Why we use the reciprocal of the probability of being selected into survey as the weighting sample?
It provides a 1-in-n scale for probabilities. For example, the reciprocal of 0.01 is 100, so an event with probability 0.01 has a 1 in 100 chance of happening. This is a useful way to represent small probabilities, such as 0.0023, which is about 1 in 435.
In probability and statistics, the reciprocal distribution is characterised by its probability density function being proportional to the reciprocal of the variable. I hope this helps :-)
An event with a small probability p of occurring is a rare event, and the rarer an event the bigger the entropy -lnp or ln(1/p). The entropy is a measure of the information conveyed by the event. No wonder the sample is being weighted by 1/p which is a measure of the information conveyed by the event within the sample.
Abdelbaset -
When you use probability-of-selection design-based methods for sampling, generally for finite populations, though being finite is not always necessary, the estimation technique is determined by the randomized sampling design. There is a school of thought which says this is too artificial, and regression modeling of the population is important in itself, not just for helping to establish the best probability-based design. There is an interesting review of these thoughts here:
Ken Brewer's Waksberg Award article:
.....
Brewer, K.R.W. (2014), “Three controversies in the history of survey sampling,” Survey Methodology,
(December 2013/January 2014), Vol 39, No 2, pp. 249-262. Statistics Canada, Catalogue No. 12-001-X.
http://www.statcan.gc.ca/pub/12-001-x/2013002/article/11883-eng.htm.
He believed in using probability sampling and models together, but he explains the different approaches, the pros and cons.
....
In your case, you appear to be referring to the most popular of the rigorous quantitative techniques for finite population sampling: a randomized design, where the probabilities of selection are used in estimates of totals, means, and sometimes proportions, and estimates of variance and bias. The simplest case is discussed in my encyclopedia entry here:
https://www.researchgate.net/publication/262971887_SURVEY_WEIGHTS
For a finite population there is this consideration:
https://www.researchgate.net/publication/262970761_FINITE_POPULATION_CORRECTION_fpc_FACTOR
.............
To understand much of this better, it may be helpful for you to study the Horvitz-Thompson estimator, as it may convey the connection between randomized selection of a sample and the use in estimating a total better than others:
https://onlinecourses.science.psu.edu/stat506/node/16
If you are new to all of this, however, you might first look at simple random sampling:
https://onlinecourses.science.psu.edu/stat506/node/5
I think that these Pennsylvania State University online materials are often very good. Here, you may just want to start with this:
https://onlinecourses.science.psu.edu/stat506/node/1
Best wishes - Jim
PS - When using auxiliary data to help improve estimates, one may use "calibration." In this context, "calibration" means that the survey (design-based) weights are modified to form calibration weights, which may also include regression weights. You can find a few notes on the concept of calibration weights in the following:
https://www.researchgate.net/publication/261508465_Use_of_Ratios_for_Estimation_of_Official_Statistics_at_a_Statistical_Agency
Article Use of Ratios for Estimation of Official Statistics at a Sta...
Article FINITE POPULATION CORRECTION (fpc) FACTOR
Article SURVEY WEIGHTS
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability