Myriam Elizabeth Saavedra López - Repositorio Digital USFQ ...

level
Advantages:
α = 2k/ (n + 1) (7)
• The p-values of the tests are exact in the sense that no approximation of the distribution of the
test statistic is required.
• The p-values are inexact in the sense that the number of possible realizations under ψ is typically
infinite. At least the number of realizations will be so large that enumeration is not possible.
• The number n of simulations must be chosen sufficiently large. For a 5% level lest n = 99 and
for a 1% level test n = 999 have been recommended. As long as the model ψ can be simulated,
the observed pattern can be compared against complex point processes by essentially the same
procedure. Simulation tests thus provide great flexibility.
Disadvantages:
• Simulations of test have several critical choices at the user side, such as the number of simulations
and the test statistic.
• Diggles (1993) cautions against “data dredging”, the selection of a non-sensible statistic test for
the sake of rejecting a particular hypothesis. Even if a sensible statistic test is chosen, the results
of test simulations may not agree. The power of this procedure is also difficult to establish, in
particular, when applied to test for point patterns. The alternative hypothesis for which the
power is to be determined is not clear.
3.3 Simulation Envelopes
According to Schabenberger (2005, 88-89), a Monte Carlos Test calculates a single statistic test for
the observed pattern and each of the simulated patterns. Often, it is used with functions of the point
patterns. For example, let ri denote the distance from event si to the nearest other event and let
I (ri ≤ r) denote the indicator function with return 1, whether ri ≤ r. Then ˆ G (r) = 1
n
10
n
i=1 I (ri ≤ r)
is an estimate of the distribution function of nearest-neighbour event distances and can be calculated
for any value of r. With a clustered pattern, we expect an excess number of short nearest-neighbour
distances (compared to a CSR pattern). The method for obtaining simulation envelopes is similar to
that used for a Monte Carlos test, but instead of evaluating a single test statistic for each simulation,
a function such as ˆ G (r) is computed. Let ˆ G0 (r) denote the empirical distribution function based
on the observed point pattern. Calculate ˆ G1 (r) , ...., ˆ Gg (r) from g point patterns simulated under
CSR (or any other hypothesis of interest). Calculate the percentiles of the investigated function from