Abstract
Conventional sampling in biostatistics and economics posits an individual in a fixed observable state (e.g., diseased or not, poor or not, etc.). Social, market, and opinion research, however, require a cognitive sampling theory which recognizes that a respondent has a choice between two options (e.g., yes versus no). This new theory posits the survey re spondent as a personal probability. Once the sample is drawn, a series of independent non-identical Bernoulli trials are carried out. The outcome of each trial is a momentary binary choice governed by this unobserved proba bility. Liapunov’s extended central limit theorem (Lehmann, 1999) and the Horvitz-Thompson (1952) theorem are then brought to bear on sampling unobservables, in contrast to sampling observations. This formulation reaf firms the usefulness of a weighted sample proportion, which is now seen to estimate a different target parameter than that of conventional design-based sampling theory
Highlights
Estimating a population proportion is commonplace in psychological assessment, experimentation, and opinion surveys
The present paper argues for the application of the more realistic Liapunov central limit theorem
A sample of random variables, with subsequent Bernoulli trials, gives a sample proportion that estimates the mean of a population of proportions
Summary
Estimating a population proportion is commonplace in psychological assessment, experimentation, and opinion surveys. The present paper argues for the application of the more realistic Liapunov central limit theorem This relaxes the status quo to independent non-identically distributed (i.n.d.) Bernoulli trials, each with an individual-specific (case) weight and response probability. Rather than revealing a predetermined individual state, each Bernoulli trial generates a momentary response (a one or zero) driven by an individual’s unobserved probability Given this alternative representation of the survey respondent, the Liapunov central limit theorem, and the Horvitz-Thompson (1952) theorem, are invoked to estimate the mean of the population of personal probabilities. This gives the conditional and unconditional expectations of the (approximately normal) sample mean of n weighted Bernoulli variates, along with its conditional variance.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
