Add Taylor Series linearisation

[smishr]

Add Taylor series linearisation for variance estimation for arbitrary SurveyDesign schemes

• [ ] Add taylor series linearisation function(s)
• [ ] Integrate with SurveyDesign

[smishr]

SAS does Taylor series for proportions and frequency tables, see documentation here. They also give example of variance estimation using ratio method here. Perhaps possible to test this formulae with ratio intially?

[smishr]

Chapter 9 of Lohr is also a very good resource with simple explanations and key formulae. Less mathematical clutter than Sarndal (1992).

[bschneidr]

I would strongly encourage taking a page from the 'survey' package in R, and using the general method of linearization based on influence functions.

https://www.practicalsignificance.com/posts/survey-covariances-using-influence-functions/#how-does-this-work

Using influence functions actually greatly simplifies the programming needed to implement linearization variance estimation. For each statistic (mean, total, regression coefficient, etc.), you just figure out how to calculate influence functions for that statistic, and then you pass the influence function values to whatever Julia function you use to estimate sampling variances/covariances for population totals.

This blog post I wrote gives a concrete example of how to calculate and use influence functions:

https://www.practicalsignificance.com/posts/how-correlated-are-survey-estimates-from-overlapping-groups/