unequal probability
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Pier Luigi Conti ◽  
Fulvia Mecatti ◽  
Federica Nicolussi

Guillaume Chauvet

Abstract Chromy (1979) proposed an unequal probability sampling algorithm, which is the default sequential method used in the SURVEYSELECT procedure of the SAS software. In this article, we demonstrate that Chromy sampling is equivalent to pivotal sampling. This makes it possible to estimate the variance unbiasedly for the randomized version of the method programmed in the SURVEYSELECT procedure.

Seyed-Farzad MOHAMMADI ◽  
Cyrus ALINIA ◽  
Ebrahim GHADERI ◽  
Alireza LASHAY ◽  
Mahmoud JABBARVAND ◽  

Background: Due to lack of information about ophthalmic economics in Iran, health policy makers unable to distribute resources optimally in terms of efficiency and equity. Therefore, we estimated the total and eye care utilization, out-of-pocket expenditures, and its association with social determinants of health in Iran in 2015. Methods: A multi-stage population-based, cross-sectional study in a random sample aged 50 yr or older in Kurdistan Province, Northwest Iran was used. The utilization rate of eye and general health care and related out-of-pocket expenditures was estimated during the recent last six months. To find the association between social factors and care out-of-pocket expenditures, we used a Heckman two-step regression model. Results: About 81% and 37% of participants were utilized the health and ophthalmic services, respectively. Statistically significant lower ophthalmic utilization rates were observed among men, middle-aged population, illiterate participants, rural residents, daily-paid workers, and the poorest participants. The average of vision and total health-related out-of-pocket expenditures among those used these services have estimated as US$43.7 (SE: 2.6) and US$439.9 (SE: 22.8), respectively. The highest (US$ 396.6) and lowest (US$ 10.4) ophthalmic out-of-pocket costs were related to patients with Glaucoma and Central Nervous System abnormalities, respectively. Multivariate analyses confirmed an unequal probability of having the ophthalmic out-of-pocket expenditures among different subgroups especially in favor of females, older, and those with more severe visual impairment. Conclusion: Ophthalmic disorders reconstituted about 10% of all health services OOP expenditures on average among individuals older than 50 yr.

2020 ◽  
Vol 36 (2) ◽  
pp. 237-249
Yves G. Berger ◽  
İklim Gedik Balay

AbstractWe propose an estimator for the Gini coefficient, based on a ratio of means. We show how bootstrap and empirical likelihood can be combined to construct confidence intervals. Our simulation study shows the estimator proposed is usually less biased than customary estimators. The observed coverages of the empirical likelihood confidence interval proposed are also closer to the nominal value.

Biometrika ◽  
2020 ◽  
Vol 107 (4) ◽  
pp. 857-873
A Yiu ◽  
R J B Goudie ◽  
B D M Tom

Summary Fully Bayesian inference in the presence of unequal probability sampling requires stronger structural assumptions on the data-generating distribution than frequentist semiparametric methods, but offers the potential for improved small-sample inference and convenient evidence synthesis. We demonstrate that the Bayesian exponentially tilted empirical likelihood can be used to combine the practical benefits of Bayesian inference with the robustness and attractive large-sample properties of frequentist approaches. Estimators defined as the solutions to unbiased estimating equations can be used to define a semiparametric model through the set of corresponding moment constraints. We prove Bernstein–von Mises theorems which show that the posterior constructed from the resulting exponentially tilted empirical likelihood becomes approximately normal, centred at the chosen estimator with matching asymptotic variance; thus, the posterior has properties analogous to those of the estimator, such as double robustness, and the frequentist coverage of any credible set will be approximately equal to its credibility. The proposed method can be used to obtain modified versions of existing estimators with improved properties, such as guarantees that the estimator lies within the parameter space. Unlike existing Bayesian proposals, our method does not prescribe a particular choice of prior or require posterior variance correction, and simulations suggest that it provides superior performance in terms of frequentist criteria.

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