Asymptotic Properties of Order Statistics Correlation

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Asymptotic Properties of Order Statistics Correlation Coef?cient in the Normal Cases
Weichao Xu, Member, IEEE, Chunqi Chang, Member, IEEE, Y. S. Hung, Senior Member, IEEE, and Peter Chin Wan Fung
Abstract—We have previously proposed a novel order statistics correlation coef?cient (OSCC), which possesses some desirable advantages when measuring linear and monotone nonlinear associations between two signals. However, the understanding of this new coef?cient is far from complete. A lot of theoretical questions, such as the expressions of its distribution and moments, remain to be addressed. Motivated by this unsatisfactory situation, in this paper we prove that for samples drawn from bivariate normal populations, the distribution of OSCC is asymptotically equivalent to that of the Pearson’s product moment correlation coef?cient (PPMCC). We also reveal its close relationships with the other two coef?cients, namely, Gini correlation (GC) and Spearman’s rho (SR). Monte Carlo simulation results agree with the theoretical ?ndings. Index Terms—Bivariate normal, concomitant, delta method, transform, Gini correlation (GC), Kurtosis, order Fisher’s statistics correlation coef?cient (OSCC), Pearson’s product moment correlation coef?cient (PPMCC), ranks, relative ef?ciency, skewness, Spearman’s rho (SR).
I. INTRODUCTION AND MOTIVATION
A
MULTITUDE of methods have been proposed in the literature to measure the intensity of correlation between two random variables with a bivariate distribution. Among these measures the Pearson’s product moment correlation coef?cient [1]–[4] (PPMCC), Spearman’s rho [5] (SR), and Kendall’s tau [5] are perhaps the most widely used [6]. The Pearson’s coef?cient is appropriate mainly for indicating linear associations, while the other two rank-based coef?cients are invariant under increasing monotone transformations [5]. Recently, the present authors proposed a novel measure of correlation called order statistics correlation coef?cient (OSCC), which bridges the gap between Pearson’s coef?cient and the other two rank-based coef?cients [7], [8]. Theoretical analyses and extensive Monte Carlo experiments have shown that OSCC has properties including 1) robustness in the presence of noise, 2) small biasedness, 3) high sensitivity to changes in correlation between signals, 4) capability to detect accurately time-delay, 5) fast computational speed,
and 6) robustness under monotone nonlinear transformations. These desirable properties make OSCC a potentially useful alternative to the three classical correlation coef?cients [8]. However, the understanding of OSCC is far from complete due to the lack of knowledge on its distribution with respect to bivariate normal populations. Such knowledge is indispensable when one performs theoretical analyses of OSCC, such as the analytical expressions of its mean, variance, skewness, and kurtosis, just to name a few. In order to gain further insight into OSCC, we derive in this paper the asymptotic distribution of OSCC when samples are drawn from a bivariate normal population with correlation (binormal model). We will also compare OSCC to PPMCC, since the latter has long served as a benchmark in the area of correlation studies. In other words, any other correlation coef?cients should preferably emulate the properties of PPMCC under the binormal model [9]. The paper is organized as follows. In Section II, we give some basic de?nitions and several lemmas needed in this paper. In Section III, we prove that OSCC is asymptotically equivalent to PPMCC in terms of distribution and moments. In Section IV, we formulate and prove two theorems on the relationship between OSCC and the other two coef?cients. Section V is devoted to the applicability of the asymptotic theories to small samples based on simulation results. Finally, in Section VI, we draw our conclusion on the order statistics correlation coef?cient. II. PRELIMINARIES In this section, we give some basic concepts and lemmas concerning normal random variables and the associated order statistics. These prerequisites are necessary in order to establish our main results in the sequel. A. De?nitions 1) Order Statistics and Concomitants: Let denote independent and identically distributed (i.i.d.) data pairs drawn from a bivariate population with continuous joint cumulative distribution function (cdf). Sorting the data pairs in ascending order with respect to the magnitudes of , we get a sequence , where are of new data pairs termed the order statistics of and the associated concomitants [10]–[12]. is at the th position in the sorted 2) Ranks: Suppose that , the number is termed sequence the rank of and is denoted by . Similarly we can get the which is denoted by [5]. rank of
Manuscript received March 30, 2007; revised August 25, 2007. This work was supported in part by the Hong Kong Innovation and Technology Commission under Grant ITS/109/02, in part by Hong Kong RGC under Grant N_HKU703/03, and in part by the University of Hong Kong under Small Project Grant 200507176052. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Patrice Abry. W. Xu, C. Chang, and Y. S. Hung are with the Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong (e-mail: wcxu@eee.hku.hk; cqchang@eee.hku.hk; yshung@eee.hku.hk). P. C. W. Fung is with the Department of Medicine, University of Hong Kong, Hong Kong (e-mail: hrspfcw@hkucc.hku.hk). Digital Object Identi?er 10.1109/TSP.2007.916127
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B. Lemmas In the sequel, we use symbols , , , and to denote the mean, variance, covariance, and correlation coef?cient of (between) random variables, reif follows a spectively. We write and , normal distribution with if and follow a and bivariate normal distribution with , , , , and . Given these notations, we now list the following lemmas which are necessary to investigate the properties of OSCC under the binormal model. be i.i.d. data pairs Lemma 1: Let . from a standard bivariate normal population , , , Write and . Then (1) and (2) for being large. Lemma 2: Let , , Then, as large
From its de?nition in (6), it is easy to verify that OSCC and . However, as pointed out in is not symmetric in [8], a symmetric version can be de?ned as if symmetry is a critical feature in practice. The well-known PPMCC is de?ned as [2]
(7)
It is quite dif?cult and maybe impossible to derive the exact directly from (6). However, as demonstrated distribution of below, we can establish the asymptotic equivalence between and by writing as the summation of and a remainder term, whose mean and variance tend to zero with great rapidity. A. Mean and Variance of Under Binormal Model
Theorem 1: Let and be de?ned as in (6) and (7) with respect to i.i.d. sample pairs from a bivariate normal popula. Then, as tion with correlation . Write (8)
,
, be the same as in Lemma 1. Let , and .
and (9) is shift and scale invariant Proof: It has been shown that [7], [8]. Therefore, without any loss of generality, we assume in the following that the parent population is of the standard bivariate normal distribution, that is (10) . We also have Write assumption (10). Let and be shown from [10] that from the . It can (11) are independent of , the latter being mutually indewhere . Then can be written pendent with distribution as the quotient of [8] (12) and
(3) Lemma 3: Let i.i.d. sample lation be the order statistics of an drawn from a standard normal popu. Then, as large (4) where Lemma 4: Let Write for , . Then, as . be de?ned as in Lemma 3. , , and large (5) The proofs of these lemmas are provided in the Appendices. III. ASYMPTOTIC PROPERTIES OF OSCC UNDER BINORMAL MODEL Let be i.i.d. data pairs drawn from a bivariate normal population with correlation . As proposed in [7], [8], the OSCC is de?ned as
(13) Let and denote the numerator and denominator of (7), respectively. Let , , , and be de?ned as in Lemma 1. After some straightforward algebra, we have (14) (6) and (15)

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where (16) (17) (18) and (19)
From (7), (18), and (19), it is obvious that and . Then we have
,
,
(27) Substituting (3) and (25) into (27) gives (28) from which and together with (3), (23) we have
Now we can write (20) By a similar procedure it follows that and denote the numerator and denominator of (20), Let respectively. It follows from the Delta method [13] that Substituting (29) and (30) into (21) and letting (21) and (30) yield (31) By virtual of the Cauthy–Schwarz inequality [14] as well as the results of (3) and (24), it follows that (29)
(22) , , To evaluate (21) and (22), it is suf?cient to ?nd , , and , whose orders of magnitude are determined by , , , , and . The asymptotic means and variances of , , and are provided by Lemma 2. Now we focus on the orders of and in the following. Taking expectation of (17) and applying the properties of as well as some elementary inequalities yield
(32) and similarly (33) Then we have from (28) and (33) that
(23) where the last step follows from Lemma 3. By Lemma 3, Lemma 4, and some elementary inequalities, we have (34) Given (3), (32)–(34), and applying the Cauthy-Schwarz inequality again, we have (35) (24) Similarly, we can obtain It follows directly from (35) and (36) that (25) and (37) A substitution of (29), (30), and (35)–(37) into (22) leads to (38) (26) whence the result. and (36)

