The problem of estimating a proportion of objects with a particular attribute in a finite population is considered. The classical estimator is compared with the estimator, which uses the information that the population is divided among two strata. Theoretical results are illustrated with a numerical example.
In this article we made an approach to use tools of the grade data analysis to scrutinize structure of the consumer expenditures. Research material was taken from the Central Statistical Office of Poland database concerning expenditure of Polish households in the period of 1999–2015. Grade data analysis is treated as one of the multivariate data analysis methods and can be perceived as a method that is focused on fine visualization of the data in order to better understanding the interdependence that is contains. To do so, we used so-called overrepresentation maps. The analysis was conducted with taking into account time, class of the locality and socio-economic group. Conducted research showed the direction of changes in the average structure of the consumer expenditure.
This article is revision-based. The aim of the research was to define the welfare measures used, which are in the form of proportion. At the beginning, a historical sketch of the welfare economy was presented. Then, the welfare domains described in the literature are presented and the corresponding welfare indicators were identified. In the next section of the article indicators classified into particular subgroups were presented. The subgroups are: traditional measures of the economy, modified national accounts, index measures and environmental indicators. The article ends with a discussion of the data sources for the welfare indicators and based on that the indicators in the form of proportion were indicated.
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