Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.

Contents

Introduction

Robust statistics seeks to provide methods that emulate classical methods[clarification needed], but which are not unduly affected by outliers or other small departures from model assumptions. In statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments, classical methods rely heavily on assumptions which are often not met in practice. In particular, it is often assumed that the data residuals are normally distributed, at least approximately, or that the central limit theorem In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed (Rice 1995). The central limit theorem also requires the random variables to be identically distributed, unless can be relied on to produce normally distributed estimates. Unfortunately, when there are outliers in the data, classical methods often have very poor performance.[citation needed]

This can be studied empirically by examining the sampling distribution In statistics, a sampling distribution is the probability distribution of a given statistic , based on a random sample. The sampling distribution depends on the distribution of the population, the statistic being considered, and the sample size used of various estimators under a mixture model In statistics, a mixture model is a probabilistic model for density estimation using a mixture distribution. A mixture model can be regarded as a type of unsupervised learning or clustering. Mixture models should not be confused with models for compositional data, i.e., data whose components are constrained to sum to a constant value, 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 (the errors).

In order to quantify the robustness of a method, it is necessary to define some measures of robustness. Perhaps the most common of these are the breakdown point and the influence function, described below.

Robust parametric statistics Parametric statistics is a branch of statistics that assumes data come from a type of probability distribution and makes inferences about the parameters of the distribution. Most well-known elementary statistical methods are parametric tends to rely on replacing the normal distribution in classical methods with 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.

Examples of robust and non-robust statistics

Trimmed estimators and Winsorised estimators are general methods to make statistics more robust. M-estimators are a general class of robust statistics.

Definition

This section requires expansion.

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 In statistics, an estimator is called efficient if it estimates the parameter of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors of different magnitudes. The most common, and reasonably small bias A cognitive bias is the human tendency to make systematic errors in certain circumstances based on cognitive factors rather than evidence. Such biases can result from information-processing shortcuts called heuristics. They include errors in judgment, social attribution, and memory. Cognitive biases are a common outcome of human thought, and often, as well as being asymptotically unbiased, meaning having a bias tending towards 0 as the sample size tends towards infinity.

One of the most important cases is distributional robustness[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). Thus, in the context of robust statistics, distributionally robust and outlier-resistant are effectively synonymous[1].

A related topic is that of resistant statistics, which are resistant to the effect of extreme scores. Most statistics are either robust and resistant, or neither.

Example: speed of light data

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 look 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 statistics, an estimator is called efficient if it estimates the parameter of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors of different magnitudes. The most common in the presence of outliers and less variable measures of location are available.

Estimation of location

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 QQ-plot (panel (b)). The outliers are clearly 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 In computing, R is a programming language and software environment for statistical computing and graphics. It is an implementation of the S programming language with lexical scoping semantics inspired by Scheme. R was created by Ross Ihaka and Robert Gentleman at the University of Auckland, New Zealand, and is now developed by the R Development and 10,000 bootstrap Bootstrapping is a statistical method for estimating the sampling distribution of an estimator by sampling with replacement from the original sample, most often with the purpose of deriving robust estimates of standard errors and confidence intervals of a population parameter like a mean, median, proportion, odds ratio, correlation coefficient or 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 note that 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

Main article: Robust measures of scale

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, median absolute deviation (MAD) and Qn estimator of scale (Rousseeuw and Croux, 1993). The plots are based on 10000 bootstrap samples for each estimator, and some normal random noise was 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

Traditionally, statisticians would manually screen data for outliers In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as:, and remove them, usually checking the source of the data to see if the outliers In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: 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.

Variety of applications

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

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

Breakdown point

Intuitively, the breakdown point of an estimator In statistics, an estimator or point estimate is a statistic that is used to infer the value of an unknown parameter in a statistical model. The parameter being estimated is sometimes called the estimand. It can be either finite-dimensional (in parametric and semi-parametric models), or infinite-dimensional (semi-nonparametric and non-parametric is the proportion of incorrect observations (i.e. arbitrarily large observations) an estimator can handle before giving an arbitrarily large result. For example, given n independent random variables and the corresponding realizations , we can use to estimate the mean. Such an estimator has a breakdown point of 0 because we can make arbitrarily large just by changing any of .

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. 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 breakdown point of X%, for the chosen level of X. Huber (1981) and Maronna et al. (2006) contain more details.

Statistics with high breakdown points are sometimes called resistant statistics.[2]

Example: speed of light data

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. Intuitively, 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. Clearly, the trimmed mean is less affected by the outliers and has a higher breakdown point.

Notice that 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

Tukey's biweight function

The empirical influence function gives us an idea of how an estimator behaves when we change one point in the sample and relies on the data (i.e. no model assumptions). 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. The context is the following:

  1. is a probability space,
  2. is a measure space (state space),
  3. Θ is a parameter space of dimension ,
  4. (Γ,S) is a measure space,
  5. is a projection,
  6. is the set of all possible distributions on Σ

For example,

  1. is any probability space,
  2. ,
  3. ,
  4. is defined by γ(x,y) = x.

The definition of an empirical influence function is: Let and are iid and is a sample from these variables. is an estimator. Let . The empirical influence function EIFi at observation i is defined by:

What this actually 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.

This notion of influence function is analogous to other notions of influence function, such as impulse response In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change. In both cases, the impulse response describes the reaction of: it measures sensitivity to the value at a point.

Influence function and sensitivity curve

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 A be a convex subset of the set of all finite signed measures on . We want to estimate the parameter of a distribution F in A. Let the functional be the asymptotic value of some estimator sequence . We will suppose that this functional is Fisher consistent, i.e. . This means that at the model F, the estimator sequence asymptotically measures the right quantity.

Let G be some distribution in A. What happens when the data doesn't follow the model F exactly but another, slightly different, "going towards" G?

We're looking at: ,

which is the directional derivative of T at F, in the direction of G.

Let . Δx is the probability measure which gives mass 1 to x. We chose G = Δx. The influence function is then defined by:

It describes the effect of an infinitesimal contamination at the point x on the estimate we are seeking, standardized by the mass 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

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

  1. Finite rejection point ρ * ,
  2. Small gross-error sensitivity γ * ,
  3. Small local-shift sensitivity λ * .

Rejection point

Gross-error sensitivity

Local-shift sensitivity

This value, which looks a lot like a Lipschitz constant, represents the effect of shifting an observation slightly from x to a neighbouring point y, i.e., add an observation at y and remove one at x.

M-estimators

Main article: M-estimator

(The mathematical context of this paragraph is given in the section on empirical influence functions.)

Historically, several approaches to robust estimation were proposed, including R-estimators and L-estimators. However, M-estimators now appear to dominate the field as a result of their generality, high breakdown point, and their efficiency. See Huber (1981).

M-estimators are a generalization of maximum likelihood estimators (MLEs). What we try to do with MLE's is to maximize or, equivalently, minimize . In 1964, Huber proposed to generalize this to the minimization of , where ρ is some function. MLE are therefore a special case of M-estimators (hence the name: "Maximum likelihood type" estimators).

Minimizing can often be done by differentiating ρ and solving , where (if ρ has a derivative).

Several choices of ρ and ψ have been proposed. The two figures below show four ρ functions and their corresponding ψ functions.

For squared errors, ρ(x) increases at an accelerating rate, whilst for absolute errors, it increases at a constant rate. When Winsorizing is used, a mixture of these two effects is introduced: for small values of x, ρ increases at the squared rate, but once the chosen threshold is reached (1.5 in this example), the rate of increase becomes constant.

Tukey's biweight (also known as bisquare) function behaves in a similar way to the squared error function at first, but for larger errors, the function tapers off.

Properties of M-estimators

Notice that M-estimators do not necessarily relate to a probability density function. Therefore, off-the-shelf approaches to inference that arise from likelihood theory can not, in general, be used.

It can be shown that M-estimators are asymptotically normally distributed, so that as long as their standard errors can be computed, an approximate approach to inference is available.

Since M-estimators are normal only asymptotically, for small sample sizes it might be appropriate to use an alternative approach to inference, such as the bootstrap. However, M-estimates are not necessarily unique (i.e. there might be more than one solution that satisfies the equations). Also, it is possible that any particular bootstrap sample can contain more outliers than the estimator's breakdown point. Therefore, some care is needed when designing bootstrap schemes.

Of course, as we saw with the speed of light example, the mean is only normally distributed asymptotically and when outliers are present the approximation can be very poor even for quite large samples. However, classical statistical tests, including those based on the mean, are typically bounded above by the nominal size of the test. The same is not true of M-estimators and the type I error rate can be substantially above the nominal level.

These considerations do not "invalidate" M-estimation in any way. They merely make clear that some care is needed in their use, as is true of any other method of estimation.

Influence function of an M-estimator

It can be shown that the influence function of an M-estimator T is proportional to ψ (see Huber, 1981 (and 2004), page 45), which means we can derive the properties of such an estimator (such as its rejection point, gross-error sensitivity or local-shift sensitivity) when we know its ψ function.

IF(x;T,F) = M − 1ψ(x,T(F)) with the given by: .

Choice of ψ and ρ

In many practical situations, the choice of the ψ function is not critical to gaining a good robust estimate, and many choices will give similar results that offer great improvements, in terms of efficiency and bias, over classical estimates in the presence of outliers (Huber, 1981).

