1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.5. Quantitative Techniques 1.3.5.17. Detection of Outliers


Purpose: Detection of Outliers 
The generalized (extreme Studentized deviate) ESD test
(Rosner 1983)
is used to detect one or more
outliers in a univariate data set
that follows an approximately
normal distribution.
The primary limitation of the Grubbs test and the TietjenMoore test is that the suspected number of outliers, k, must be specified exactly. If k is not specified correctly, this can distort the conclusions of these tests. On the other hand, the generalized ESD test (Rosner 1983) only requires that an upper bound for the suspected number of outliers be specified. 

Definition 
Given the upper bound, r, the generalized ESD test
essentially performs r separate tests: a test for one
outlier, a test for two outliers, and so on up to r
outliers.
The generalized ESD test is defined for the hypothesis:
Note that although the generalized ESD is essentially Grubbs test applied sequentially, there are a few important distinctions:


Generalized ESD Test Example 
The Rosner paper
gives an example with the following data.
0.25 0.68 0.94 1.15 1.20 1.26 1.26 1.34 1.38 1.43 1.49 1.49 1.55 1.56 1.58 1.65 1.69 1.70 1.76 1.77 1.81 1.91 1.94 1.96 1.99 2.06 2.09 2.10 2.14 2.15 2.23 2.24 2.26 2.35 2.37 2.40 2.47 2.54 2.62 2.64 2.90 2.92 2.92 2.93 3.21 3.26 3.30 3.59 3.68 4.30 4.64 5.34 5.42 6.01 As a first step, a normal probability plot was generated
This plot indicates that the normality assumption is questionable. Following the Rosner paper, we test for up to 10 outliers: H_{0}: there are no outliers in the data H_{a}: there are up to 10 outliers in the data Significance level: α = 0.05 Critical region: Reject H_{0} if R_{i} > critical value Summary Table for TwoTailed Test  Exact Test Critical Number of Statistic Value, λ_{i} Outliers, i Value, R_{i} 5 %  1 3.118 3.158 2 2.942 3.151 3 3.179 3.143 * 4 2.810 3.136 5 2.815 3.128 6 2.848 3.120 7 2.279 3.111 8 2.310 3.103 9 2.101 3.094 10 2.067 3.085For the generalized ESD test above, there are essentially 10 separate tests being performed. For this example, the largest number of outliers for which the test statistic is greater than the critical value (at the 5 % level) is three. We therefore conclude that there are three outliers in this data set. 

Questions 
The generalized ESD test can be used to answer the following
question:


Importance 
Many statistical techniques are sensitive to the presence
of outliers. For example, simple calculations of the mean
and standard deviation may be distorted by a single grossly
inaccurate data point.
Checking for outliers should be a routine part of any data analysis. Potential outliers should be examined to see if they are possibly erroneous. If the data point is in error, it should be corrected if possible and deleted if it is not possible. If there is no reason to believe that the outlying point is in error, it should not be deleted without careful consideration. However, the use of more robust techniques may be warranted. Robust techniques will often downweight the effect of outlying points without deleting them. 

Related Techniques  Several graphical techniques can, and should, be used to help detect outliers. A simple normal probability plot, run sequence plot, a box plot, or a histogram should show any obviously outlying points. In addition to showing potential outliers, several of these graphics also help assess whether the data follow an approximately normal distribution.  
Software  Some general purpose statistical software programs support the generalized ESD test. Both Dataplot code and R code can be used to generate the analyses in this section. 