Frequency Guided Tutorials

In some situations, we only observe a single outcome but can conceptualize an outcome as resulting from a two (or more) stage process. These are called two-stage, or “hierarchical,” type situations. Some special cases include:

1. problems where the parameters of the distribution are random variables,
2. mixture problems, where stage 1 represents the type of subpopulation and stage 2 represents a random variable with a distribution that depends on population type
3. an aggregate distribution, where stage 1 represents the number of events and stage two represents the amount per event.

In these situations, the law of iterated expectations can be useful. The law of total variation is a special case that is particularly helpful for variance calculations.

To apply these rules,

• Identify the random variable that is being conditioned upon, typically a stage 1 outcome (that is not observed).
• Conditional on the stage 1 outcome, calculate summary measures such as a mean, variance, and the like.
• There are several results of the step (ii), one for each stage 1 outcome. Then, combine these results using the iterated expectations or total variation rules.

For ( (a,b,0) ) problems:

• Examine the recursive expression and identify the distribution as ( (a,b,0) ) or ( (a,b,1) )
• Based on the sign of ( a ), identify the distribution as either binomial ( (a lt 0) ), Poisson ( (a=0) ), or negative binomial ( (a gt 0) ).
• Using the values of ( a ) and ( b ), identify the parameters of the distribution
• With the entire parametric distribution, you can compute any quantity desired, such as the starting value (p_0), expectation, and so forth.
• For some problems, you may need to interchange these steps. For example, in question 166, you can use the starting value to help determine the parameters of the distribution

For ( (a,b,1) ) problems, the strategy is similar. Here, one needs extra information (typically, the starting value (p_0^M)) in order to determine the proportionality constant
$$ c=frac{1-p_0^M}{1-p_0^0}$$
You can then use the relationship ( p_0^M = c p_0^0 ) to determine the modified distribution.

• Retrieve the general expression for the log-likelihood
• Using the problem information, identify the count variables
• Using the problem information, identify the probabilities
• Differentiate the log-likelihood
• Set the log-likelihood equal to zero and solve for the parameter estimate

In mixture problems, we only observe a single outcome but can conceptualize an outcome as resulting from a two (or more) stage process. These are called two-stage, or “hierarchical,” type situations. Specifically, stage 1 represents the type of subpopulation and stage 2 represents a random variable with a distribution that depends on population type.

In special instances such as Poisson-gamma, we know that a gamma mixture of Poisson variables has a negative binomial distribution. For other problems, one applies general rules from probability such as the law of iterated expectations, by using conditional probabilities through the hierarchical nature of the problem. To apply these rules,

• Identify the random variable that is being conditioned upon, typically a stage 1 outcome (that is not observed).
• Conditional on the stage 1 outcome, calculate summary measures such as a mean, variance, and the like.
• There are several results of the step (ii), one for each stage 1 outcome. Then, combine these results using, for example, the iterated expectations.

For model comparison/selection problems, consider the following outline of steps:

1. Identify the probabilities of outcomes under a model. For some problems, you will need to compute a maximum likelihood (or other type of) estimator to get these probabilities
2. From the outcome probabilities, determine how many counts one expects for each outcome under the model
3. Compare the actual and expected outcomes by computing a chi-square goodness of fit statistics
4. For some problems, you are asked to compare the goodness of fit statistic to a chi-square reference distribution. For this distribution, the parameter is known as a degree of freedom, say, df. Use df = the number of cells minus 1 minus the number of estimated parameters. (If there are no estimated parameters, just use df = the number of cells minus 1 )