Statistical quality control (SQC) is used by companies and industries for many reasons. For example, the process capability of machines is an important aspect of SQC, which consists in evaluating the ability of a production process to perform with the required specifications. In other words, the process capability measures the ability of a process of producing acceptable products according to the established specifications. The most common indicator used to measure the process capability is the process capability index, which depends on the process standard deviation. In practice, the standard deviation is unknown, and the process capability index is thus estimated by using an estimator of the process standard deviation. In this paper, we describe the most common estimators of the process standard deviation, and define the corresponding estimators of the process capability index. A bound for the bias ratio of the various estimators is obtained. Monte Carlo simulation studies are carried out to analyze the empirical performance of the various estimators of the process capability index. Empirical results indicate that biases can be obtained, specially in the presence of small samples. We also observe that the estimators of the process capability index based on sample ranges are less accurate than the alternative estimators

The problem of missing data is a common feature in any study, and a single imputation method is often applied to deal with this problem. The first contribution of this paper is to analyse the empirical performance of some traditional single imputation methods when they are applied to the estimation of the Gini index, a popular measure of inequality used in many studies. Various methods for constructing confidence intervals for the Gini index are also empirically evaluated. We consider several empirical measures to analyse the performance of estimators and confidence intervals, allowing us to quantify the magnitude of the non-response bias problem. We find extremely large biases under certain non-response mechanisms, and this problem gets noticeably worse as the proportion of missing data increases. For a large correlation coefficient between the target and auxiliary variables, the regression imputation method may notably mitigate this bias problem, yielding appropriate mean square errors. We also find that confidence intervals have poor coverage rates when the probability of data being missing is not uniform, and that the regression imputation method substantially improves the handling of this problem as the correlation coefficient increases.

The ratio method is commonly used to the estimation of means and totals. This method was extended to the problem of estimating the distribution function. An alternative ratio estimator of the distribution function is defined. A result that compares the variances of the aforementioned ratio estimators is used to define optimum design-based ratio estimators of the distribution function. Different empirical results indicate that the optimum ratio estimators can be more efficient than alternative ratio estimators. In addition, we show by simulations that alternative ratio estimators can have large biases, whereas biases of the optimum ratio estimators are negligible in this situation.

Many poverty measures are estimated by using sample data collected from social surveys. Two examples are the poverty gap and the poverty severity indices. A novel method for the estimation of these poverty indicators is described. Social surveys usually contain different variables, some of which can be used to improve the estimation of poverty measures. The proposed estimation methodology is based on this idea. The variance estimation and the construction of confidence intervals are also topics addressed in this article. Real survey data, extracted from the European Union Survey on Income and Living Conditions and based on various countries, are used to investigate some empirical properties of our estimators via Monte Carlo simulation studies. Empirical results indicate that the suggested methods can be more accurate than the customary estimator. Desirable results are also obtained for the proposed variances and confidence intervals. Various populations generated from the Gamma distribution also support our findings.

Estimation of a proportion is commonly used in areas such as medicine, biopharmaceutical experiments, etc. Estimation of a proportion using auxiliary information has not been investigated in the literature. Ratio estimators of the population proportion and two-sided confidence intervals based upon auxiliary information are derived in this paper. Real data extracted from the Spanish National Health Survey are used to demonstrate the application of the proposed methods in the estimation of prevalences. Results derived from simulation studies show that proposed estimators are more efficient than the traditional estimator. Proposed confidence intervals outperform the alternative methods, especially in terms of interval width. A study on patients with hypertension is also considered to calculate various estimators and confidence intervals.