Mathematical statistics for economics and business (second edition) part 2
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Mathematical statistics for economics and business (second edition) part 2
Point Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 timation of Probability Models7.3Estimators and Estimator Properties7.4Sufficient Statistics7.5Minimum Variance Unbiased Estimation"I IfThe problem of point estimation examined in this chapter is concerned with the estimation of the values of unknown parameters, or functions of parameters, that repr Mathematical statistics for economics and business (second edition) part 2 esent characteristics of interest relating to a probability model of some collection of economic, sociological, biological, or physical experiments. TMathematical statistics for economics and business (second edition) part 2
he outcomes generated by the collection of experiments arc assumed to be outcomes of a random sample with some joint probability density function /(X]Point Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 epts we will examine in this chapter can be applied to the case of general random sampling, as well as simple random sampling and random sampling with replacement, Í.C., all of the random sampling types discussed in Chapter 6. The objective of point estimation will be to utilize functions of the ran Mathematical statistics for economics and business (second edition) part 2 dom sample outcome to generate good (in some sense} estimates of the unknown characteristics of interest.7.1Parametric, Semi parametric, and NonparameMathematical statistics for economics and business (second edition) part 2
tric Estimation ProblemsThe types of estimation problems that will be examined in this (and the next) chapter are problems of parametric estimation anPoint Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 th the estimates of the values of unknown parameters that characterize parametric probability models or semiparametric probability models of the population, process, or general364 Chapter 7 Point Lstiriialion Ihwryexperiments under study. Both 0Í these models have specific parametric func tional str Mathematical statistics for economics and business (second edition) part 2 ucture to them that becomes fixed and known once values of parameters arc numerically specified, The difference between the two models lies in whetherMathematical statistics for economics and business (second edition) part 2
a particular parametric family or class of probability distributions underlies the probability model and is fully determined by setting the values ofPoint Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 metric functional structure that becomes fixed when parameter values are specified. We discuss these models in more detail below.Given the prominence of parameters in the estimation problems we will be examining, and the need to distinguish their appearance and effect in specifying parametric, semip Mathematical statistics for economics and business (second edition) part 2 arametric, and nonparametric probability models, we extend the scope of the term probability model to explicitly encompass the definition of parameterMathematical statistics for economics and business (second edition) part 2
s and their admissible values. Note, because it is possible that the range of the random variable can change with changing values of the parameter vecPoint Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 efinition below by including the parameter vector in the definition of the range of X.Definition 7.1Probability ModelA probability model for the random variable X is defined by the set {J?(X:0),/(x;0). 0 e 0}, where ÍÌ defines the admissible values of the parameter vector 0.In the context of point e Mathematical statistics for economics and business (second edition) part 2 stimation problems, and later hypothesis testing and confidence interval estimation problems, X will refer to a random sample relating to some populatMathematical statistics for economics and business (second edition) part 2
ion, process, or general set of experiments having characteristics that are the interest of estimation, and /(*: 0) will be the joint probability densPoint Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 F, and thus represents all of the possible values for the unknowns one may be interested in estimating. The objective of point estimation is to increase know ledge of 0 beyond simply knowing all of its admissible values. It can be the case that prior knowledge exists regarding the values <-) can ass Mathematical statistics for economics and business (second edition) part 2 ume in a given empirical application, in which case ÍÌ can lx: specified to incorporate that knowledge.We note, given our convention that the range anMathematical statistics for economics and business (second edition) part 2
d the support of the random variable X arc equivalent (recall Definition 2.13), that explicitly listing the range of the random variable as part of thPoint Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 values implies the range of the random variable asKfX; Wi - (x : /(x: 01 >0) fori-) < ÍÌ We will sec ahead that in point estimation problems an explicit specification of the range of a random sample X is important for a number of reasons, including determining the types of estimation procedures tha Mathematical statistics for economics and business (second edition) part 2 t can be used in a given estimation problem, and for defining7,1 Parametric, Scmipatametric, .111(1 Nonparametric Estimation Problems365rhe range of eMathematical statistics for economics and business (second edition) part 2
stimates that arc possible to generate from a particular point estimator specification. We will therefore continue to explicitly include the range of Point Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 G ÍÌ} when emphasizing the range of the random variable is not germane to the discussion.Definition 7.2 Probability Model: Abbreviated NotationAn abbreviated notation for the probability model of random variable X is {/(x;0), <•) e 0}, where = {x : /t'x.H)>0} is taken as implicit in the definition o Mathematical statistics for economics and business (second edition) part 2 f the model.7.1.1Parametric ModelsA parametric model is one in which the functional form of the joint probability density function, /(X|,...,xn;0), coMathematical statistics for economics and business (second edition) part 2
ntained in the probability model for the observed sample data, X, is fully specified and known once the value of the parameter vector, (•), is given aPoint Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 of the random sample X, as/(Xj.... ,xn; <->). for 0 E ÍỈ, with the implication that if the appropriate value of the parameter vector, say H|), were known, then /(xi......xn:0o) would represent true probability density functionunderlying the observed outcome X of the random sample. We note that in ap Mathematical statistics for economics and business (second edition) part 2 plications the analyst may not feel fully confident in the specification of the probability model {/(x:0), 0 € Í2}, and view it as a tentative workingMathematical statistics for economics and business (second edition) part 2
model, in which case the adequacy of the model may itself be an issue in need of further statistical analysis and testing. However, use of parametricPoint Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 tric model is one in which the functional form of the joint probability density function component of the probability model for the observed sample data, X, is not fully specified and is not known when the value of the parameter vector of the model, H, is given a specific numerical value. Instead of Mathematical statistics for economics and business (second edition) part 2 defining a collection of explicit parametric functional forms for the joint density of the random sample X, when defining the model, as in the parameMathematical statistics for economics and business (second edition) part 2
tric case, the analyst defines a number of properties that the underlying true sampling density f(x\.....xr,;0(i) is thought to possess. Such informatPoint Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 contained in the random sample exhibit independence or not. Given a numerical value for the parameter vector <->, any parametric structural components of the model arc given an explicit fully specified functional form, but other components of the model, most notably the underlying joint density fun Mathematical statistics for economics and business (second edition) part 2 ction366 Chapter 7 Point Lstirnalion Itx-oryfor the random sample, /(X|............Xn:(-)), remains unknown and not fullyspecified.7.1.3NonparametricMathematical statistics for economics and business (second edition) part 2
ModelsA nonparametric model is one in which neither the functional form of the joint probability density function component of the probability model fPoint Estimation Theory■7.1 Parametric, Semiparametric, and NonparametricEstimation Problems7.2Additional Considerations for the Specification and Est Mathematical statistics for economics and business (second edition) part 2 parameters (-). These models proceed with minimal assumptions on rhe structure of the probability model, with the analyst simply acknowledging rhe existence of some general characteristics and relationships relating to the random variables in the random sample, such as the existence of a general reg Mathematical statistics for economics and business (second edition) part 2 ression relationship, or the existence of a population probability distribution if the sample were generated through simple random sampling.Gọi ngay
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