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2024 | OriginalPaper | Buchkapitel

2. Order Statistics

verfasst von : Iickho Song, So Ryoung Park, Wenyi Zhang, Seungwon Lee

Erschienen in: Fundamentals of Order and Rank Statistics

Verlag: Springer Nature Switzerland

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Abstract

In this chapter, recollecting the notion of order statistics introduced in Sect. 1.4, we discuss the distributions and properties of order statistics in detail. As mentioned in Sect. 1.4, in dealing with the order statistics, we assume that the dimension, or size, of the random vectors in this chapter is \(n \in \mathbb {J}_{2, \infty }\) unless specified otherwise. When it becomes appropriate, we denote the size n of the random vector more explicitly.

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Vorheriges Kapitel Preliminaries

Nächstes Kapitel Rank Statistics

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Note that the cdf of the binomial distribution \(b(n,p)\) is \(F(x) = \sum \limits _{k=0}^{\lfloor x \rfloor } \, { }_{n}\mbox{C}_{k} p^k(1-p)^{n-k}\) for \(\lfloor x \rfloor \le x < \lfloor x \rfloor +1\).

In describing the pmf and cdf in the discrete space, we often use a simpler notation by just specifying the values at non-zero points of the pmf instead of specifying over all real numbers unless the specification for all real points is required.

Exercise 2.15 discusses this observation more specifically in the gamma distribution, a generalization of the exponential distribution.

Here, \(\sum \limits _{j=r}^n \, { }_n\mbox{C}_j x^j (1-x)^{n-j}= \sum \limits _{j=r}^{n-1} \, { }_n\mbox{C}_j x^j (1-x)^{n-j} + x^n= 1\) when \(x=1\).

The pdf (2.3.14) can be obtained also by differentiating the cdf (2.3.13).

Note that \(f_{X_{[r]},X_{[s]}} (x,x) =0\) for \(s \ge r+2\). For \(s=r+1\), the values of \(f_{X_{[r]},X_{[s]}} (x,y)\) expressed as (2.3.31) along the line \(x=y\) depends on \(u(0)\): nonetheless, the values of a two-dimensional function along a line of discontinuities on the two-dimensional plane are practically negligible, which is similar to the fact that the value of a one-dimensional function at discontinuities is practically negligible.

In this expression, \(\int _{-1}^y \int _w^y dv dw\), \(\int _{-1}^x \int _w^y dv dw\), and \(\int _{-1}^x \int _w^1 dv dw\) can be replaced with \(\int _{-1}^y \int _{-1}^v dw dv\), \(\left ( \int _{-1}^x \int _{-1}^v + \int _x^y \int _{-1}^x \right ) dw dv\), and \(\left ( \int _{-1}^x \int _{-1}^v + \int _x^1 \int _{-1}^x \right ) dw dv\), respectively, when the order of integrations over v and w are interchanged.

Note that the value of \(\sum \limits _{s=2}^{n} \sum \limits _{r=1}^{s-1} f_{r,s} (x,x)\) depends on \(u(0)\): nonetheless, the difference is practically negligible.

Here, recollecting that \(n_0 = 0\) and \(n_{k+1} = n+1\), the numbers of integrations in the j-th group of integrations are \(n_1 -1 = n_1-n_0-1 \) from 1 to \(n_1 -1\) when \(j=0\), \(n_2-n_1-1 \) from \(n_1 +1\) to \(n_2 -1\) when \(j=1\), \(\cdots \), \(n-n_k = n_{k+1}-n_k-1\) from \(n_k +1\) to n when \(j=k\). Thus, the total number of integrations is \(\left (n_1-1 \right ) + \left ( n_2 - n_1 -1 \right ) + \cdots + \left ( n - n_k \right ) = n-k\).

When obtaining (2.3.55) from (2.3.52) and (2.3.54), note that \(\frac {u\left (x_{i+1}-x_i \right )}{u\left (x_{i+1}-x_i \right )} =u\left (x_{i+1}-x_i \right )\). In addition, \(\frac {u\left ( x_{r+1}-x_r \right )}{u\left ( x_s - x_r \right )}\) is defined only when \(x_s > x_r\) and is equal to \(u\left ( x_{r+1}-x_r \right )\) because \(x_s \ge x_{r+1}\).

The conditional joint pdf (2.3.55) is meaningful only when \(x_1 < x_2 < \cdots < x_n\). Meanwhile, the joint pdf of the order statistics of an \((s-r-1)\)-dimensional i.i.d. random vector with the marginal pdf \(\bar {f} (x) = \frac { f(x) u\left ( x - x_r \right )u\left ( x_s - x \right )} { F\left ( x_s \right ) - F\left ( x_r \right ) } \) is meaningful only when \(x_{r} < x_{r+1} < \cdots < x_{s}\).

Refer to (1.E.7).

Refer to (1.E.6).

Specifically, \(f_{\left . X_{[s]} \right | \boldsymbol {X}_{[r]}} \left (\left . x_s \right | \boldsymbol {x}^r \right )\) has the additional term of \(\prod \limits _{i=1}^{r-1} u\left ( x_{i+1} - x_i \right )\). In other words, \(f_{\left . X_{[s]} \right | \boldsymbol {X}_{[r]}} \left (\left . x_s \right | \boldsymbol {x}^r \right )\) is zero unless \(x_1 < x_2 < \cdots < x_r < x_s\) while \(f_{\left . X_{[s]} \right | X_{[r]}} \left (\left . x_s \right | x_r \right )\) is zero unless \(x_r < x_s\).

Refer to Exercise 2.28 as well.

