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2013 | OriginalPaper | Buchkapitel

7. Point Estimation Theory

verfasst von : Ron C. Mittelhammer

Erschienen in: Mathematical Statistics for Economics and Business

Verlag: Springer New York

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Abstract

The problem of point estimation examined in this chapter is concerned with the estimation of the values of unknown parameters, or functions of parameters, that represent characteristics of interest relating to a probability model of some collection of economic, sociological, biological, or physical experiments. The outcomes generated by the collection of experiments are assumed to be outcomes of a random sample with some joint probability density function \( f({{x}_{1}},\ldots, {{x}_n};\vec{\Theta} ) \). The random sample need not be from a population distribution, so that it is not necessary that X ₁,…,X _n be iid. The estimation concepts 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, i.e., all of the random sampling types discussed in Chapter 6. The objective of point estimation will be to utilize functions of the random sample outcome to generate good (in some sense) estimates of the unknown characteristics of interest.

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Vorheriges Kapitel Sampling, Sample Moments and Sampling Distributions

Nächstes Kapitel Point Estimation Methods

There is not universal agreement on the meaning of the terms parametric, nonparametric, and distribution-free. Sometimes nonparametric and distribution-free are used synonymously, although the case of distribution-free parametric estimation is pervasive in econometric work. See J.D. Gibbons, (1982), Encyclopedia of Statistical Sciences, Vol. 4. New York: Wiley, pp. 400–401.

Note, as always, that the function q can be the identity function q(\( {\mathbf{\Theta}} \)) ≡ \( {\mathbf{\Theta}} \), in which case we could be referring to estimating the vector \( {\mathbf{\Theta}} \) itself. Henceforth, it will be understood that since q(\( {\mathbf{\Theta}} \)) ≡ \( {\mathbf{\Theta}} \) is a possible choice of q(\( {\mathbf{\Theta}} \)), all discussion of estimating q(\( {\mathbf{\Theta}} \)) could be referring to estimating the vector \( {\mathbf{\Theta}} \) itself.

A concise review and comparison of a number of alternative criteria is given by T. Amemiya (1994), Introduction to Statistics and Econometrics, Cambridge, MA, Harvard University Press, pp. 118–121.

By definition, A is negative semidefinite iff \(\boldsymbol{\ell}\)′A \(\boldsymbol{\ell}\) ≤ 0 ∀\(\boldsymbol{\ell}\). Then the i ^th diagonal entry of A must be ≤ 0 since this entry can be defined by \(\boldsymbol{\ell}\)′A \(\boldsymbol{\ell}\) with \(\boldsymbol{\ell}\) being a zero vector except for a 1 in the i ^th position.

A nonzero matrix has at least unit rank. The rank of a negative semidefinite symmetric matrix is equal to the number of negatively valued eigenvalues, and all eigenvalues of a negative semidefinite matrix are ≤ 0. The trace of a negative semidefinite symmetric matrix is equal to the sum of its eigenvalues. Since all diagonal entries in a negative semidefinite matrix must be ≤ 0, it follows that a nonzero negative semidefinite symmetric matrix must have one or more negative diagonal entries.

Some analysts use a weak mean square error (WMSE) criterion that relates to only expected squared distance considerations. T ^* is WMSE superior to T iff E_Θ ( d ²(T ^*,q(Θ)) ≤ E_Θ(d ²(T,q(Θ)) ∀Θ∈Ω, and < for some Θ∈Ω. Relative efficiency and admissibility can be defined in the context of WMSE superiority and are left to the reader.

By “smallest MSE matrix,” we mean that MSE _Θ(T ^*)-MSE _Θ(T) is a negative semidefinite matrix for all estimators T of q(Θ) and for all Θ.

This is alternatively referred to in the literature by the term uniformly minimum variance unbiased estimator (UMVUE), where the adverb “uniformly” is used to emphasize the condition “∀Θ∈Ω.” In our usage of the terms, MVUE and UMVUE will be interchangeable.

LeCam, L., (1953) “On Some Asymptotic Properties of Maximum Likelihood Estimates and Related Bayes Estimates”, University of California Publications in Statistics, 1:277–330, 1953.

Note that the conditional density function referred to in this definition is degenerate in the general sense alluded to in Section 3.10, ?footnote 20. That is since (x ₁,…,x _n) satisfies the r restrictions s _i(x ₁,…,x _n) = s _i, for i = 1,…,r by virtue of the event being conditioned upon, the arguments x ₁,…,x _n of the conditional density are not all free to vary but rather are functionally related. If one wanted to utilize the conditional density for actually calculating conditional probabilities of events for (X ₁,…,X _n), and if the random variables were continuous, then line integrals would be required as discussed previously in Chapter 3 concerning the use of degenerate densities. This technical problem is of no concern in our current discussion of sufficient statistics since we will have no need to actually calculate conditional probabilities from the conditional density.

