# Robust statistics – Wikipedia

Type of statistics

**Robust statistics** are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such as estimating location, scale, and regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly.

## Introduction[edit]

Robust statistics seek to provide methods that emulate popular statistical methods, but are not unduly affected by outliers or other small departures from model assumptions. In statistics, classical estimation methods rely heavily on assumptions that are often not met in practice. In particular, it is often assumed that the data errors are normally distributed, at least approximately, or that the central limit theorem can be relied on to produce normally distributed estimates. Unfortunately, when there are outliers in the data, classical estimators often have very poor performance, when judged using the *breakdown point* and the *influence function*, described below.

The practical effect of problems seen in the influence function can be studied empirically by examining the sampling distribution of proposed estimators under a mixture model, where one mixes in a small amount (1–5% is often sufficient) of contamination. For instance, one may use a mixture of 95% a normal distribution, and 5% a normal distribution with the same mean but significantly higher standard deviation (representing outliers).

Robust parametric statistics can proceed in two ways:

- by designing estimators so that a pre-selected behaviour of the influence function is achieved
- by replacing estimators that are optimal under the assumption of a normal distribution with estimators that are optimal for, or at least derived for, other distributions: for example using the
*t*-distribution with low degrees of freedom (high kurtosis; degrees of freedom between 4 and 6 have often been found to be useful in practice^{[citation needed]}) or with a mixture of two or more distributions.

Robust estimates have been studied for the following problems:

## Definition[edit]

This section
needs expansion. You can help by adding to it. (July 2008) |

There are various definitions of a “robust statistic.” Strictly speaking, a **robust statistic** is resistant to errors in the results, produced by deviations from assumptions^{[1]} (e.g., of normality). This means that if the assumptions are only approximately met, the **robust estimator** will still have a reasonable efficiency, and reasonably small bias, as well as being asymptotically unbiased, meaning having a bias tending towards 0 as the sample size tends towards infinity.

Usually, the most important case is **distributional robustness** – robustness to breaking of the assumptions about the underlying distribution of the data.^{[1]} Classical statistical procedures are typically sensitive to “longtailedness” (e.g., when the distribution of the data has longer tails than the assumed normal distribution). This implies that they will be strongly affected by the presence of outliers in the data, and the estimates they produce may be heavily distorted if there are extreme outliers in the data, compared to what they would be if the outliers were not included in the data.

By contrast, more robust estimators that are not so sensitive to distributional distortions such as longtailedness are also resistant to the presence of outliers. Thus, in the context of robust statistics, *distributionally robust* and *outlier-resistant* are effectively synonymous.^{[1]} For one perspective on research in robust statistics up to 2000, see Portnoy & He (2000).

Some experts prefer the term **resistant statistics** for distributional robustness, and reserve ‘robustness’ for non-distributional robustness, e.g., robustness to violation of assumptions about the probability model or estimator, but this is a minority usage. Plain ‘robustness’ to mean ‘distributional robustness’ is common.

When considering how robust an estimator is to the presence of outliers, it is useful to test what happens when an extreme outlier is **added** to the dataset and to test what happens when an extreme outlier **replaces** one of the existing data points, and then to consider the effect of multiple additions or replacements.

## Examples[edit]

The mean is not a robust measure of central tendency. If the dataset is e.g. the values {2,3,5,6,9}, then if we add another datapoint with value -1000 or +1000 to the data, the resulting mean will be very different to the mean of the original data. Similarly, if we replace one of the values with a datapoint of value -1000 or +1000 then the resulting mean will be very different to the mean of the original data.

The median is a robust measure of central tendency. Taking the same dataset {2,3,5,6,9}, if we add another datapoint with value -1000 or +1000 then the median will change slightly, but it will still be similar to the median of the original data. If we replace one of the values with a datapoint of value -1000 or +1000 then the resulting median will still be similar to the median of the original data.

Described in terms of breakdown points, the median has a breakdown point of 50%, meaning that half the points must be outliers before the median can be moved outside the range of the non-outliers, while the mean has a breakdown point of 0, as a single large observation can throw it off.

The median absolute deviation and interquartile range are robust measures of statistical dispersion, while the standard deviation and range are not.

Trimmed estimators and Winsorised estimators are general methods to make statistics more robust. L-estimators are a general class of simple statistics, often robust, while M-estimators are a general class of robust statistics, and are now the preferred solution, though they can be quite involved to calculate.

### Speed-of-light data[edit]

Gelman et al. in Bayesian Data Analysis (2004) consider a data set relating to speed-of-light measurements made by Simon Newcomb. The data sets for that book can be found via the Classic data sets page, and the book’s website contains more information on the data.

Although the bulk of the data looks to be more or less normally distributed, there are two obvious outliers. These outliers have a large effect on the mean, dragging it towards them, and away from the center of the bulk of the data. Thus, if the mean is intended as a measure of the location of the center of the data, it is, in a sense, biased when outliers are present.

