Simultaneous inference for proportions in multivariate stratified random sampling without replacement for service quality control using multiple choice questions

Determining the asymptotic distribution of the estimator for parameters of interest when employing complex sampling designs is a methodological issue of extreme importance. This paper addresses the problem of determining the asymptotic distribution of the vector of sample proportions when the survey questionnaire employs multiple-choice questions, as often done in service quality evaluations. Specifically, we consider a stratified sampling design where units within each stratum are selected without replacement, and the multivariate random variable from which the sample is drawn follows a multivariate Bernoulli distribution. The main goal is to obtain a valid asymptotic result under the specific conditions imposed by this design, so as to make statistical inference about the vector of specific parameters of interest, such as the proportion of responses assigned to each subdimension considered in the survey questionnaire.