The reader may wonder why we define the conditional density “from the definition of conditional probability,” instead of using the rather straightforward methods for defining conditional densities presented in Chapter 2, Section 2.6. The problem is that here we are conditioning on an event that involves all of the random variables X ₁,…,X _n, whereas in Chapter 2 we were dealing with the usual case where the event being conditioned upon involves only a subset of the random variable X ₁,…,X _n having fewer than n elements.

Lehmann, E.L. and H. Scheffe’ (1950). Completeness, Similar Regions, and Unbiased Estimation, Sankhyā, 10, pp. 305.

See E.W. Barankin and M. Katz, (1959) Sufficient Statistics of Minimal Dimension, Sankhya, 21:217–246; R. Shimizu, (1966) Remarks on Sufficient Statistics, Ann. Inst. Statist. Math., 18:49–66; D.A.S. Fraser, (1963) On Sufficiency and the Exponential Family, Jour. Roy. Statist. Soc., Series B, 25:115–123.

By nonredundant, we mean that none of the constraints are implied by the others. Redundant constraints are constraints that are ineffective or unnecessary in defining sets.

It may be more appropriate to assume finite lower and upper bounds for a and b, respectively. Doing so will not change the final result of the example.

If one (or more) c _i(Θ) were linearly dependent on the other c _j(Θ)’s, then “only if” would not apply. To see this, suppose c _k(Θ) = \( \sum\nolimits_{{i = 1}}^{{k - 1}} {{{a}_i}{{c}_i}\left( {\mathbf{\Theta}} \right)} \). Then the exp term could be rewritten as\( {\exp \left[ {\mathop{\Sigma}\limits_{{i = 1}}^{{k - 1}} \mathop{c}\nolimits_i ({\mathbf{\Theta}} )\left[ {\mathop{g}\nolimits_i ({\bf x}) - \mathop{g}\nolimits_i ({\bf y}) + \mathop{a}\nolimits_i [\mathop{g}\nolimits_k ({\bf x}) - \mathop{g}\nolimits_k ({\bf y})]} \right]} \right]} \)and so g _i(x) = g _i(y), i = 1,…,k, is sufficient but not necessary for the term to be independent of Θ, and thus s(X) would not be minimal.

The reader will recall that the random sample, (X ₁,…,X _n), is by definition a set of sufficient statistics for f(x;Θ). However, it is clear that no improvement (decrease) in the MSE of an unbiased estimator will be achieved by conditioning on (X ₁,…,X _n), i.e., the reader should verify that this is a case where E(t*(X) − t(X))² = 0 and MSE equality is achieved in the Rao–Blackwell theorem.

Recall that by open rectangle, we mean that the parameter space can be represented as Ω = {(Θ ₁,…, Θ _k): a _i < Θ _i < b _i, i = 1,…, k}, where any of the a _i’s could be −∞ and any of the b _i’s could be ∞. This condition can actually be weakened to requiring only that the parameter space be an open subset of ℝ^k, and not necessarily an open rectangle, and the CRLB would still apply.

This can be easily seen, since \(\boldsymbol{\ell}\)′D Cov(Z) D’ \(\boldsymbol{\ell}\) = \(\boldsymbol{\ell}\)′_* Cov(Z) \(\boldsymbol{\ell}\) _* ≥ 0, where \(\boldsymbol{\ell} ^*\) = D′ \(\boldsymbol{\ell} ,\) ∀ \(\boldsymbol{\ell}.\)

Note, this rules out the degenerate cases p = 1 or p = 0, in which all sample observations would then be 1s and 0s, respectively. If such were the case for the outcome of any given random sample from the Bernoulli distribution, the best, and in fact only reasonable estimates of the parameter p would be 1 and 0, respectively, since there would be no sample variability with which to conclude anything different.

LeCam, L., (1953) On Some Asymptotic Properties of Maximum Likelihood Estimates and Related Bayesx Estimates, University of California Publications in Statistics, pp. 1:277–330.

Titel: Point Estimation Theory
verfasst von: Ron C. Mittelhammer
Verlag: Springer New York
Buch: Mathematical Statistics for Economics and Business
Print ISBN: 978-1-4614-5021-4

Electronic ISBN: 978-1-4614-5022-1

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-5022-1_7

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