STT 430/530, Nonparametric Statistics

1
3 pivot quantities on which to base bootstrap
confidence intervals

• Note that the first has a t(n-1) distribution
  when
• sampling from a normal population.
• The second has a chi-square(n-1) distribution
• when sampling from a normal pop.
• The third is Fishers transformation of the
• correlation coefficient (rxy) and it is approx.
• normal with mean 0 and variance1/(n-3).
• These pivot quantities share 3 important
  properties
• They have known distributions
• They have a parameter that controls their shape
• This parameter varies with sample size, but the
• shape will not vary with the value of the
  parameter
• being estimated....

2

• As the Students t distribution takes away the
  mean and divides by the s.e. of the sample mean,
  so we codify this process and call it
  Studentization for any estimator t of q .
• Well use the bootstrap form of the Studentized
  estimate to construct various bootstrap
  confidence intervals...

3

• So heres the way to do this for the mean.
• compute the sample mean and sample s.d. from
  the original data
• obtain a bootstrap sample of size n from the
  original data and compute the bootstrap mean,
  and the bootstrap standard
• deviation, sb.
• now compute the bootstrap t-pivot quantity
• now repeat this step at least 1000 times to get
  the bootstrap distribution of tb .
• let tb, .025 and tb, .975 be the .025 and
  .975 quantiles of the bootstrap distribution of
  tb . Construct the 95 confidence interval for m
  as
• Now implement this in R... try with the latch
  failure data in Table 1.2.1

4

• Bootstrap confidence intervals for the population
  standard deviation, s, can be done with the
  c2-pivot
• If the sample is from a normal distribution, then
  the c2-pivot has a chi-square distribution with
  n-1 degrees of freedom and a 95 confidence
  interval for sigma can be computed as
• If normality is not a reasonable assumption, then
  well use the bootstrap as follows
• compute the sample variance s2 for the original
  sample data.
• bootstrap the sample of size n and compute the
  bootstrap sample variance and the bootstrap
  c2-pivot quantity
• (here, sb2 is the bootstrap sample variance)
• now repeat 1000 times or more to get the
  bootstrap distribution of the bootstrapped
  c2-pivots

5

• then the bootstrap 95 CI for the population
  variance is
• take the square root if you want confidence
  intervals for sigma...
• Implement this in R... again on the Table 1.2.1
  data.
• Here are the problems from the textbook that are
  due at the final exam time. Be sure to write up
  the solution so I can tell that you understand
  the methods and the context of the problem.
  Include appropriate R output to illustrate your
  solutions...
• p.106 6
• p.141ff 4,5,6 (second part), 7,9
• p.189ff 1,3,4,5,7,8,9(a),12,13
• p.298ff 1,5,7
• On the last class day I will give you some data
  that will form the basis for the final exam the
  next week.
• See page 258 for a chart of Coverage Percentages
  in a simulation done by the author for various
  underlying distributions - make sure you
  understand the conclusions of that discussion

6

• See section 8.3 for a discussion of three
  additional methods for computing bootstrap
  confidence intervals in more generality The
  three methods are
• Percentile method
• Residual method
• BCA method (BCAbias corrected and accelerated)
• Suppose we are trying to estimate the unknown
  population parameter ? - use based on a
  sample of size n from the population.
• The percentile method constructs the CI by
  bootstrapping the estimator to get its
  distribution - in particular its .025 and .975
  quantiles. Construct the 95 CI for ? as
• The residual method estimates ? by first getting
  the bootstrap distribution of the residuals
• and then constructing the 95 CI for ? as

7

• The BCA method is more complex and a discussion
  of it is given in section 8.3.2. If youre
  interested in pursuing this method look at the R
  package called boot . More details can be found
  in Davison, A.C. and Hinkley, D.V (1997)
  Bootstrap Methods and Their Applications and
  Lunneborg, C.E. (2000) Data Analysis by
  Resampling Concepts and Applications.