A Comparison of Nonparametric Methods in the Graduation of Mortality: Application to Data from the Valencia Region (Spain)

[EN] The nonparametric graduation of mortality data aims to estimate death rates by carrying out a smoothing of the crude rates obtained directly from original data. The main difference with regard to parametric models is that the assumption of an age-dependent function is unnecessary, which is advantageous when the information behind the model is unknown, as one cause of error is often the choice of an inappropriate model. This paper reviews the various alternatives and presents their application to mortality data from the Valencia Region, Spain. The comparison leads us to the conclusion that the best model is a smoothing by means of Generalised Additive Models (GAM) with splines. The most interesting part of this paper is the development of a plan that can be applied to mortality data for a wide range of age groups in any geographical area, allowing the most appropriate table to be chosen for the data in hand.The authors are indebted to the anonymous referees whose suggestions improved the original manuscript. This work was partially supported by a grant from MEyC (Ministerio de Educación y Ciencia, Spain, project MTM-2004-06231).The research of Francisco Montes has also been partially supported by a grant from DGITT (Direcció General d Investigació i Transferència Tecnològica de la Generalitat Valenciana, Project GRUPOS03/189).Debón Aucejo, AM.; Montes-Suay, F.; Sala-Garrido, R. (2006). A Comparison of Nonparametric Methods in the Graduation of Mortality: Application to Data from the Valencia Region (Spain). International Statistical Review. 74(2):215-233. https://doi.org/10.1111/j.1751-5823.2006.tb00171.xS215233742Bloomfield, D. S. F., & Haberman, S. (1987). Graduation: some experiments with kernel methods. Journal of the Institute of Actuaries, 114(2), 339-369. doi:10.1017/s0020268100019090Breiman, L., & Peters, S. (1992). Comparing Automatic Smoothers (A Public Service Enterprise). International Statistical Review / Revue Internationale de Statistique, 60(3), 271. doi:10.2307/1403679Brockmann, M., Gasser, T., & Herrmann, E. (1993). Locally Adaptive Bandwidth Choice for Kernel Regression Estimators. Journal of the American Statistical Association, 88(424), 1302-1309. doi:10.1080/01621459.1993.10476411Chan,, F. Y., Chan, L. K., Falkenberg, J., & Yu, M. H. (1986). Applications of linear and quadratic programmings to some cases of the Whittaker-Henderson graduation method. Scandinavian Actuarial Journal, 1986(3-4), 141-153. doi:10.1080/03461238.1986.10413805Clements, M. S. (2005). Lung cancer rate predictions using generalized additive models. Biostatistics, 6(4), 576-589. doi:10.1093/biostatistics/kxi028Cleveland, W. S. (1979). Robust Locally Weighted Regression and Smoothing Scatterplots. Journal of the American Statistical Association, 74(368), 829-836. doi:10.1080/01621459.1979.10481038Copas, J. B., & Haberman, S. (1983). Non-parametric graduation using kernel methods. Journal of the Institute of Actuaries, 110(01), 135-156. doi:10.1017/s0020268100041275Epanechnikov, V. A. (1969). Non-Parametric Estimation of a Multivariate Probability Density. Theory of Probability & Its Applications, 14(1), 153-158. doi:10.1137/1114019Felipe, A., Guillen, M., & Perez-Marin, A. M. (2002). Recent Mortality Trends in the Spanish Population. British Actuarial Journal, 8(4), 757-786. doi:10.1017/s1357321700003901Forfar, D. O., McCutcheon, J. J., & Wilkie, A. D. (1988). On graduation by mathematical formula. Journal of the Institute of Actuaries, 115(1), 1-149. doi:10.1017/s0020268100042633Gasser, T., Kneip, A., & Köhler, W. (1991). A Flexible and Fast Method for Automatic Smoothing. Journal of the American Statistical Association, 86(415), 643-652. doi:10.1080/01621459.1991.10475090Gasser, T., & Müller, H.-G. (1979). Kernel estimation of regression functions. Smoothing Techniques for Curve Estimation, 23-68. doi:10.1007/bfb0098489Gavin, J. B., Haberman, S., & Verrall, R. J. (1994). On the choice of bandwidth for kernel graduation. Journal of the Institute of Actuaries, 121(1), 119-134. doi:10.1017/s0020268100020102Guillen, M., Nielsen, J. P., & Perez-Marin, A. M. (2006). Multiplicative Hazard Models for Studying the Evolution of Mortality. Annals of Actuarial Science, 1(1), 165-177. doi:10.1017/s1748499500000099Heligman, L., & Pollard, J. H. (1980). The age pattern of mortality. Journal of the Institute of Actuaries, 107(1), 49-80. doi:10.1017/s0020268100040257Hoem, J. M., & Linnemann, P. (1988). The tails in moving average graduation. Scandinavian Actuarial Journal, 1988(4), 193-229. doi:10.1080/03461238.1988.10413848McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models. doi:10.1007/978-1-4899-3242-6Nadaraya, E. A. (1964). On Estimating Regression. Theory of Probability & Its Applications, 9(1), 141-142. doi:10.1137/1109020Nielsen, J. P. (2003). Smoothing and Prediction with a View to Actuarial Science, Biostatistics and Finance. Scandinavian Actuarial Journal, 2003(1), 51-74. doi:10.1080/03461230308484Silverman, B. W. (1984). A Fast and Efficient Cross-Validation Method for Smoothing Parameter Choice in Spline Regression. Journal of the American Statistical Association, 79(387), 584-589. doi:10.1080/01621459.1984.10478084Verrall, R. J. (1993). Graduation by dynamic regression methods. Journal of the Institute of Actuaries, 120(1), 153-170. doi:10.1017/s002026810003688xWang, J.-L., Müller, H.-G., & Capra, W. B. (1998). Analysis of oldest-old mortality: lifetables revisited. The Annals of Statistics, 26(1), 126-163. doi:10.1214/aos/1030563980Whittaker, E. T. (1922). On a New Method of Graduation. Proceedings of the Edinburgh Mathematical Society, 41, 63-75. doi:10.1017/s0013091500077853Zhang, H. (2004). Mixed effects multivariate adaptive splines model for the analysis of longitudinal and growth curve data. Statistical Methods in Medical Research, 13(1), 63-82. doi:10.1191/0962280204sm353r

