jwij; (12)... In PAGE 10: ...1 M- lters Before we can extend the general class of M-smoothers, it is useful to examine the special cases of the weighted mean and median smoothers. Referring to Table 1, the output of a weighted mean smoother, with input vector x and a weight vector w of non-negative weights, is given by ^ (w; x) = mean (wi xi)jN i=1 = N X i=1 wixi = N X i=1 wi: (11) Notice from Table1 that ^ minimizes the cost function N X i=1 wi(xi ? )2. When the weights are allowed to be both positive and negative, fwi 2 RgN i=1, a natural extension of (11) would be to write ~ (w; x) 4 = N X i=1 wixi = N X i=1 jwij; (12)... In PAGE 11: ... The weighted mean lter can therefore be thought of as a weighted mean smoother applied to a modi ed set of samples fzigN i=1, using the non-negative weights fgi 0gN i=1. Re- ferring to the cost function for the weighted mean smoother in Table1 , we can infer that the weighted mean lter output ~ (w; x) minimizes the cost function N X i=1 jwij (sgn(wi)xi ? )2. Following the above approach, the weighted median smoother was recently extended to a class of weighted median lters with real-valued weights [16].... In PAGE 12: ... Note that the weighted median lter can also be described as a weighted median smoother applied to a modi ed input vector z = [xT ; ?xT ]T and a modi ed weight vector of non-negative weights, g = [wT ; jhjT ]T . Referring to the weighted median smoother cost function in Table1 , we can show that the weighted median lter output minimizes the cost function N X i=1 wi jxi ? j + N X i=1 jhij j ? xi ? j. At rst glance, the double weighting in (16) seems redundant and the structure in (15), with a single real-valued weight for each sample, appears to be su cient as in the case of linear FIR lters.... ..."

Cited by 4

### Table 4: WSA (m ) estimates for selected tests.

"... In PAGE 7: ...Table 4: WSA (m ) estimates for selected tests. We see that WSA estimates in Table4 are similar to the cush- ion area ( m ). Thus, the minimization of WSA pro- vided by a SES is mostly negated by being a catamaran.... ..."

### Table II: GARCH-M Estimation Results

in Parameterizing Currency Risk in the EMS: The Irish Pound and Spanish Peseta against the German Mark

### Table III: ICAP-M Estimation Results:

in Parameterizing Currency Risk in the EMS: The Irish Pound and Spanish Peseta against the German Mark

### Table 1. The averages of the biases and MSEs of the 250 M (LS)-estimators Distr- M-Estimators LS Estimators

"... In PAGE 6: ...08377 5.51905 The averages of the biases (RBL) of the M-estimators of the parametric components and the averages of the mean squared errors (MSE) of the B-spline M-estimators for the unknown smooth function are listed in Table1 . The corresponding medians are given in Table 2.... ..."

### Table 1 Comparison between LS and M-estimation (with no or one outlier)

2007

"... In PAGE 8: ...33 -10.24 Number of zero weight observations null null PRN18, PRN225 null null PRN18, PRN203 PRN216, PRN225 As expected, in the scenario with no outliner, both LS and the M-estimation schemes have produced the same navigation solutions (see Table1 ). As the number of the outliers increases, the navigation solutions based on the LS scheme are significantly biased, but the same time, both M-estimation schemes (Huber and IGGIII) have effectively mitigated the effect of these outliers (see Tables 1, 2, 3).... ..."

### Table 1: A few commonly used M-estimators type (x) (x) w(x)

1997

"... In PAGE 35: ... This avoids having to search through the complete parameter space. Table1 lists a few commonly used in uence functions. They are graphically dipicted in Fig.... ..."

Cited by 117