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Asymptotic formula for sum of moment mean deviation for order statistics from uniform distribution

Rafał Kapelko

Department of Computer Science, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

E-mail Address: rafal.kapelko@pwr.edu.pl

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https://doi.org/10.1142/S1793830919500150Cited by:2 (Source: Crossref)

Abstract

Assume that $n$ mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors $i$ from $1$ to $n,$ where the contribution of the $i^{th}$ sensor is its displacement from the current location to the anchor equidistant point $t_{i} = \frac{i}{n} - \frac{1}{2 n},$ raised to the $a^{th}$ power, when $a$ is an odd natural number.

As a consequence, we derive the following asymptotic identity. Fix $a$ positive integer. Let $X_{i : n}$ denote the $i^{th}$ order statistic from a random sample of size $n$ from the Uniform $(0, 1)$ population. Then

n∑i=1E[|Xi:n−E[Xi:n]|a]=Γ(a2+1)2a2(1+a)nna2+O(√nna2),<math display="block" altimg="eq-00014.gif"><mrow><munderover accentunder="true" accent="true"><mrow><mo class="MathClass-op">∑</mo></mrow><mrow><mi>i</mi><mo class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mstyle mathvariant="bold"><mi>E</mi></mstyle><mo class="MathClass-open" stretchy="false">[</mo><mo class="MathClass-rel">|</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mspace width=".17em" class="thinspace"></mspace><mo class="MathClass-punc">:</mo><mspace width=".17em" class="thinspace"></mspace><mi>n</mi></mrow></msub><mo stretchy="false" class="MathClass-bin">−</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mo class="MathClass-open" stretchy="false">[</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mspace width=".17em" class="thinspace"></mspace><mo class="MathClass-punc">:</mo><mspace width=".17em" class="thinspace"></mspace><mi>n</mi></mrow></msub><mo class="MathClass-close" stretchy="false">]</mo><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>a</mi></mrow></msup><mo class="MathClass-close" stretchy="false">]</mo><mo class="MathClass-rel">=</mo><mfrac><mrow><mi mathvariant="normal">Γ</mi><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo stretchy="false" class="MathClass-bin">+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mn>1</mn><mo stretchy="false" class="MathClass-bin">+</mo><mi>a</mi><mo class="MathClass-close" stretchy="false">)</mo></mrow></mfrac><mfrac><mrow><mi>n</mi></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo stretchy="false" class="MathClass-bin">+</mo><mi>O</mi><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced><mspace width="-.17em" class="negativethinspace"></mspace><mo class="MathClass-punc">,</mo></mrow></math>

where

$Γ (z)$ is the Gamma function.

Keywords:

AMSC: 68R05, 05A10, 62E20

References

1. B. Arnold, N. Balakrishnan and H. Nagaraja, A First Course in Order Statistics, Vol. 54 (SIAM, 1992). Google Scholar
2. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1 (John Wiley, NY, 1968). Google Scholar
3. P. Flajolet and B. Sedgewick, An Introduction to the Analysis of Algorithms (Addison-Wesley, 1995). Google Scholar
4. H. Gould and J. Quaintance, Combinatorial Identities for Stirling Numbers (World Scientific Publishing, Singapore, 2015). Google Scholar
5. R. Graham, D. Knuth and O. Patashnik, Concrete Mathematics A Foundation for Computer Science (Addison-Wesley, Reading, MA, 1994). Google Scholar
6. R. Kapelko and E. Kranakis, On the displacement for covering a unit interval with randomly placed sensors, Inform. Process. Lett. 116 (2016) 710–717. Crossref, Google Scholar
7. NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/8.17. Google Scholar