Assesing the influence of environmental variables on the performance of water companies: An efficiency analysis tree approach

Producción CientíficaEfficiency assessment is a valuable tool for industries that are regulated, such as the provision of drinking water. Hence, past research on this topic is wide. However, current, widely used approaches such as parametric, non-parametric and partial frontier methods present several limitations and pitfalls. Thus, here, the Efficiency Analysis Tree (EAT) method was trialled on a sample of water companies. This method overcomes overfitting issues, because it employs a combination of classification, regression tree methods, and non-parametric analyses. For comparative purposes, efficiency was also estimated using Data Envelopment Analysis (DEA) and Free Disposal Hull (FDH) non-parametric methods. The approach was applied empirically using a sample of English and Welsh water companies during 1991–2020. Average efficiency was estimated at 0.489, showing that water companies could save 51.1% of their costs if efficient. Except for the 2011–2015 period, efficiency increased over time, indicating that price reviews by the English and Welsh water regulator contributed to improving water company performance. The application of bootstrap regression analysis techniques showed that the main source of raw water, percentage of metered properties, population density, and percentage of water leakage represented environmental variables that significantly influenced the efficiency scores of water companies. The approach introduced here could be of use to water regulators, as it overcomes the existing limitations of traditional approaches employed to assess the performance of water companies, facilitating sound decision-making

Eco‑efficiency assessment under natural and managerial disposability : an empirical application for Chilean water companies

Getting a good understanding regarding the economic and environmental performance of water utilities is of great importance to achieve the goal of an efcient and sustainable industry. In this study, we apply the range adjusted measure (RAM) data envelopment analysis (DEA) model to evaluate the integrated (production and environmental) efciency of several water utilities located in Chile. Integrated efciency is evaluated using the concepts of natural and managerial disposability. This approach further allows us to quantify the contribution of each input and undesirable product on efciency scores. The results highlighted that the Chilean water industry showed high levels of production and environmental efciency over time. Under natural disposability, water utilities could control production costs to reduce water leakage and unplanned water supply interruptions by 3.3% on average. Under managerial disposability, water utilities could further cut down undesirable outputs by 1.4% on average by adopting best managerial practices. On average, potential savings in operating costs, employment, water leakage, and unplanned water supply interruptions were higher for concessionary utilities as they showed slightly lower efciency scores than full private utilities

