April 21, 2010 STAT 950 Chris Wichman

Motivation Every ten years, the U.S. government conducts a population census, and every five years the U. S. National Agricultural Statistics Service conducts an Agriculture Census. Notice, that for the given “moment in time” that the census is taken, the total population, N, is known. In the intervening years, the numbers from each census are used to make inferences. For example, mean population in urban areas, and farm output (average bushels/acre).

Motivation Of interest is an intervening year population average: Two statistics commonly employed in these situations: The ratio estimator: The regression estimator:

Sample Average Without Replacement Samples Population Average, where the unbiased estimator of μ is When is based on a sample taken without replacement, the true variance of is: the unbiased estimator of which is:

The Problem with the Ordinary Bootstrap Recall, when a resample, is taken with replacement from the original sample then: Note that the only matches the form of if the sampling fraction,. In other words, the ordinary bootstrap fails to realize the “contraction” in.

Proposed Resampling Methods Modified Sample Size With replacement Without replacement Mirror Match Population Superpopulation

Modified Sample Size Find a resampling size such that the is approximately matched by. Process: Find the form of Take the expected value of and set equal to Solve for

Modified Sample Size With-Replacement For with replacement resampling, the bootstrapped variance of is: this leads to a modified sample size > than n

Modified Sample Size Without-replacement For without-replacement resampling, notice that the effective N for each resample is really n. The making the obvious choice for one in which

Mirror Match Goals: Capture the dependence due to sampling without- replacement Minimize the instability of the resampled statistic, by matching the original sample size Process: Suppose Then simply concatenate k resamples of size m together to form an

Mirror Match When m and k are not integers: Round m = nf to the nearest whole number Choose k such that Randomly select either k or (k+1) without-replacement resamples of size m from. Sampling probabilities should be chosen to match f

Population Bootstrap If is an integer: create a fake population Y*, by repeating k times. Generate R replicate samples of size n, by sampling without-replacement from Y*. Each resample will have the same sampling fraction as the original sample.

Population Bootstrap If is not an integer: Find k and l such that N = nk + l, and. create a fake population Y*, by repeating k times and joining it with a without replacement sample of size l from. This step is repeated R times. Generate R replicate samples of size n, by sampling without-replacement from Y*. Each resample will have the same sampling fraction as the original sample.

Superpopulation Bootstrap For each resample, 1,...,R Create a fake population, Y*, of size N, by resampling with replacement from, N times. From each Y 1 *,..., Y N * take a without replacement sample of size n. Each resample will have the same sampling fraction as the original sample.

Example 3.15: City Population Data A Comparison of Confidence Intervals In this example, the normal approximation C.I. refers to the bias corrected interval: The remaining intervals are Studentized confidence intervals :

Example 3.15: City Population Data Table 3.7 Resampling Scheme RatioRegression Normal Modified Size, n' = NA Modified Size, n' = Mirror Match, m = Population Superpopulation Resampling Scheme RatioRegression Normal Modified Size, n' = NA Modified Size, n' = Mirror Match, m = Population Superpopulation

Example 3.15: City Population Data Table 3.8 Recreated in R Coverage Length LowerUpperOverall AverageSD Normal Modified Size, n' = Modified Size, n' = Mirror Match, m = Population Superpopulation From BMA pg 96 Coverage Length LowerUpperOverall AverageSD Normal Modified Size, n' = Modified Size, n' = Mirror Match, m = Population Superpopulation

Example 3.15: City Population Data Figure 3.6

How Well does the Normal Approximation fit the Distribution of t reg and t rat ?

Conclusions About t rat and t reg The normal approximation for the ratio and regression estimators performs poorly. The estimated expected length of confidence intervals based on the normal approximation are very short relative to the other resampling methods. The estimated variance of the regression estimator is unstable, potentially causing huge swings in z* ultimately affecting the bounds of Studentized confidence intervals.

Stratified Sampling Suppose the population of interest is divided into k strata, then the population total, Each strata now has it’s own sampling fraction, Each strata represents proportion of the population.

t rat for a Stratified Sample Of interest is the overall mean: The ratio estimator for a stratified population becomes:

Example 3.17: Stratified Ratio Here, Davison and Hinkley drop the regression estimator, due to the potential instability of the variance affecting the bootstrapped confidence intervals. They also drop the Modified Sample, because they felt it was a “less promising” finite population resampling scheme.

Example 3.17: Methodology Simulate N pairs (u, x) divided into k strata of sizes “small-k”: k = 3, N i = 18, n i = 6 “small-k”:k = 5, N i = 72, n i = 24 “large-k”: k = 20, N i = 18, n i = different samples of size were taken from the dataset(s) produced above. For each sample, R=199 resamples were used to compute confidence intervals for θ.

All methods were used on the sample as described in example 3.15, with the exception of superpopulation resampling, which was conducted for each strata. Example 3.17: Methodology BMA Table 3.9k=20, N=18 k=5, N=72 k=3, N=18 LUO LUO LUO Normal Modified Sample Size Mirror-match Population Superpopulation

Conclusions: Stratified Sample The estimated coverage for Normal, Modified Sample Size, and Population resampling methods are all close to the nominal 90% desired. The “tail” probabilities are each roughly 5%. Neither the Mirror-match (estimated coverage of 83%), nor the Superpopulation (estimated coverage of 95%) performed very well. Due to their ease of calculation, Davison and Hinkley conclude that the Population and Modified Sample Size perform the best.