The Generalized Bootstrap GB

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The Generalized Bootstrap GB

• Edward J. Dudewicz
• Department of Mathematics
• Syracuse University
• Syracuse, New York 13244

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• The Generalized Bootstrap GB History, Theory,
  Comparison (with the Naïve Bootstrap NB of
  Efron), and Practice
• or
• I Fitted A Distribution to My Data, Now What Do
  I Do With It? (Dont Use Your Daddys Bootstrap)

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History

• I. Suppose one has a dataset consisting of n
  observations X1,X2,,Xn where
• a. the observations are independent random
  variables, i.e.
• P(X1 x1, X2 x2,, Xn xn)
• P(X1 x1)P(X2 x2)P(Xn xn) for all real
  numbers x1,x2,,xn and,
• b. each of the observations has the same
  probability distribution, i.e.
• P(X1 x) P(X2 x) P(Xn x) for every
  real number x.
• (We often denote the common value P(Xi x) by
  F(x), called the distribution function (d.f.) of
  Xi.)
• We then say the dataset X1,X2,,Xn consists of
  independent (a.) and identically distributed (b.)
  (i.i.d.) random variables.

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Examples

• Examples of this situation include many
  situations encountered in statistical practice.
  For example, some examples in Environmental
  Quality cited by Michael E. Ginevan and Douglas
  E. Splitstone (Statistical Tools for
  Environmental Quality Measurement, CRC Press,
  2003, Ch. 6) include these
• Ex. 1 On a residential site, two hundred and
  twenty-four samples were collected and the
  arsenic concentration in the surface soil was
  measured.
• Ex. 2 265 daily values were measured of the
  concentration of copper in the effluent of a
  treatment plant discharged into a river.

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• In such examples, often it is assumed that the
  d.f. is one of the classical statistical models
  such as the normal, log-normal, or Weibull
  distribution.
• In Ex. 1, it often would be assumed that the
  sample mean follows the normal distribution
  (relying on the Central Limit Theorem). This
  might allow one to assess the risk to an
  individual who moves around a residential lot at
  random.
• However, the distribution of the original data is
  highly skewed towards the upper tail, and it
  turns out that even has a skewed distribution.

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• In Ex. 2, it often would be assumed (USEPA
  assumption for effluent concentration) that the
  data follow a log-normal distribution. This
  might allow one to estimate the 99th percentile
  of effluent concentration, which might be used in
  a permit for the amount allowed in effluent (and
  with the 99th percentile in the permit, one would
  expect a 1 chance of violating the permit on any
  one day). However, the Shapiro-Wilk test
  strongly rejects a log-normal model. And since
  penalties for violating such a permit from the
  state Department of Environmental Protection can
  range from 10,000 per day and a possible jail
  term, one might be ill-advised to assume a
  log-normal model in this setting.

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Problem

• So, we have a Problem the data often do not
  follow such classical models as the normal,
  log-normal, or Weibull distribution.

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What might be a Solution?

• An approach, in use as early as 1967, is based on
  drawing samples at random with replacement from
  the data at hand.
• For instance, in Ex. 1, draw 10,000 samples of
  size 224 at random with replacement from the
  dataset. for each of the samples find the sample
  mean. To estimate the 95th percentile of the
  exposure distribution, take the 9500th from
  smallest of the 10,000 sample means.
• In Ex. 2, generate (e.g.) 5,000 samples of size
  265 at random with replacement from the dataset,
  and for each find the 99th percentile of copper
  concentration in the effluent. Take the 95th
  percentile of the 5,000 samples, i.e. the
  (.95)(5000) 4750th from smallest as a number
  for which we are 95 percent confident that there
  is no more than a one-percent chance of an
  unintentional permit violation if the permit
  limit for effluent copper is set at this 4750th
  value.

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bootstrapping

• This method gained wide use when it was given
  the name bootstrap by B. Efron in 1979 (or, the
  bootstrap method, BM). We note that the
  Merriam-Webster Online Dictionary defines a
  bootstrap as a looped strap sewed at the side or
  the rear top of a boot to help in pulling it on
  I have boots I sometimes use in winter, and they
  have bootstraps. As an aside, according to the
  Wikipedia, Karl Friedrich Hieronymus, Baron von
  Munchhausen (17201797), was a German nobleman
  who supposedly told tall tales about his
  adventures, one of them being that he used his
  own bootstraps to pull himself out of the ocean,
  which gave rise to the term bootstrapping.

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Theory

• The Theory behind the bootstrap resampling method
  is essentially that of the Central Limit Theorem
  (or of the Glivenko-Cantelli Theorem) it will
  perform well as the sample size approaches
  infinity.