Note here that the distributions of \(X_k\) and \(1-X_k\) are the same because \(X_k\) has the distribution \(U(0, 1)\). Consequently, the distributions of \(-\ln X_k\) and \(-\ln \left (1-X_k \right )\) are also the same.

Detailed steps in the evaluation of the joint cdf’s are provided in section “Proofs and Calculations” in Appendix 1, Chap. 2.

Here, \(f_{|X|{ }_{[q]}} (0) = 0\) for \(q \in \mathbb {J}_{2,\infty }\) because \(G_F(0)=0\). In addition, letting \(u(0)=0\), we have \(f_{|X|{ }_{[1]}} (0) = ng_F(0)u(0)=0\) also.

If \(x > y\) when \(r < s\) with \(r, s \in \mathbb {J}_{1,n}\), we have \(p_{r, s}(x,y)=0\). In section “Order Statistics from Uniform Distributions” in Appendix 2, Chap. 2, we discuss another method of obtaining (2.4.23): refer to (2.A2.29).

The result obtained by changing all the negative signs into positive signs in the determinant of a square matrix is called the permanent of the matrix. The permanent of a matrix \(\boldsymbol {A}\) is usually denoted by \(\,{ }^{+}\left |\boldsymbol {A}\right |{ }^{+}\) or \(\mbox{per}\left [ \boldsymbol {A}\right ]\).

M. Abramowitz, I.A. Stegun (ed.), Handbook of Mathematical Functions (Dover, New York, 1972)

N. Balakrishnan, A.C. Cohen, Order Statistics and Inference: Estimation Methods (Academic, San Diego, 1991)

K.E. Barner, G.R. Arce, Order-statistic filtering and smoothing of time-series: Part II. in Handbook of Statistics, vol. 17 (Elsevier, Amsterdam, 1998), pp. 555–602

P.J. Bickel, On some robust estimates of location. Ann. Math. Stat. 36, 847–858 (1965)MathSciNetCrossRef

R.J. Crinon, The Wilcoxon filter: A robust filtering scheme, in Proceeding of the International Conference on Acoustics, Speech, and Signal Processing (1985), pp. 18.5.1–18.5.4

H.A. David, Order Statistics, 2nd edn. (John Wiley and Sons, New York, 1980)

H.A. David, H.N. Nagaraja, Order Statistics, 3rd edn. (John Wiley and Sons, New York, 2003)CrossRef

I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980)

J. Hajek, Z. Sidak, P.K. Sen, Theory of Rank Tests, 2nd edn. (Academic, New York, 1999)

10.

Y. Han, I. Song, Y. Park, Some root properties of recursive weighted median filters. Signal Process. 25(3), 337–344 (1991)CrossRef

11.

S. Lee, Y. Zhang, S. Yoon, I. Song, Order statistics and recursive updating with aging factor for cooperative cognitive radio networks under SSDF attacks. ICT Express 6(1), 3–6 (2020)CrossRef

12.

Y.H. Lee, S.A. Kassam, Generalized median filtering and related nonlinear filtering techniques. IEEE Trans. Acoust. Speech Signal Process. 33(3), 672–683 (1985)CrossRef

13.

E.H. Loyd, Least-squares estimation of location and scale parameters using order statistics. Biometrika 39, 88–95 (1952)MathSciNetCrossRef

14.

C.L. Mallows, Some theory of nonlinear smoothers. Ann. Stat. 8(4), 695–715 (1980)MathSciNetCrossRef

15.

S.R. Park, H. Kwon, S.Y. Kim, I. Song, Relative frequency of order statistics in independent and identically distributed random vectors. Commun. Stat. Appl. Methods 13(2), 243–254 (2006)

16.

S.R. Park, I. Song, S. Yoon, T. An, H.-K. Min, On the sums of the probability functions of order statistics. J. Korean Stat. Soc. 42(2), 257–265 (2013)MathSciNetCrossRef

17.

S. Peltonen, P. Kuosmanen, K. Egiazarian, M. Gabbouj, J. Astola, Analysis and optimization of weighted order statistic and stack filters, in Nonlinear Image Processing, ed. by S.K. Mitra, G.L. Sicuranza. (Academic, San Diego, 2001), pp. 1–26

18.

I. Pitas, A.N. Venetsanopoulos, Digital filters based on order statistics, in Nonlinear Digital Filters, vol. 84 (Springer, Boston, 1990)CrossRef

19.

H.V. Poor, J.B. Thomas, Advances in Statistical Signal Processing: Signal Detection, vol. 2 (JAI Press, Greenwich, 1993)

20.

I. Song, J. Bae, S.Y. Kim, Advanced Theory of Signal Detection (Springer-Verlag, Berlin, 2002)CrossRef

21.

I. Song, S. Lee, S.R. Park, S. Yoon, Asymptotic value of the probability that the first order statistic is from null hypothesis. Appl. Math. 4(12), 1702–1705 (2013)CrossRef

22.

I. Song, C.H. Park, K.S. Kim, S.R. Park, Random Variables and Stochastic Processes (in Korean) (Freedom Academy, Paju, 2014)

23.

I. Song, S.R. Park, S. Yoon, Probability and Random Variables: Theory and Applications (Springer-Verlag, Berlin, 2022)CrossRef

24.

A. Stuart, J.K. Ord, Advanced Theory of Statistics: Distribution Theory, vol. 1, 5th edn. (Oxford University, New York, 1987)

Titel: Order Statistics
verfasst von: Iickho Song
So Ryoung Park
Wenyi Zhang
Seungwon Lee
Verlag: Springer Nature Switzerland
Buch: Fundamentals of Order and Rank Statistics
Print ISBN: 978-3-031-50600-0

Electronic ISBN: 978-3-031-50601-7

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-50601-7_2

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