Calculation of Fractional Weights

The Fractional Weights option is based on the Bayesian bootstrap (Rubin 1981). The number of times that an observation occurs in a given bootstrap sample is called its bootstrap weight. In the simple bootstrap, the bootstrap weights for each bootstrap sample are determined using simple random sampling with replacement.

In the Bayesian approach, sampling probabilities are treated as unknown parameters and their posterior distribution is obtained using a non-informative prior. Estimates of the probabilities are obtained by sampling from this posterior distribution. These estimates are used to construct the bootstrap weights, as follows:

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Randomly generate a vector of n values from a gamma distribution with shape parameter equal to (n - 1)/n and scale parameter equal to 1.

Note: Rubin (1981) uses 1 as the gamma shape parameter. The shape parameter that is used in JMP Pro ensures that the mean and variance of the fractional weights are equal to the mean and variance of the simple bootstrap weights.

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Compute S = sum of the n values.

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Compute the fractional weights by multiplying the vector of n values by n/S.

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If a Freq or Weight variable is specified for the analysis, multiply the fractional weights by the Freq or Weight values on a row-by-row basis. The sum of the values of the Freq or Weight variable must be greater than 1.

This procedure scales the fractional weights to have mean and variance equal to those of the simple bootstrap weights. The fractional bootstrap weights are positive, sum to n, and have a mean of 1.