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In statistics, a unit of observation is the unit described by the data that one analyzes. A study may treat groups as a unit of observation with a country as the unit of analysis, drawing conclusions on group characteristics from data collected at the national level. For example, in a study of the demand for money, the unit of observation might be chosen as the individual, with different observations (data points) for a given point in time differing as to which individual they refer to; or the unit of observation might be the country, with different observations differing only in regard to the country they refer to. The unit of observation should not be confused with the unit of analysis. A study may have a differing unit of observation and unit of analysis: for example, in community research, the research design may collect data at the individual level of observation but the level of analysis might be at the neighborhood level, drawing conclusions on neighborhood characteristics from data collected from individuals. Together, the unit of observation and the level of analysis define the population of a research enterprise. A data point or observation is a set of one or more measurements on a single member of the unit of observation. For example, in a study of the determinants of money demand with the unit of observation being the individual, a data point might be the values of income, wealth, age of individual, and number of dependents. Statistical inference about the population would be conducted using a statistical sample consisting of various such data points. In addition, in statistical graphics, a "data point" may be an individual item with a statistical display; such points may relate to either a single member of a population or to a summary statistic calculated for a given subpopulation. The measurements contained in a unit of observation are formally typed, where here type is used in a way compatible with datatype in computing; so that the type of measurement can specify whether the measurement results in a Boolean value from {yes, no}, an integer or real number, the identity of some , or some vector or array.

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Robust Parameter Estimation for the Ornstein-Uhlenbeck Process

Sonja Rieder

In this thesis, we treat robust estimation for the parameters of the Ornstein–Uhlenbeck process, which are the mean, the variance, and the friction. We start by considering classical maximum likelihood estimation. For the simulation study, where we also investigate the choice of the time lag, we use the method of moment (MoM) estimator as initial estimator for the friction parameter of the maximum likelihood estimator (MLE). However, in several aspects the MLE is not robust. For robustification, we first derive elementary M-estimates by extending the method of M-estimation from Huber (1981). We use an intuitively robustified MoM estimate as initial estimate and compare by means of simulation the M-estimate with the MLE. This approach is, however, only ad-hoc since Huber’s minimum Fisher information and minimax asymptotic variance theory remains incomplete for simultaneous location and scale, and does not cover more general models (as for example the Ornstein–Uhlenbeck process). A more general robustness concept due to Kohl et al. (2010), Rieder (1994), and Staab (1984) is based on local asymptotic normality (LAN), asymptotically linear (AL) estimates, and shrinking neighborhoods. We then apply this concept to the Ornstein–Uhlenbeck process. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. For two kind of neighborhoods (average and average square neighborhoods) we obtain optimally robust ICs. In case of average neighborhoods, their graph exhibits surprising, redescending behavior. For average square neighborhoods the graph is between the one of the elementary M-estimates and the MLE. Finally, we discuss the estimator construction, that is, the problem of constructing an estimator from the family of optimal ICs. We carry out in our context the One-Step construction dating back to LeCam and use both an intuitively robustified MoM estimate and the elementary M-estimate as initial estimate. This results in optimally AL estimates (for average and average square neighborhoods). By means of simulation we then compare the different estimators: MLE, elementary M-estimates, and optimally AL estimates. In addition, we give an application to electricity prices.

EPFL2012

Robust parameter estimation for the Ornstein-Uhlenbeck process

Sonja Rieder

In this paper, we derive elementary M- and optimally robust asymptotic linear (AL)-estimates for the parameters of an Ornstein-Uhlenbeck process. Simulation and estimation of the process are already well-studied, see Iacus (Simulation and inference for stochastic differential equations. Springer, New York, 2008). However, in order to protect against outliers and deviations from the ideal law the formulation of suitable neighborhood models and a corresponding robustification of the estimators are necessary. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. In a first step, we extend the method of M-estimation from Huber (Robust statistics. Wiley, New York, 1981). In a second step, we apply the general theory based on local asymptotic normality, AL estimates, and shrinking neighborhoods due to Kohl et al. (Stat Methods Appl 19:333-354, 2010), Rieder (Robust asymptotic statistics. Springer, New York, 1994), Rieder (2003), and Staab (1984). This leads to optimally robust ICs whose graph exhibits surprising behavior. In the end, we discuss the estimator construction, i.e. the problem of constructing an estimator from the family of optimal ICs. Therefore we carry out in our context the One-Step construction dating back to LeCam (Asymptotic methods in statistical decision theory. Springer, New York, 1969) and compare it by means of simulations with MLE and M-estimator.

Springer-Verlag2012

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In common usage and statistics, data (USˈdætə; UKˈdeɪtə) is a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a collection of data. Data is usually organized into structures such as tables that provide additional context and meaning, and which may themselves be used as data in larger structures.

Statistics (from German: Statistik, () "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal".