Have you used this estimator, or know of it being used? Are you familiar with its properties? Do you agree or disagree with the analysis given in the attached notes, and why or why not? When would you use a chain ratio-type estimator or related estimator?
(Note that this is a replacement for a question submitted a couple of days ago, which received a good reply regarding use of the chain ratio-type estimator for estimating crop yield with biometric auxiliary data, but ResearchGate dropped the question and Naveen's reply. The crop yield reply seemed to me to reinforce the idea of a nonlinear model, as noted in the attached paper.)
Technical Report Note on Ratio and Chain Ratio-Type Estimators: Gamma and Alp...
For example,
1) hazard rate function in reliability theory
2) logarithmic density derivative in signal processing
I had thought that the chain ratio-type estimator was only used in sample surveys to estimate totals for finite populations, but now three very different additional areas have been mentioned: crop yield, reliability, and signal processing. Surprising to me! - Well, crop yield could be from a survey, but I do not see how this works with reliability or signal processing.
Many modifications of the classical ratio estimator have been made in the past quarter century. The alternative ratios discussed in Sarndal, Swensson, and Wretman and in Brewer's work and others rely on differing degrees of heteroscedasticity, but I refer here to the chain ratio-type estimator, exponential estimator, statistics added to terms in numerators and denominators, and combinations of these ratio and product estimators, and the like, which have been produced in many papers, and as far as I have seen, it appears that little thought has been given to interpretation and logic behind these modifications of the classical ratio estimator. The 'exponential' ratio-type estimator appears to have appeared in 1991 where it seems to have been justified in terms of a transformation and its MSE, but the logic of the estimator is not clear to me.
As an example, note that in STATISTICS IN TRANSITION-new series, June 2008, Vol. 9, No. 1,
pts.stat.gov.pl/cps/rde/xbcr/pts/PTS_sit_9_1.pdf,
there is an estimator on page 106 which is said to be a linear combination (Stein/shrinkage type) of a ratio and a product estimator (exponential type). However, note that the same auxiliary variable is being used in each term. A ratio estimator requires a positive correlation between x and y and a product estimator requires a negative correlation, here for the same x and y. How can the same two data sets be both positively and negatively correlated?
Many estimators seem to be justified by optimizing the variance estimate, but I suspect that this may be at the expense of bias. Also note that just because a calculated variance estimate is low, does not mean that the estimator was correct in the first place. (Many use an OLS estimator when the data are demonstrably and logically WLS.)
When more than one auxiliary variable is used in a multiplicative manner, it seems that often, not enough attention might be given to interaction between these variables.
Perhaps I am missing something, but the logic escapes me for these modified classical ratio estimators. In the notes attached through the link from my question above, I show that a 'repeated substitution' justification for the chain ratio-type estimator does not make sense, but perhaps considering it as based on nonlinear regression instead of linear regression does make sense. Are there other justifications?
Any thoughts on any of these modifications of the classical ratio estimator, and how they should be interpreted?
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