The optimization problem of quantile and poverty measures estimation based on calibration

Abstract

New calibrated estimators of quantiles and poverty measures are proposed. These estimators combine the incorporation of auxiliary information provided by auxiliary variables related to the variable of interest by calibration techniques with the selection of optimal calibration points under simple random sampling without replacement. The problem of selecting calibration points that minimize the asymptotic variance of the quantile estimator is addressed. Once the problem is solved, the definition of the new quantile estimator requires that the optimal estimator of the distribution function on which it is based verifies the properties of the distribution function. Through a theorem, the nondecreasing monotony property for the optimal estimator of the distribution function is established and the corresponding optimal estimator can be defined. This optimal quantile estimator is also used to define new estimators for poverty measures.Simulation studies with real data from the Spanish living conditions survey compares the performance of the new estimators against various methods proposed previously, where some resampling techniques are used for the variance estimation. Based on the results of the simulation study, the proposed estimators show a good performance and are a reasonable alternative to other estimators.

Reduction of optimal calibration dimension with a new optimal auxiliary vector for calibrated estimators of the distribution function

“…In this article, we investigate whether the optimal estimator in the proposal 17 (that can be applied direclty in the estimation of qunatile and poverty measures 21 ) based on the calibration method for estimating the distribution function can be improved by reducing the dimension of the optimal vector used in the calibration process. Working with a reduced number of variables may reduce numerical problems related to optimization procedures and also limit the presence of negative, very large and unstable calibration weights.…”

“…Under simple random sampling, the problem of optimal selection points in order to obtain the best estimation has been treated in previous works 17–20 . In fact, the work 17 obtained the optimal dimension and the optimal auxiliary vector for the estimator of the distribution function proposed in the work, 6 and although this proposal do not generate a unique weight system that is optimal for each point $$$ t $$$, it produces an estimator that is computationally simple and is a genuine distribution function that can be used directly in the estimation of quantiles and poverty measures 21 …”

“…[17][18][19][20] In fact, the work 17 obtained the optimal dimension and the optimal auxiliary vector for the estimator of the distribution function proposed in the work, 6 and although this proposal do not generate a unique weight system that is optimal for each point t, it produces an estimator that is computationally simple and is a genuine distribution function that can be used directly in the estimation of quantiles and poverty measures. 21 In many situations, the optimal auxiliary vector has a very high dimension, which makes the calibration process difficult and can also affect the efficiency of the estimator. Performing calibration with a high-dimensional auxiliary dataset can be several problems: The variance of the calibration estimator can be increased, and the optimization procedure may fail.…”

Estimating General Parameters from Non-Probability Surveys Using Propensity Score Adjustment

“…In practice, and as described in [7], the use of jackknife [34] and bootstrap techniques [35] in the variance estimation for nonlinear parameters should be more advantageous because of their wide applicability for different cases and conditions. Direct applications of bootstrap methods for estimating the variance-covariance matrix ofψ involve solving the equation U(θ, λ) = 0 repeatedly for each bootstrap sample.…”

“…It only requires a vector of auxiliary variables available for each individual of the sample and the population totals of those variables. Calibration is able to remove selection bias in nonprobability samples if the selection mechanism is ignorable [4], and despite being originally developed for parametric estimation, further work [5][6][7] has extended calibration to distribution function, quantile and poverty measures estimation.…”