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• But, in practice, this method often fails to
  perform well. As Ginevan and Splitstone note
  for the two examples given above, application of
  the bootstrap to the estimation of extreme
  percentiles is not as robust as its application
  to the estimation of summary statistics like the
  meanIt requires large original sample sizes and
  the results are more sensitive to outlying
  observations.

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• So, the bootstrap is not the solution. What to
  do? Ginevan and Splitstone say
• What can I do that would be any better? In
  this case a better alternative is not obvious,
  so, despite its limitations, the bootstrap
  provides a reasonable solution.

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• Lack of a better alternative does not make a
  reasonable solution.

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• But, there is a better alternative to the naïve
  bootstrap.

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drawbacks of the BM

• Consider the drawbacks of the BM. For example,
  if we have 50 years of data on rainfall in an
  area and we are designing a levee, the BM will
  never yield observations more extreme than the
  most extreme in the dataset of 50. This can
  easily lead to design of a levee that will not
  withstand the100-year ?ood (a typical design
  criterion in the U.S.) or the1000-year ?ood (a
  design criterion in Europe) with high
  probability. (For more details, see Dudewicz
  (1992) and Dudewicz and Mishra (1988, Section
  5.6).) Thus
• the Bootstrap Method is fraught with danger of
  seriously inadequate results.

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The Generalized Bootstrap (GB) Method

• The Generalized Bootstrap alternative was
  introduced formally in Dudewicz (1992) (and
  discussed earlier in Karian and Dudewicz (1991,
  Section 6.6)).

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essence of the Generalized Bootstrap (GB) Method

• The essence of the Generalized Bootstrap (GB)
  Method is to ?t an EGLD to the available data,
  and then take samples from the ?tted distribution
  (and work with them as the BM does with its
  samples from the data itself). This method has
  been shown to do better than the BM when the
  number of data points is not very large, and do
  as well as the BM when the number of data points
  is large. Sun and MüllerSchwarze (1996) have an
  excellent exposition with real-data examples.
  More recent examples in datamining, including
  interactive computer systems, consumer purchases,
  crop damage, and pathogens, are considered by
  Dudewicz and Karian (1999b).

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A Recent Example of use of the GB Method

• In the article A Rainfall-Based Model for
  Predicting the Regional Incidence of Wheat Seed
  Infection by Stagonospora nodorum in New York
  (Phytopathology 92 (2002), 511-518), Denis A.
  Shah and Gary C. Bergstrom of the Department of
  Plant Pathology at Cornell University developed a
  predicted distribution of seed infection
  incidence. They modeled the relationship between
  incidence of wheat seed infection (by S. nodorum)
  and rainfall.

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they had Dataavailable for 7 years only

• In their study, they had Data on the incidence
  of wheat seed infection by S. nodorum in New
  Yorkavailable for 7 years only. Because least
  square parameter estimates have a bias inversely
  proportional to sample size, there is reason for
  concern with such a small data set.

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a solution

• As a solution, we generated data sets for the
  regression analysis by the generalized bootstrap
  methodIt differs from the familiar
  bootstrapin that samples are generated from
  probability distributions fitted to the original
  data rather than sampling from the observed data
  itself.

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The bootstrap is not capable of observations
that may be rare

• The bootstrap is not capable of generating
  samples containing observations that may be rare
  because it is based on the actual observed data,

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drawbackovercome by the generalized bootstrap

• but this drawback was overcome by the
  generalized bootstrap, which is based on the
  probability distribution assumed to describe the
  population from which the actual sampled data
  were obtained. For the multiple regression, we
  generated 1,000 generalized bootstrap samples for
  each year (7,000 total samples) from the
  probability distributions fitted to seed
  infection incidence and rainfall.

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Comparisons

• One could base a GB method on any distribution,
  not only the EGLD, so let us call the GB
  introduced in 1992 the GB(EGLD).

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the GB(N)

• For example, one could base a method on the
  Normal distribution, say the GB(N). But the N
  distribution has only a mean and variance to be
  fitted, the skewness is fixed at 0 and the
  kurtosis at 3. So, this method would not be able
  to adapt to the wide range of distributions one
  finds in applications.

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the GB(EGLD) suitable forapplications

• In contrast, the EGLD fits all means and
  variances, and a wide range of skewness and
  kurtosis, which makes the GB(EGLD) suitable for
  many applications.

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Good fittingis key

• Good fitting of the basic EGLD is key to good GB
  resultsand this is a thrust of many of the
  papers to be presented at this Symposium.

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this Symposiumof importance

• Other distributions and distribution systems
  could also be used as a basis for a GB, and
  results with them are thus also of importance in
  this use of the fitted distribution, as are
  results with multivariate distributions.