Also, the distribution of the mean is known to be asymptotically normal due to the central limit theorem. However, outliers can make the distribution of the mean non-normal even for fairly large data sets. Besides this non-normality, the mean is also inefficient in the presence of outliers and less variable measures of location are available.

### Estimation of location[edit]

The plot below shows a density plot of the speed-of-light data, together with a rug plot (panel (a)). Also shown is a normal Q–Q plot (panel (b)). The outliers are visible in these plots.

Panels (c) and (d) of the plot show the bootstrap distribution of the mean (c) and the 10% trimmed mean (d). The trimmed mean is a simple robust estimator of location that deletes a certain percentage of observations (10% here) *from each end* of the data, then computes the mean in the usual way. The analysis was performed in R and 10,000 bootstrap samples were used for each of the raw and trimmed means.

The distribution of the mean is clearly much wider than that of the 10% trimmed mean (the plots are on the same scale). Also whereas the distribution of the trimmed mean appears to be close to normal, the distribution of the raw mean is quite skewed to the left. So, in this sample of 66 observations, only 2 outliers cause the central limit theorem to be inapplicable.

Robust statistical methods, of which the trimmed mean is a simple example, seek to outperform classical statistical methods in the presence of outliers, or, more generally, when underlying parametric assumptions are not quite correct.

Whilst the trimmed mean performs well relative to the mean in this example, better robust estimates are available. In fact, the mean, median and trimmed mean are all special cases of M-estimators. Details appear in the sections below.

### Estimation of scale[edit]

The outliers in the speed-of-light data have more than just an adverse effect on the mean; the usual estimate of scale is the standard deviation, and this quantity is even more badly affected by outliers because the squares of the deviations from the mean go into the calculation, so the outliers’ effects are exacerbated.

The plots below show the bootstrap distributions of the standard deviation, the median absolute deviation (MAD) and the Rousseeuw–Croux (Qn) estimator of scale. The plots are based on 10,000 bootstrap samples for each estimator, with some Gaussian noise added to the resampled data (smoothed bootstrap). Panel (a) shows the distribution of the standard deviation, (b) of the MAD and (c) of Qn.

The distribution of standard deviation is erratic and wide, a result of the outliers. The MAD is better behaved, and Qn is a little bit more efficient than MAD. This simple example demonstrates that when outliers are present, the standard deviation cannot be recommended as an estimate of scale.

### Manual screening for outliers[edit]

Traditionally, statisticians would manually screen data for outliers, and remove them, usually checking the source of the data to see whether the outliers were erroneously recorded. Indeed, in the speed-of-light example above, it is easy to see and remove the two outliers prior to proceeding with any further analysis. However, in modern times, data sets often consist of large numbers of variables being measured on large numbers of experimental units. Therefore, manual screening for outliers is often impractical.

Outliers can often interact in such a way that they mask each other. As a simple example, consider a small univariate data set containing one modest and one large outlier. The estimated standard deviation will be grossly inflated by the large outlier. The result is that the modest outlier looks relatively normal. As soon as the large outlier is removed, the estimated standard deviation shrinks, and the modest outlier now looks unusual.

This problem of masking gets worse as the complexity of the data increases. For example, in regression problems, diagnostic plots are used to identify outliers. However, it is common that once a few outliers have been removed, others become visible. The problem is even worse in higher dimensions.

Robust methods provide automatic ways of detecting, downweighting (or removing), and flagging outliers, largely removing the need for manual screening. Care must be taken; initial data showing the ozone hole first appearing over Antarctica were rejected as outliers by non-human screening.^{[3]}

### Variety of applications[edit]

Although this article deals with general principles for univariate statistical methods, robust methods also exist for regression problems, generalized linear models, and parameter estimation of various distributions.

## Measures of robustness[edit]

The basic tools used to describe and measure robustness are the *breakdown point*, the *influence function* and the *sensitivity curve*.

### Breakdown point[edit]

Intuitively, the breakdown point of an estimator is the proportion of incorrect observations (e.g. arbitrarily large observations) an estimator can handle before giving an incorrect (e.g., arbitrarily large) result. Usually, the asymptotic (infinite sample) limit is quoted as the breakdown point, although the finite-sample breakdown point may be more useful.^{[4]} For example, given

independent random variables

${displaystyle (X_{1},dots ,X_{n})}$and the corresponding realizations

${displaystyle x_{1},dots ,x_{n}}$, we can use

${displaystyle {overline {X_{n}}}:={frac {X_{1}+cdots +X_{n}}{n}}}$to estimate the mean. Such an estimator has a breakdown point of 0 (or finite-sample breakdown point of

${displaystyle 1/n}$) because we can make

${displaystyle {overline {x}}}$arbitrarily large just by changing any of

${displaystyle x_{1},dots ,x_{n}}$.

