# How to Improve the Design of a Survey of Multiple Rare Variables Using Rare Variable Location Information

I am attempting to update the design of an annual survey which measures the percent-deficient of 26 different asset inventories. Most of these assets don't occur on every sampling unit and some of them are quite rare (occur on around 5% of sampling units). The asset deficiency rates can be estimated by either a simple proportion estimator, a domain proportion (for those that either exist or do not exist on a sampling unit and, if they exist, are either deficient or not), or a ratio estimator (for assets that have 0 or more units of inventory on a sampling unit and some or none of that inventory is deficient). The sample size that has been used historically is around 1,200 (~1%), which has been proportionally allocated into 72 strata.

Most estimates from past samples have indicated useful accuracy (standard errors of 1% or less). However, for assets that are rare and have high deficiency rates the accuracy is much worse and not useful for their purpose (e.g. standard errors of 5%).

Knowledge of which sampling units contain which assets is available (not necessarily the amount of that asset) and I would like to use this information to alter the sample inclusion probabilities within the same stratification scheme. Ideally, all assets would be estimated with a certain degree of accuracy. One way to do this might set inclusion probabilities such that the standard error would likely fall below some threshold for each asset (perhaps unique for different assets). If this is impossible for all 26 assets, then minimize the deviation above the threshold with more weight given to more important assets.

My initial thought was to create a linear program to determine inclusion probability adjustment factors for a sampling unit based on which assets existed on the unit (e.g. if asset 1 was on a sampling unit increase the inclusion probability by some amount). A feasible solution would ensure that a sample's expected number of units which contain each asset would meet a requirement (e.g. at least 300 units containing asset 1 are selected). This requirement could be determined based on (1) the number of sample units containing the feature from the previous year and (2) the asset's standard error from the previous year. The LP objective might be to minimize the difference between the smallest and largest weights.

Does this seem reasonable/possible? Are there better approaches? Are there references available?