Theoretically, ψ functions are to be preferred, and Tukey's biweight (also known as bisquare) function is a popular choice. Maronna et al. (2006) recommend the biweight function with efficiency at the normal set to 85%.

Robust parametric approaches

M-estimators do not necessarily relate to a density function and so are not fully parametric. Fully parametric approaches to robust modeling and inference, both Bayesian and likelihood approaches, usually deal with heavy tailed distributions such as Student's t-distribution.

For the t-distribution with ν degrees of freedom, it can be shown that

For ν = 1, the t-distribution is equivalent to the Cauchy distribution. Notice that the degrees of freedom is sometimes known as the kurtosis parameter. It is the parameter that controls how heavy the tails are. In principle, ν can be estimated from the data in the same way as any other parameter. In practice, it is common for there to be multiple local maxima when ν is allowed to vary. As such, it is common to fix ν at a value around 4 or 6. The figure below displays the ψ-function for 4 different values of ν.

Example: speed of light data

For the speed of light data, allowing the kurtosis parameter to vary and maximizing the likelihood, we get

Fixing ν = 4 and maximizing the likelihood gives

Robust decision theory

For more details on this topic, see Decision theory#Alternatives to probability theory Decision theory in Philosophy, mathematics and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision. It is very closely related to the field of game theory.

Decision theory Decision theory in Philosophy, mathematics and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision. It is very closely related to the field of game theory based on maximizing expected value In probability theory and statistics, the expected value of a random variable is the integral of the random variable with respect to its probability measure or the expected utility hypothesis In economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes is represented by a function of the payout (whether in money or other goods), the probability of occurrence, risk aversion, and the different utility of the same is sensitive to assumptions about probabilities of various outcomes, particularly if expectation is dominated by rare extreme events.

By contrast, non-probabilistic decision theories like minimax Minimax is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss while maximizing the potential gain. Alternatively, it can be thought of as maximizing the minimum gain (maximin). Originally formulated for two-player zero-sum game theory, covering both the cases where players take and minimax regret are independent of assumptions about the probabilities of outcomes, depending only on evaluating possible outcomes and their desirabilities. Scenario analysis Scenario analysis can also be used to illuminate "wild cards." For example, analysis of the possibility of the earth being struck by a large celestial object suggests that whilst the probability is low, the damage inflicted is so high that the event is much more important (threatening) than the low probability (in any one year) alone and stress testing Stress testing is a form of testing that is used to determine the stability of a given system or entity. It involves testing beyond normal operational capacity, often to a breaking point, in order to observe the results. Stress testing may have a more specific meaning in certain industries, such as fatigue testing for materials are informal non-probabilistic methods, while info-gap decision theory is a formal robust decision theory. Possibility theory and Dempster–Shafer theory are other non-probabilistic methods.

Advocates of probabilistic approaches to decision theory argue that in fact all decision rules can be derived or dominated by Bayesian methods, appealing to results such as the complete class theorems, which show that all admissible decision rules are equivalent to a Bayesian decision rule with some prior distribution (possibly improper) and some utility function In economics, utility is a measure of the relative satisfaction from, or desirability of, consumption of various goods and services. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one's utility.

Related concepts

A pivotal quantity In statistics, a pivotal quantity or pivot is a function of observations whose distribution does not depend on unknown parameters. Note that a pivot quantity need not be a statistic—the function and its value can depend on parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic is a function of data, whose underlying population distribution is a member of a parametric family, that is not dependent on the values of the parameters. An ancillary statistic is such a function that is also a statistic, meaning that it is computed in terms of the data alone. Such functions are robust to parameters in the sense that they are independent of the values of the parameters, but not robust to the model in the sense that they assume an underlying model (parametric family), and in fact such functions are often very sensitive to violations of the model assumptions. Thus test statistics In statistical hypothesis testing, a test statistic is a numerical summary of a set of data that reduces the data to one or a small number of values that can be used to perform a hypothesis test. Given a null hypothesis and a test statistic T, we can specify a "null value" T0 such that values of T close to T0 present the strongest, frequently constructed in terms of these to not be sensitive to assumptions about parameters, are still very sensitive to model assumptions.

Key contributors

Key contributors to the field of robust statistics include Frank Hampel, Peter J. Huber and John Tukey Tukey was born in New Bedford, Massachusetts in 1915, and obtained a B.A. in 1936 and M.Sc. in 1937, in chemistry, from Brown University, before moving to Princeton University where he received a Ph.D. in mathematics.

See also

References

  1. ^ a b c Robust Statistics, Peter. J. Huber, Wiley, 1981 (republished in paperback, 2004), page 1.
  2. ^ Resistant statistics, David B. Stephenson

External links

Categories: Statistical theory | Robust statistics

 

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