The higher the breakdown point of an estimator, the more robust it is. Intuitively, we can understand that a breakdown point cannot exceed 50% because if more than half of the observations are contaminated, it is not possible to distinguish between the underlying distribution and the contaminating distribution Rousseeuw & Leroy (1986)

. Therefore, the maximum breakdown point is 0.5 and there are estimators which achieve such a breakdown point. For example, the median has a breakdown point of 0.5. The X% trimmed mean has a breakdown point of X%, for the chosen level of X. Huber (1981) and Maronna, Martin & Yohai (2006) contain more details. The level and the power breakdown points of tests are investigated in He, Simpson & Portnoy (1990).Statistics with high breakdown points are sometimes called **resistant statistics.**^{[5]}

#### Example: speed-of-light data[edit]

In the speed-of-light example, removing the two lowest observations causes the mean to change from 26.2 to 27.75, a change of 1.55. The estimate of scale produced by the Qn method is 6.3. We can divide this by the square root of the sample size to get a robust standard error, and we find this quantity to be 0.78. Thus, the change in the mean resulting from removing two outliers is approximately twice the robust standard error.

The 10% trimmed mean for the speed-of-light data is 27.43. Removing the two lowest observations and recomputing gives 27.67. The trimmed mean is less affected by the outliers and has a higher breakdown point.

If we replace the lowest observation, −44, by −1000, the mean becomes 11.73, whereas the 10% trimmed mean is still 27.43. In many areas of applied statistics, it is common for data to be log-transformed to make them near symmetrical. Very small values become large negative when log-transformed, and zeroes become negatively infinite. Therefore, this example is of practical interest.

### Empirical influence function[edit]

The empirical influence function is a measure of the dependence of the estimator on the value of any one of the points in the sample. It is a model-free measure in the sense that it simply relies on calculating the estimator again with a different sample. On the right is Tukey’s biweight function, which, as we will later see, is an example of what a “good” (in a sense defined later on) empirical influence function should look like.

In mathematical terms, an influence function is defined as a vector in the space of the estimator, which is in turn defined for a sample which is a subset of the population:

For example,

The empirical influence function is defined as follows.

Let

${displaystyle nin mathbb {N} ^{*}}$and

${displaystyle X_{1},dots ,X_{n}:(Omega ,{mathcal {A}})rightarrow ({mathcal {X}},Sigma )}$are i.i.d. and

${displaystyle (x_{1},dots ,x_{n})}$is a sample from these variables.

${displaystyle T_{n}:({mathcal {X}}^{n},Sigma ^{n})rightarrow (Gamma ,S)}$is an estimator. Let

${displaystyle iin {1,dots ,n}}$. The empirical influence function

${displaystyle EIF_{i}}$at observation

${displaystyle i}$is defined by:

What this means is that we are replacing the *i*-th value in the sample by an arbitrary value and looking at the output of the estimator. Alternatively, the EIF is defined as the effect, scaled by n+1 instead of n, on the estimator of adding the point

to the sample.^{[citation needed]}

### Influence function and sensitivity curve[edit]

Instead of relying solely on the data, we could use the distribution of the random variables. The approach is quite different from that of the previous paragraph. What we are now trying to do is to see what happens to an estimator when we change the distribution of the data slightly: it assumes a *distribution,* and measures sensitivity to change in this distribution. By contrast, the empirical influence assumes a *sample set,* and measures sensitivity to change in the samples.

Let

${displaystyle A}$be a convex subset of the set of all finite signed measures on

${displaystyle Sigma }$. We want to estimate the parameter

${displaystyle theta in Theta }$of a distribution

${displaystyle F}$in

${displaystyle A}$. Let the functional

${displaystyle T:Arightarrow Gamma }$be the asymptotic value of some estimator sequence

${displaystyle (T_{n})_{nin mathbb {N} }}$. We will suppose that this functional is Fisher consistent, i.e.

${displaystyle forall theta in Theta ,T(F_{theta })=theta }$. This means that at the model

${displaystyle F}$, the estimator sequence asymptotically measures the correct quantity.

Let

${displaystyle G}$be some distribution in

${displaystyle A}$. What happens when the data doesn’t follow the model

${displaystyle F}$exactly but another, slightly different, “going towards”

${displaystyle G}$?

We’re looking at:

which is the one-sided Gateaux derivative of

${displaystyle T}$at

${displaystyle F}$, in the direction of

${displaystyle G-F}$.

Let

${displaystyle xin {mathcal {X}}}$.

${displaystyle Delta _{x}}$is the probability measure which gives mass 1 to

${displaystyle {x}}$. We choose

${displaystyle G=Delta _{x}}$. The influence function is then defined by:

It describes the effect of an infinitesimal contamination at the point

${displaystyle x}$on the estimate we are seeking, standardized by the mass

${displaystyle t}$of the contamination (the asymptotic bias caused by contamination in the observations). For a robust estimator, we want a bounded influence function, that is, one which does not go to infinity as x becomes arbitrarily large.

### Desirable properties[edit]

Properties of an influence function that bestow it with desirable performance are:

- Finite rejection point
- Small gross-error sensitivity
- Small local-shift sensitivity

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