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Repo Overview

The likelihood ratio test’s asymptotic sampling distribution occurs when two conditions are true:

  • N approaches infinity. (1)
  • The tested parameter is not close to the boundary. (2)

For the geometric distribution, satisfying the first condition causes the second condition to be violated. This repo explores how this affects the asymptotic sampling distributions and type I error rates.

Geometric Distribution

An Example

The geometric distribution gives the probability of the number of failures until the first success. As an example, imagine selling fruit door-to-door. If there is a 15% chance a home owner purchases, what is the probability of needing to visit exactly 11 homes for the first sale?

round(dgeom(x = 10, prob = .15), 3)
#> [1] 0.03

What is the probability of needing to visit 11 or more homes for the first sale?

round(pgeom(q = 10, prob = .15, lower.tail = FALSE), 3)
#> [1] 0.167

Note there are a few definitions of the geometric distribution. R’s probability functions count the number of failures before the first success. Other definitions count the number of independent trials. Hence 10 and not 11 in the code.

Asymptotic Theory

Intuitively, if the first home owner purchases the probability of purchase is high. Roughly,a small N implies p is much larger than zero. Condition 2 is satisfied and condition 1 is violated.

If one thousand homes are visited before the first purchase, the probability of purchase is low. Roughly, a large N implies p is near or equal to zero. Condition 1 is satisfied and condition 2 is violated.

This means satisfying one condition for the asymptotic $\chi^2$ sampling distribution causes the other condition to be violated.

Sampling Distribution

Simulation Process

  • Step 1: Generate the number of failures before the first success with a given probability of success.
  • Step 2: Calculate likelihood test statistic.
  • Step 3: Repeat for b 1 to 10,000.
  • Step 4: Repeat with probability of success between .05 and .95.

The simulation is done for a one sample test and a two sample test.

Results: Sampling Distribution

For both the one sample case and the two sample case, the sampling distribution of the test statistic does not follow the $\chi^2$ distribution.

Results: Type I Error Rate

For the likelihood tests, p values are calculated under the assumption the test statistic follows a $\chi^2$ distribution. Considering this is not true, what is the type I error rate?

In general, type I error rates are below the target 5 percent. Interestingly, the likelihood test is closer to the target .05 than the exact test.

The exact test fails to have exactly .05 type I error rate because the test statistic (number of failures) is discrete and not continuous.