Peirce's criterion
Encyclopedia
In robust statistics
, Peirce's criterion is a rule for eliminating outlier
s from data sets, which was devised by Benjamin Peirce
.
s containing real-numbered measurements, the suspected outlier
s are the measured values that appear to lie outside the cluster of most of the other data values. The outliers would greatly change the estimate of location if the arithmetic average were to be used as a summary statistic of location. The problem is that the arithmetic mean is very sensitive to the inclusion of any outliers; in statistical terminology, the arithmetic mean is not robust.
In the presence of outliers, the statistician has two options. First, the statistician may remove the suspected outlier
s from the data set and then use the arithmetic mean to estimate the location parameter. Second, the statistician may use a robust statistic, such as the median
statistic.
Peirce's criterion is a statistical procedure for eliminating outliers.
:
Peirce's criterion is derived from a statistical analysis of the Gaussian distribution. Unlike some other criteria for removing outliers, Peirce's method can be applied to identify two or more outliers.
Hawkins provides a formula for the criterion.
Peirce's criterion was used for decades at the United States Coast Survey.
Peirce's criterion was discussed in William Chauvenet
's book.
Robust statistics
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.- Introduction :...
, Peirce's criterion is a rule for eliminating outlier
Outlier
In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....
s from data sets, which was devised by Benjamin Peirce
Benjamin Peirce
Benjamin Peirce was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics....
.
The problem of outliers
In data setData set
A data set is a collection of data, usually presented in tabular form. Each column represents a particular variable. Each row corresponds to a given member of the data set in question. Its values for each of the variables, such as height and weight of an object or values of random numbers. Each...
s containing real-numbered measurements, the suspected outlier
Outlier
In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....
s are the measured values that appear to lie outside the cluster of most of the other data values. The outliers would greatly change the estimate of location if the arithmetic average were to be used as a summary statistic of location. The problem is that the arithmetic mean is very sensitive to the inclusion of any outliers; in statistical terminology, the arithmetic mean is not robust.
In the presence of outliers, the statistician has two options. First, the statistician may remove the suspected outlier
Outlier
In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....
s from the data set and then use the arithmetic mean to estimate the location parameter. Second, the statistician may use a robust statistic, such as the median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
statistic.
Peirce's criterion is a statistical procedure for eliminating outliers.
Peirce's criterion
The statistician and historian of statistics Stephen M. Stigler wrote the following about Benjamin PeirceBenjamin Peirce
Benjamin Peirce was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics....
:
"In 1852 he published the first significance test designed to tell an investigator whether an outlier should be rejected (Peirce 1852, 1878). The test, based on a likelihood ratio type of argument, had the distinction of producing an international debate on the wisdom of such actions (Anscombe, 1960, Rider, 1933, StiglerStephen StiglerStephen Mack Stigler is Ernest DeWitt Burton Distinguished Service Professor at the Department of Statistics of the University of Chicago. His research has focused on statistical theory of robust estimators and the history of statistics...
, 1973a)."
Peirce's criterion is derived from a statistical analysis of the Gaussian distribution. Unlike some other criteria for removing outliers, Peirce's method can be applied to identify two or more outliers.
"It is proposed to determine in a series ofobservations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as
such observations. The principle upon which it is proposed to solve this problem is, that the proposed observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many, and no more, abnormal observations."
Hawkins provides a formula for the criterion.
Peirce's criterion was used for decades at the United States Coast Survey.
"From 1852 to 1867 he served as the director of the longitude determinations of the U. S. Coast Survey and from 1867 to 1874 as superintendent of the Survey. During these years his test was consistently employed by all the clerks of this, the most active and mathematically inclined statistical organization of the era."
Peirce's criterion was discussed in William Chauvenet
William Chauvenet
William Chauvenet was an early American educator. A professor of mathematics, astronomy, navigation, and surveying, he was always known and well liked among students and faculty....
's book.