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2013 | OriginalPaper | Buchkapitel

5. Basic Asymptotics

verfasst von : Ron C. Mittelhammer

Erschienen in: Mathematical Statistics for Economics and Business

Verlag: Springer New York

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Abstract

In this chapter we establish some results relating to the probability characteristics of functions of n-variate random variables X _(n)=(X ₁,…,X _n) when n is large. In particular, certain types of functions Y _n=g(X ₁,…,X _n) of an n-variate random variable may converge in various ways to a constant, its probability distribution may be well-approximated by a so-called asymptotic distribution as \( n \) increases, or the probability distribution of g(X _(n)) may converge to a limiting distribution as n → ∞.

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Vorheriges Kapitel Parametric Families of Density Functions

Nächstes Kapitel Sampling, Sample Moments and Sampling Distributions

As an example of this situation, let \( {{X}_n} \) have the discrete uniform distribution on the range \( \left\{ {\frac{1}{n},\frac{2}{n},\ldots, \frac{{n - 1}}{n},1} \right\},\rm{for}n = 1,2,3,\ldots \), and let X have the continuous uniform distribution on \( \left[ {0,1} \right] \). The probability distribution of \( {{X}_n} \) converges to the distribution of X as n → ∞. However, the sequence of PDFs are such that \( {{f}_n}(x)\to 0 \,{\rm {as}}\, n \to \infty \forall x\in \left[ {0,1} \right] \).

Note that \( {{\lim}_{{n\to \infty }}} \) F _n(y) is not even a cumulative distribution function for \( y\in \mathbb{R} \).

The reader may be wondering why we do not simply replace the uncountably infinite collection of probability statements \( {P(|\mathop{y}\nolimits_n - y| <\varepsilon ), \forall \varepsilon> 0} \), in the definition of convergence in probability with the definition \( {\mathop{{\lim }}\limits_{{n\to \infty }} {P(\mathop{y}\nolimits_n = y) = 1}} \). The problem is that such a convergence definition would not be useful in the examination of any sequence of nondegenerate continuous random variables, since P(y _n = y) = 0 ∀ n ⇒ \( {{{{\lim }}_{{n\to \infty }}}P(\mathop{y}\nolimits_n = y) = 0} \), no matter how close the outcomes of Y _n become to outcomes of Y as n → ∞.

In the event that max does not exist, max is replaced by sup (supremum, i.e., the smallest upper bound) in the statement of the theorem.

Gut, Allan (2005). Probability: A Graduate Course . Springer-Verlag, New York, Theorem 3.4.

The concepts of SLLNs and WLLNs can be generalized to the case where {μ _n} is a sequence of constants that are not necessarily the means of the X _i’s. See Y.S. Chow and H. Teicher (1978), Probability Theory, p. 121.

Some authors use the terminology “asymptotically uncorrelated” for this concept (e.g., H. White, Asymptotic Theory, pp. 49). However, the concept does not rule out negative correlation.

This is an example of a stochastic process that we will revisit in our discussion of the general linear model in Chapter 8. “Stochastic process” means any collection of random variables {X _t; t∈T}, where T is some index set that serves to order the random variables in the collection. Special cases of stochastic processes include a scalar random variable when T = {1}, an n-variate random vector when T = {1,2,…,n}, and a random sequence when T = {1,2,3,…}.

The reader who wishes to read about central limit theory in its most general form can examine Chapter 5 of R.G. Laha, and V.K. Rohatgi (1979), Probability Theory, New York: John Wiley.

M. Kendall and A. Stuart (1977), The Advanced Theory of Statistics, Volume I, New York: Macmillan, pp. 115. Note that any random variable for which an MGF exists is such that its moment sequence uniquely identifies its probability distribution.

P. Van Beeck (1972), An application of Fourier methods to the problem of sharpening the Berry-Esseen Inequality, Z. Wahrschein-lichkeits Theorie und Verw. Gebiete 23, pp. 187–196.

For example, see Y.S. Chow and H. Teicher, Probability Theory, (1978) Chapter 9, and R.G. Laha and V.K. Rohatgi, (1976) Probability Theory, Chapter 5).

M. Zolotarev (1967), A Sharpening of the Inequality of Berry-Esseen. Z. Wahrscheinlichkeits Theorie und Verw. Gebiete, 8 pp. 332–342.

A double array is one where the second subscript of the random variables in the ith row of the array ends with the value k _i, rather than with the value i as in the case of the triangular array, and k _n → ∞ as n → ∞. See Serfling, Approximation Theorems, pp. 31.

Note the W _ni, i = 1,…,n, are independent because the ε _i’s are independent.

H. Cramer and H. Wold, (1936) Some Theorems on Distribution Functions, J. London Math. Soc., 11(1936), pp. 290–295.

Recall that \( {{\bf V}_n^{{1/2}} } \) is the symmetric square root matrix of V _n, and \( {{\bf V}_n^{{1/2}} } \) is the inverse of \( {{\bf V}_n^{{1/2}} } \). The defining property of \( {{\bf V}_n^{{1/2}} } \) is that \( {{\bf V}_n^{{1/2}} {\bf V}_n^{{1/2}} = {\bf V}_n } \), while \( {{\bf V}_n^{{ - 1/2}} {\bf V}_n^{{ - 1/2}} = {\bf V}_n^{{ - 1}} } \).

If max does not exist, max is replaced by sup (the supremum).

\( \sum\nolimits_{{i = 1}}^n {{{i}^{{ - p}}}} \) is the so-called p series that converges for p>1 and diverges for p∈(0,1]. Since the series converges for p = 2, it must be the case that \( \sum\nolimits_{{i = n + 1}}^{\infty } {{{i}^{{ - 2}}}} \to 0 \ {\rm{as}} \ n\to \infty \). See Bartle, The Elements of Real Analysis, 2^nd Ed. pp. 290–291.

Titel: Basic Asymptotics
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_5

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