Immigration and Naturalization Service
Management of Property
Report No. 0109
March 2001
Office of the Inspector General
STATISTICAL SAMPLING MODEL
The statistical sampling universe for our model was defined as all items in the AMIS database with an acquisition cost greater than $2 and less than $1 million. The defined universe contained 300,236 items with a total dollar cost of $629,885,361. The items were located at over 500 different offices primarily in the United States with a few offices located outside the United States. We used a stratified, multistage clustered, stratified random sample model to provide both effective coverage of items as well as precise estimates. Probabilityproportionaltosize sampling methods were applied to obtain effective coverage of larger units.
From the universe of 300,236 items we tested a random sample of 1,205 items. The randomly sampled items were spread across INS Districts, Border Patrol Sectors, Service Centers, Headquarters, and other offices (please refer to Appendix II for a list of sites reviewed and associated sample allocations). Of the 1,205 sampled items, 311 items could not be accounted for. Using our statistical model and the formulae referenced by (A) and (B) below, we projected the sample test results to the entire universe with 95% confidence.
The INS could not account for at least 61,028 items with a total cost of $68,907,085 and as many as 81,723 items with a total cost of $107,645,022. To be conservative, we stated our projections at the lower values when questioning costs in our audit.
To arrive at these results, the mathematical model notations and formulae used to compute the unbiased estimates of expected values and the variances leading to (A) and (B) are as follows.
H The number of primary unit strata in the population Nh The number of primary units in the h^{th} stratum nh The number of primary units sampled from the h^{th} stratum Mhi The number of secondary units in the i^{th} sampled primary unit from the h^{th} stratum nhi The number of secondary units sampled from the i^{th} sampled primary unit within the h^{th} stratum Khij The number of secondary unit strata in the j^{th} sampled secondary unit from the i^{th} sampled primary unit within the h^{th} stratum Mhijk The number of items in the k^{th} stratum of the j^{th} sampled secondary unit within i^{th} sampled primary unit from the h^{th }stratum mhijk The number of items sampled from the k^{th} stratum of the j^{th} secondary unit sampled from the i^{th} sampled primary unit within the h^{th} stratum yhijku Random variable and its value corresponding to u^{th} sampled item from the k^{th} stratum of the j^{th} secondary unit sampled from the i^{th} sampled primary unit within the h^{th} stratum thijku Cost of the unit yhijku
yhijku  0 If uth sampled item from the kth stratum of the jth secondary unit sampled from the ith sampled primary unit within the hth stratum is accounted for 1 Otherwise 
Thus
Estimate of the total number of items not accounted for in the population
Sample proportion of hits (items not accounted for) in the kth stratum of the jth secondary unit

To compute the variance of the estimate the formulae used are as follows.
Where indicates variance of the ith unit at the next lower level; , are the inclusion probabilities of the units i and j respectively in the sample at the current level; and is the inclusion probability of both the units i and j together being selected at the current level sample.
The 95% lower confidence limit on the estimate is given by
(A)
and 95% upper confidence limit on the estimate is given by
(B)
Similarly, the lower and upper 95% confidence limits for the estimate were also computed. The lower confidence limits for (1) the items valued at greater than $2, but less than or equal to $5,000, and (2) Headquartersonly items are obtained by adjusting the appropriate indexes in the above formulae.