(i) to find the Lagrange multiplier λ as the solution to (6) with a single non-stratified
sample;
(ii) to obtain the MPEL solutions for stratified sampling, raking ratio estimation
and other “irregular” cases; and
(iii) to construct the pseudo EL ratio confidence intervals through profiling.
We assume that the population mean X is an inner point of the convex hull formed by the sample observations {xi, i ∈ s} so that a unique solution to (6) exists. Chen et al. (2002) proposed a simple algorithm for solving (6) with guaranteed convergence if the solution exists.
The uniqueness of the solution and the convergence of the algorithm are proved based on a duality argument: maximizing lns(p) with respect to p subject to pi > 0, Pi∈s pi = 1 and the benchmark constraints (5) is a dual problem of maximizing H(λ) = Pi∈s ˜ di(s) log(1 + λ0ui) with respect to λ with no restrictions on λ. In both
cases the solution λ solves the equation system (6)
In both cases the solution λ solves the equation system (6). Since H(λ) is a concave function of λ with the matrix of second order derivatives negative definite, a unique maximum point to H(λ) exists and can be found using the Newton-Raphson search algorithm. Denoting xi −X by ui, the algorithm of Chen et al. (2002) for solving (6) is as follows
6 Discussion
The EL and PEL approaches are flexible enough to handle a variety of other problems.
In particular, data from two or more independent surveys from the same target population can be combined naturally through the PEL approach and efficient point estimators and pseudo EL ratio confidence intervals for the population mean can be obtained.
The PEL approach is flexible in combining data from different sources as demonstrated above. Depending on
what is available, new constraints can be added to and existing ones can be removed from the system of constraints.
Another problem of interest is to make inference on the population parameters of interest in the presence of imputation for item non-response.
Again, the EL and pseudo EL approaches can be applied in a systematic manner to handle imputed data and any auxiliary population information. Recent work has focused on EL inference on the mean, distribution function and quantiles of a variable of interest y, assuming that an iid sample {(yi, xi), i = 1, · · · , n} subject to missing yi is available. The missing y-values are imputed using regression imputation, assuming a missing at random (MAR) response mechanism and a linear regression model (Wang and Rao, 2002a; Qin, Rao and Ren, 2006). EL inference using kernel regression imputation, assuming only that the conditional expectation of y given x is a smooth function of x, has likewise been studied (Wang and Rao, 2002b; Wang and Chen, 2006). Various extensions have also been analyzed. The pseudo EL approach can be applied to extend the above work to survey data