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Robust Regression and Outlier Detection

Peter J. Rousseeuw, Annick M. Leroy

Chapter 4

The Special Case of One-Dimensional Location - all with Video Answers

Educators

Chapter Questions

02:22

Problem 1

Assuming that the observations $y_i$ are normally distributed with standard deviation $\sigma$, show that both $1.4826 \mathrm{med}_f\left|y_f-\operatorname{med}_k y_k\right|$ and $\left.1.2533 \frac{1}{n} \sum_{f=1}^n \right\rvert\, y_f-$ med $_k y_k \mid$ tend to $\sigma$ when $n$ increases.

Ameer Said
Ameer Said
Numerade Educator
02:30

Problem 2

Show that the minimization of $\sum_{i=1}^n w_i\left(y_i-T_w\right)^2$ yields the solution
$$
T_w=\frac{\sum_{i=1}^n w_i y_i}{\sum_{i=1}^n w_i},
$$
which proves that the one-step $W$-estimator (1.15) is a reweighted least squares procedure.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 3

Show that the sum of the $a_i$ in (1.16) must be equal to 1 to make the corresponding $L$-estimator location equivariant. How about scale equivariance? Can the median be written as an $L$-estimator?

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Problem 4

Generate four random numbers according to the standard normal distribution $N(0,1)$. Does this sample look symmetric? Calculate the arithmetic mean, sample median, and LMS, and compute the classical third moment for testing symmetry.

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05:10

Problem 5

Make a plot of the objective functions of $(2.1)$ and $(2.5)$ as a function of $\theta$ for the sample $\{1,3,4,6,7,10\}$.

Jacquelinne S. Mejia Sandoval
Jacquelinne S. Mejia Sandoval
Numerade Educator
01:29

Problem 6

Compute the LTS and the LWS for the Rosner data (2.2).

Manik Pulyani
Manik Pulyani
Numerade Educator
02:11

Problem 7

Show that the LMS of three observations equals the midpoint of the closest pair. What happens to the shorth and the LTS in this situation?

Carson Merrill
Carson Merrill
Numerade Educator

Problem 8

Show that for any affine equivariant location estimator $T$, it holds that $T\left(y_1\right)=y_1$ and $T\left(y_1, y_2\right)=\frac{1}{2}\left(y_1+y_2\right)$. Things become more interesting when there are three observations (assuming that they are not all equal to the same value). By equivariance, we may assume that $y_1=-1$ and $y_2=1$, so the whole estimator may be described by means of the univariate function $h(y)=T(-1,1, y)$. Draw the function $h$ for the arithmetic mean, the sample median, and the LMS.

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00:21

Problem 9

Analyze the monthly payments of exercise 8 of Chapter 1 (i.e., the response variable of Table 1) in a one-dimensional way by means of PROGRESS. (Motivation: In exercise 6 of Chapter 2 this variable was regressed against time, and the RLS slope was not significantly different from zero.) How many outliers do you find now?

Raymond Matshanda
Raymond Matshanda
Numerade Educator
01:17

Problem 10

Apply PROGRESS to the one-dimensional data set of Cushny and Peebles (1905), first analyzed by Student (1908), consisting of the following 10 observations:
$$
0.0,0.8,1.0,1.2,1.3,1.3,1.4,1.8,2.4,4.6 \text {. }
$$

Explain why the arithmetic mean is larger than the robust estimates of location. What happens to the standard deviation? Are there any points with standardized LS residuals outside the $\pm 2.5$ region? Compare this with the residuals of LMS and RLS.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
00:28

Problem 11

Grubbs (1969) considered the following one-dimensional data sets.
(a) Strengths of hard-drawn copper wire $(n=10)$ :
$$
568,570,570,570,572,572,572,578,584,596
$$
(b) Residuals of observed vertical semidiameters of Venus $(n=15)$ :
$$
\begin{gathered}
-1.40,-0.44,-0.30,-0.24,-0.22,-0.13,-0.05, \\
0.06,0.10,0.18,0.20,0.39,0.48,0.63,1.01
\end{gathered}
$$
(c) Percent elongations of plastic material $(n=10)$ :
$$
2.02,2.22,3.04,3.23,3.59,3.73,3.94,4.05,4.11,4.13
$$
(d) Ranges of projectiles $(n=8)$ :
$$
\text { 4420, 4549, 4730, 4765, 4782, 4803, 4833, } 4838 .
$$

Analyze these data sets by means of PROGRESS. How many outliers are identified?

Evan Schroeder
Evan Schroeder
Numerade Educator

Problem 12

Write down the proof of Theorem 4 for the variant of the LTS in which $n(1-\alpha)$ of the squared residuals are used, with $0<\alpha<50 \%$.

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Problem 13

(Research problem) In Table 1 it seems that the spread of the LWS lies between the LTS (which is known to converge like $n^{-1 / 2}$ ) and the LMS (which converges like $n^{-1 / 3}$ ). We do not yet know the convergence rate of the LWS estimator.

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Problem 14

Is there a relation between the function $h$ of exercise 8 and sensitivity curves?

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