Exploring recursion for optimal estimators under cascade rotation - ARCHIVED

We are concerned with optimal linear estimation of means on subsequent occasions under sample rotation where evolution of samples in time is designed through a cascade pattern. It has been known since the seminal paper of Patterson (1950) that when the units are not allowed to return to the sample after leaving it for certain period (there are no gaps in the rotation pattern), one step recursion for optimal estimator holds. However, in some important real surveys, e.g., Current Population Survey in the US or Labour Force Survey in many countries in Europe, units return to the sample after being absent in the sample for several occasions (there are gaps in rotation patterns). In such situations difficulty of the question of the form of the recurrence for optimal estimator increases drastically. This issue has not been resolved yet. Instead alternative sub-optimal approaches were developed, as K - composite estimation (see e.g., Hansen, Hurwitz, Nisselson and Steinberg (1955)), AK - composite estimation (see e.g., Gurney and Daly (1965)) or time series approach (see e.g., Binder and Hidiroglou (1988)).

In the present paper we overcome this long-standing difficulty, that is, we present analytical recursion formulas for the optimal linear estimator of the mean for schemes with gaps in rotation patterns. It is achieved under some technical conditions: ASSUMPTION I and ASSUMPTION II (numerical experiments suggest that these assumptions might be universally satisfied). To attain the goal we develop an algebraic operator approach which allows to reduce the problem of recursion for the optimal linear estimator to two issues: (1) localization of roots (possibly complex) of a polynomial Q_p defined in terms of the rotation pattern (Q_p happens to be conveniently expressed through Chebyshev polynomials of the first kind), (2) rank of a matrix S defined in terms of the rotation pattern and the roots of the polynomial Q_p. In particular, it is shown that the order of the recursion is equal to one plus the size of the largest gap in the rotation pattern. Exact formulas for calculation of the recurrence coefficients are given - of course, to use them one has to check (in many cases, numerically) that ASSUMPTIONs I and II are satisfied. The solution is illustrated through several examples of rotation schemes arising in real surveys.