A comprehensive assessment of energy efficiency of wastewater treatment plants: An efficiency analysis tree approach

Wastewater treatment plants (WWTPs) are energy intensive facilities. Controlling energy use in WWTPs could bring substantial benefits to people and environment. Understanding how energy efficient the wastewater treatment process is and what drives efficiency would allow treating wastewater in a more sustainable way. In this study, we employed the efficiency analysis trees approach, that combines machine learning and linear programming techniques, to estimate energy efficiency of wastewater treatment process. The findings indicated that considerable energy inefficiency among WWTPs in Chile existed. The mean energy efficiency was 0.287 suggesting that energy use should cut reduce by 71.3 % to treat the same volume of wastewater. This was equivalent to a reduction in energy use by 0.40 kWh/m3 on average. Moreover, only 4 out of 203 assessed WWTPs (1.97 %) were identified as energy efficient. It was also found that the age of treatment plant and type of secondary technology played an important role in explaining energy efficiency variations among WWTPs

Monetary valuation of unsorted waste: A shadow price approach

Producción CientíficaImproving the management of municipal solid waste (MSW) is fundamental to promote circular economy and sustainability. Unsorted waste involves negative environmental impacts which often are ignored in economic feasibility studies due to its difficult valuation. In this study the shadow price of unsorted waste using the directional distance function was estimated. This methodological approach also allowed us to compute eco-efficiency scores of a set of municipalities in the provision of MSW services. The empirical application focused on a sample of 119 Chilean municipalities. The results showed that the average shadow price of unsorted waste was 297.66 €/ton which means that the environmental cost of left waste as unsorted was 297.66 € per ton. A regression tree model illustrated that population density, tourism intensity and the generation of waste per capita significantly influenced the shadow price of unsorted waste. Moreover, it was illustrated that Chilean municipalities were very inefficient in the management of MSW since the average eco-efficiency score was 0.272. The findings from this study reveal that additional and alternative policies should be adopted to improve the management of MSW and increase its recycling rate.Agencia Nacional de Investigación y Desarrollo -Fondo Nacional de Desarrollo Científico y Tecnológico (project 1210077

Cost and quality of service performance in the Chilean water industry: A comparison of stochastic approaches

Producción CientíficaThe evaluation of efficiency can be of great value to water companies and regulators to adopt policies and design incentives to enhance performance. This study delves into the implications of employing distinct methodologies, namely the classical Stochastic Frontier Analysis (SFA), Bayesian SFA, and Stochastic non-parametric Envelopment of Data (StoNED), to evaluate cost and quality of service efficiency within the water industry. Chilean water companies reported average efficiencies of 0.623, 0.583, and 0.522 using the SFA, BSFA, and StoNED approaches, respectively. Furthermore, the SFA analysis suggested that the performance of water companies experienced a decline of −0.59% per year from 2010 to 2018. In contrast, the BSFA and StoNED estimations indicated an opposite trend, with annual performance improvements of 0.51% and 0.17% respectively, over the same period. These findings underscore the critical role of selecting appropriate methodologies when interpreting and comparing efficiency results for making informed long-term decisions

Understanding water energy nexus in drinking water provision: an eco-efficiency assessment of water companies

Producción CientíficaUnderstanding water-energy nexus in the provision of drinking water services is a challenge which has outstanding relevance in the current climatic emergency. Environmental efficiency and eco-efficiency assessment of water companies are two useful tools to address this challenge. In this study, we estimated hyperbolic and enhanced hyperbolic distance functions to compute the potential reduction in greenhouse gas (GHG) emissions and energy costs in the provision of drinking water. The empirical application focused on the English and Welsh water companies over 2011-2019. Average environmental efficiency and eco-efficiency scores were 0.920 and 0.962, respectively which indicates that water companies performed well but there is room for improvement. Moreover, due to the economies of scale, the cost of reducing GHG emissions was higher for water and sewerage companies than for water only companies. The results and conclusions of this study allow better understanding of the relationship between the provision of drinking water, energy costs and GHG emissions

