Zum Problem der Verteilungen mit Shortfall bei der Nutzung des Indikators 'Durchschnittliche Körpergröße'

'Für das Problem des Shortfall bei der Nutzung des Indikators 'Durchschnittliche Körpergröße' wurden in den vergangenen 12 Jahren einige Lösungsvorschläge gemacht. Leider produzieren die verschiedenen Schätzer nach wie vor unterschiedliche Ergebnisse. Im vorliegenden Beitrag wird zunächst ein Überblick über die prinzipielle Arbeitsweise der unterschiedlichen Schätzer gegeben. Die zentrale Frage dabei ist: Wie kommt es zu unterschiedlichen Schätzergebnissen? Das am besten geeignete Verfahren scheint die von Heintel vorgestellte Methode (TPE) zu sein. Für reine Trendanalysen dürfte die robuste Komlos-und-Kim-Methode in Kombination mit TPE/RMSLE die sicherte Methode zu sein, mit dem Problem des Shortfalls umzugehen. Zur Überprüfung dieser Methoden werden in der vorliegenden Studie auch erste Schätzergebnisse für Trends der bayrischen Körpergrößenentwicklung vorgelegt, die auf eine Verschlechterung der Ernährungsstandards im späten 18. Jahrhundert hindeuten.' (Autorenreferat)'Research in economic history frequently uses human heights as a proxy for net nutrition. This anthropometric method enables historians to measure time trends and regional differences in nutritional status. However, the most widely used data sources for historical height measurements cannot be regarded as random samples of their underlying populations. In personnel records of volunteer armies, the lower side of the otherwise normal distribution is eroded by a phenomenon called 'shortfall'. Because recruiting practices favoured especially tall soldiers, shorter individuals are underrepresented below a certain threshold ('truncation point'). This article explains and compares different methods of estimating the true mean and standard deviation of the underlying population from these biased data sets. We conclude that an estimator called TPE/RSMLE controlled by the K/K method has better statistical features than the frequently used QBE. As an example, the height trend in Bavaria during the late 18th century is estimated using both the TPE/RSMLE and K/K method.' (author's abstract

Improvements in Maximum Likelihood Estimators of Truncated Normal Samples with Prior Knowledge of σ

Researchers analyzing historical data on human stature have long sought an estimator that performs well in truncated-normal samples. This paper reviews that search, focusing on two currently widespread procedures: truncated least squares (TLS) and truncated maximum likelihood (TML). The first suffers from bias. The second suffers in practical application from excessive variability. A simple procedure is developed to convert TLS truncated means into estimates of the underlying population means, assuming the contemporary population standard deviation. This procedure is shown to be equivalent to restricted TML estimation. Simulation methods are used to establish the mean squared error performance characteristics of the restricted and unconstrained TML estimators in relation to several population and sample parameters. The results provide general insight into the bias-precision tradeoff in restricted estimation and a specific practical guide to optimal estimator choice for researchers in anthropometrics

Improvements in Maximum Likelihood Estimators of Truncated Normal Samples with Prior Knowledge of σ

Researchers analyzing historical data on human stature have long sought an estimator that performs well in truncated-normal samples. This paper reviews that search, focusing on two currently widespread procedures: truncated least squares (TLS) and truncated maximum likelihood (TML). The first suffers from bias. The second suffers in practical application from excessive variability. A simple procedure is developed to convert TLS truncated means into estimates of the underlying population means, assuming the contemporary population standard deviation. This procedure is shown to be equivalent to restricted TML estimation. Simulation methods are used to establish the mean squared error performance characteristics of the restricted and unconstrained TML estimators in relation to several population and sample parameters. The results provide general insight into the bias-precision tradeoff in restricted estimation and a specific practical guide to optimal estimator choice for researchers in anthropometrics.truncated least squares; truncated maximum likelihood (TML); simulation methods; bias-precision trade-off; anthropometrics