This question is about ratio estimation. I'd like to know whether the
following problem is amenable to statistical techniques. If it is, can you
help me by giving any relevant references?
I wish to estimate the ratio of two variables in a population.
There are two factors affecting the population, one with n levels and one with
m levels. The objects in the population can be regarded as being laid out in
the form of an n*m grid or matrix where the levels of one factor form the
columns and the levels of the other factor form the rows. A small minority of
the entries in the matrix are missing - i.e no object exists at that position
in the matrix. There is no more than one object at any position in the matrix.
Each object has a pair of values (x,y) associated with it. I wish to estimate
sum(x)/sum(y) where the sums are over the whole population.
The weight of each row (or column) is defined as the sum of the denominators
of the objects in that row (or column) - that is, the sum of the y values in
that row (or column). I have estimates for the weights of each row and each
column. These estimates are used to pick a weighted random sample of r of the
rows and a weighted random sample of c of the columns.
Now consider those objects which belong both to a row from the sample of
rows and a column from the sample of columns. These objects can be regarded
as forming a smaller r*c "sample" matrix. This sample matrix has missing
entries only where the population matrix has missing entries (so this is
not true missing data). For each object in the sample matrix I can obtain
an estimate of the ratio x/y.
What is the best way to use this information to estimate the population
ratio?
Rachel
Rachel Gladstone
Statistics Division
Department of Health
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