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Freie Universität Berlin

1. Hingst, Hans-Ulrich. Contributions to the random sample theory and remarks to a Bootstrap limit theorem.

Degree: 2003, Freie Universität Berlin

URL: http://dx.doi.org/10.17169/refubium-7956

In the 1st chapter we look at a set ΩN , that consists of N objects with the
same choice probability, respectively. We investigate a real feature X , which
provides N values, within m pairwise distinct values; λN is the maximum of the
absolute frequencies of the m values. We draw a random sample out of ΩN of
size n without replacement. Theorem I.1 provides a bound B(n, N, λN / N ) ;
this bound estimates the Euclidean distance between the joint probability
functions of the sample variates at n-fold drawing with andm without
replacement out of ΩN to the top. A pre-version of this estimation we've
proposed at the International Congress of the Deutsche Statistische
Gesellschaft in 2000 already. The estimation may be used for every scaling of
X and for varying m = m (N) . The estimation is sharp, since for the class of
those X that are mapping ΩN bijectively (m=N) , the equation sign is true - in
this case the Euclidean distance of the joint probability functions (see
above) is even the Euclidean distance of the choice probabilities of n objects
between drawing with and without replacement out of ΩN.
Corollaries from the estimation are formulations of two limit theorems:
(i) Uniform convergence of the Euclidean distance of the joint probability
functions (see above) against zero with increasing N , permitting varying m =
m (N).
(ii) Uniform convergence of the Euclidean distance between the probability
functions of the m-dimensional Hypergeometric distribution and the m-variate
Multinomialdistribution against zero with increasing N , permitting varying m
= m (N).
The convergence order of both limits is O(1/N) at least.
In comparison with related results by Feller (1950 and later) and
Johnson/Kotz/Balakrishnan (1997) we obtain better and more generalized
statements.
In the 2nd chapter we estimate the max. relative frequency (λN /N) under some
conditions, that enable us to use the Kolmogorov / Smirnov - distance IDN ,
IDN := supt | IFN ( t ) - F ( t ) | with the empirical distribution function
IFN and her almost sure limit function F . Applying the DKW-inequality or the
Smirnov-LIL to the estimation of IDN , we get some different versions of the
estimation of (λN /N). We show that, under some conditions, (λN /N) tends to
the maximally jump of F.
In the 3rd chapter we are concerned with a fundamental requirement of the
Bootstrap limit theorem of Strobl (1995): The integral representation of a
particular Frechet differential. The concept of a decomposition of the measure
space \W allows us to give the conditions, under which this requirement is
filled, for a norm in \W indexed by a special function space.
The proved result includes a relevant result to this of Serfling (1981) as a
special case.
*Advisors/Committee Members: n (gender), Prof. Dr. H. Skarabis (firstReferee), Prof. Dr. J. Gordesch (furtherReferee).*

Subjects/Keywords: Degree of dependence of sample variates; uniform convergence of the Euclidean distance of the m-dimensional Hypergeometric distribution and the m-variate Multinomial distribution against zero when the number of pairwise distinct values m = m (N) may vary; maximum of the relative frequencies; maximum of the jumps of a distribution function; Bootstrap limit theorem; representation of a Fréchet differential of a statistical functional; decomposition of a particular space of probability measures.; 300 Sozialwissenschaften::300 Sozialwissenschaften, Soziologie::300 Sozialwissenschaften

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APA (6^{th} Edition):

Hingst, H. (2003). Contributions to the random sample theory and remarks to a Bootstrap limit theorem. (Thesis). Freie Universität Berlin. Retrieved from http://dx.doi.org/10.17169/refubium-7956

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Hingst, Hans-Ulrich. “Contributions to the random sample theory and remarks to a Bootstrap limit theorem.” 2003. Thesis, Freie Universität Berlin. Accessed December 05, 2019. http://dx.doi.org/10.17169/refubium-7956.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Hingst, Hans-Ulrich. “Contributions to the random sample theory and remarks to a Bootstrap limit theorem.” 2003. Web. 05 Dec 2019.

Vancouver:

Hingst H. Contributions to the random sample theory and remarks to a Bootstrap limit theorem. [Internet] [Thesis]. Freie Universität Berlin; 2003. [cited 2019 Dec 05]. Available from: http://dx.doi.org/10.17169/refubium-7956.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Hingst H. Contributions to the random sample theory and remarks to a Bootstrap limit theorem. [Thesis]. Freie Universität Berlin; 2003. Available from: http://dx.doi.org/10.17169/refubium-7956

Not specified: Masters Thesis or Doctoral Dissertation