shrinkage due to breakage and pilferage (this component of bias would depend on the mode of data collection); and
coverage errors in the store sampling frame (i.e., missing stores in which consumers shop and including stores in which they do not).
Regrettably, it is not possible to estimate the democratic index ILp exclusively from store-level data, at least not without additional assumptions. The democratic weights, , are population means per HH, and HH data are necessary to estimate the means unbiasedly; such data are not usually available from stores (some store chains have adopted ID card programs that allow tracking of purchases by consumer).
It may be possible to approximate the democratic index from store-level data with periodic adjustment of the weights. This possibility exploits the relationship between plutocratic and democratic weights set forth above. From store-level data, one can construct an estimator of the plutocratic weights
Then we define the estimator of the democratic weights as
where the adjustment factor is the second term on the right side, developed from an independent HH survey, such as the Consumer Expenditure Survey. In this factor, c(Dgi0,Y+i0) is an estimator of the covariance between HH share and total HH consumption, and is an estimator of mean total consumption per HH in the base period. It does not seem necessary to estimate the adjustment factor for each time period (month) the price index is produced. Perhaps it might be acceptable to maintain the adjustment factor only on an infrequent basis.
Without question, one can imagine other hybrid schemes for estimating plutocratic or democratic price indexes. BLS’s current method is an outstanding example, with quantity weights coming from one survey and monthly prices from another.