Carbon efficiency analysis in the provision of drinking water: Estimation of optimal greenhouse gas emissions

Producción CientíficaAssessing carbon efficiency (CE) in the provision of drinking water services is essential to achieve a net-zero greenhouse gas (GHG) urban water cycle. Previous studies evaluating the CE of water companies are very scarce and employed parametric and non-parametric. Both methodological approaches present limitations such as overfitting issues or require assumptions about the production technology which could lead to less reliable efficiency scores. To overcome these limitations, in this study, and for the first time, we estimated CE of English and Welsh water companies using the Efficiency Analysis Trees (EAT) approach. This technique brings together machine learning and non-linear programming techniques to estimate production frontier and efficiency scores. It also allowed us to quantify the optimal level of GHG emissions in the provision of water services and estimate potential GHG savings. Bootstrap truncated regression methods were employed to quantify the impact of operating characteristics on CE of water companies. The optimal level of GHG emissions was estimated to be between 0.062 and 133.03 tons of CO2 equivalent (CO2eq) per year and per connected property. The average CE was at the level of 0.632. This means that GHG emissions could reduce by 36.8% to maintain the same level of water services. Equivalently, this corresponds to a reduction of 488,321 tons of CO2eq per year. Water only companies exhibited a better performance than water and sewerage companies with an average CE of 0.785 and 0.540, respectively. The performance of the English and Welsh water companies decreased over time. In 2011 the average CE was 0.772 whereas it went down to 0.534 in 2020. It was also estimated that on average water companies could reduce 0.034 tons of CO2eq per cubic meter of drinking water supplied and 16.16 tons of CO2eq/ connected property per year. The regression results showed that topography and water treatment complexity had a significant impact on CE. The conclusions of this study are relevant for policy makers to define policies toward a low-carbon urban water cycle

Accounting for service quality to customers in the efficiency of water companies: evidence from England and Wales

This paper investigates the role of service quality to customers in the efficiency assessment of water companies in England and Wales. To achieve this, data envelopment analysis techniques are employed to compute the technical efficiency of the water companies following two approaches: (i) traditional assessment based on quantity variables (without the inclusion of service quality variables) and (ii) alternative assessment considering quantity and service quality variables as undesirable outputs. The analysis covers 22 water and sewerage companies and water only companies providing drinking water services. The results indicate that the traditional efficiency assessment reveals a high level of technical efficiency, suggesting that the English and Welsh water industry is mature and that one of the challenges it faces is improving service quality. When introducing service quality variables in the evaluation, the average scores of technical efficiency slightly decrease. This suggests that, on average, water companies do not necessarily provide high quality of service, meaning that the traditional efficiency assessment favours their performance. Quality of service supplied to customers within a water sector matters and should be taken into account during the benchmarking process as it could assist regulated companies and regulators to improve performance and incentives, respectively

Assessing eco-efficiency of wastewater treatment plants: A cross-evaluation strategy

Producción CientíficaEvaluating the eco-efficiency of wastewater treatment plants (WWTPs) is crucial for enhancing environmental and economic performance in the water utility sector. Previous studies in this area estimated WWTP eco-efficiency through self-evaluation, which might have led to overestimation and biased policy recommendations. To address these issues, this study applies a cross-evaluation strategy, combining self-evaluation and peer-evaluation, to assess the eco-efficiency of WWTPs. The empirical application focuses on a sample of Spanish WWTPs, yielding the following key findings. Average eco-efficiency scores were 0.353 and 0.230, for self-evaluation and global peer-evaluation approaches, respectively, confirming the overestimation of eco-efficiency scores based on self-evaluation. If WWTPs were eco-efficient, they could potentially reduce greenhouse gas (GHG) emissions by up to 0.39 kg CO2eq/year. The application of reliable methods, such as peer-evaluation, for eco-efficiency assessment of WWTPs provides water regulators with a comprehensive understanding of the environmental and economic performance of WWTPs. This knowledge guides decision-making, policy development, and resource allocation, facilitating sustainable and efficient wastewater management practices.Junta de Castilla y León - UE-FEDER (CL-EI-2021-07