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Remark 1: Since and tend to zero as , it follows from [15] that converges in probability to 0, which . On the other hand, it is well known that is denoted by . Then from [16] we have , namely, is an asymptotic consistent estimator of with respect to the bivariate normal population. . It follows directly from (38) Remark 2: Write . By the Chebyshev inequality [13], that for any positive we have . This shows how the probability of falling in can be made arbitrarily close to unity by choosing suitably. In other words, we can write (39) as well as (40)
For
, we have
(46) By an argument similar to that given in Remark 2, it follows that (47) with probability arbitrarily close to unity. The proof is then concluded by substituting (39) and (47) into (46) and neglecting the higher order in?nitesimal terms. Given Theorem 2 and the results of Hotelling [17], we can summarize the asymptotic mean, variance, skewness, and kurfor large as tosis of (48)
B. Asymptotic Distribution of OSCC Having proven Theorem 1 above, we are capable of ?nding the asymptotic distribution of OSCC based on the relation (41) It can be shown that [1]. Then we have (42) Recall that also decreases to zero as becomes large. is negligible for suf?ciently large Then the residual term [15]. In other words, when is large enough, the distribution is dominated by the distribution of , whose density of function is, for any
(49) (50) (51) which are measures of location, scatter, symmetry, and longtailedness of the distribution [15]. D. Relative Ef?ciency of to
Since both and are unbiased estimators of for large, we can compare the performance of the two estimators by means of the relative ef?ciency (RE), which can be de?ned as [18]
(52) It follows obviously from Theorem 2 that RE approaches 100% as becomes suf?ciently large. E. Fisher’s -Transform It can be shown that, when is large [16] (53) where the symbol “ ” reads “coverges in probability to” [15], and estab[16]. The asymptotic equivalence between lished above allows us to assert that (54) for large. However, as pointed out in [13], the use of the relations (53) and (54) is not recommended to test a hypothetical nonzero value of due to the slow convergence speed of (and ) to normality and dependence between the standard hence
(43) where [2].
C. Convergence Rate of the Moments of OSCC Theorem 2: Let
for
and
. Then (44)
for Proof: For Theorem 1 that
and
being suf?ciently large. and large, it follows from [8] and
(45)

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error and . In one of his pioneering papers [3], Fisher introduced the extremely useful transformation (55) and showed that (56) with great rapidity. Write that . Then we can expect
(57) is also true from the continuous mapping theorem [16]. IV. RELATIONS WITH OTHER COEFFICIENTS It can be shown that OSCC is closely related to two other coef?cients, namely, SR and Gini correlation (GC) and denote [10], [19]. Let and , respectively. Then the the ranks of following two relationships hold. Theorem 3: The order statistics correlation coef?cient of , where denotes the ranks Spearman’s rho. for Proof: It follows obviously that . Substituting these into (6), we have (58) Now we evaluate the numerator respectively, as follows: and denominator of (58),
Fig. 1. Illustration of convergence speeds of E (e ) and V (e ) against n = 20; 30; . . . ; 100 and  = 0:1, 0.3, 0.5, 0.8, 0.9, respectively. Note that the legends of (C) and (D), which are the same as those of (A) and (B), are not plotted for the purpose of clarity.
V. NUMERIC RESULTS In this section, we investigate the applicability of our asymptotic theories developed in previous sections for small samples. Monte Carlo experiments are performed for samples of size . The number of trials is set to for the purpose of accuracy. A. Convergence Rates of Fig. 1 illustrates the convergence speeds of and versus the sample size with an increment in each step. The relationships between the decreasing rates of and and the magnitude of are also revealed by , 0.3, 0.5, 0.8, 0.9, respectively. curves associated with It can be observed that with increase of , the magnitudes of and decrease downward with quite fast speed. The and are less than and , respecvalues of tively, for even as small as 30. As for the effect of on the and behave rather differently. convergence rates, In Fig. 1(A), we can see that for any ?xed , increases . On the with at ?rst, but the relation reverses after relates negatively to other hand, as shown in Fig. 1(B), in a consistent manner, that is, the larger the intensity of , the lesser the values of for any ?xed . Fig. 1(C) and (D) and depict respectively the ratios against for , 0.3, 0.5, 0.8, 0.9. We can observe that the curves of are already approximately horizontal when is small, suggesting converges to constant values with great speed. On the that other hand, the convergenc speed of is rather slow, especially and . However, despite the magnitude of when , all curves of become level when . These observations verify our theoretical results of (8) and (9) established by Theorem 1.
(59) and (60) Substituting (59) and (60) into (58), we have (61) which is the expression of the Spearman’s rho [5]. Theorem 4: The OSCC of the ranks of and the values of is the sample Gini correlation. by in (6) yields Proof: Replacing (62) which is the sample Gini correlation [19].

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TABLE I OBSERVED MEAN VALUES OF w AND r FOR n
= 20, 50, 100
TABLE III OBSERVED AND THEORETICAL VALUES OF
(w ) AND (r )
COMPARISON RESULTS OF V
(w ), V (r ) AND THEORETICAL RESULT
TABLE II
TABLE IV OBSERVED AND THEORETICAL VALUES OF
(w ) AND (r )
B. Comparative Results for Moments of
and shown that GC and SR can also serve as estimators of . For convenience, let and denote SR and GC, respectively. Then the four estimators are (63) (64) (65) (66) where comes from [19] and from [5]. Having (63) – (66), we are able to compute the relative ef?ciencies of OSCC, GR, , and ) to Pearson’s coef?cient and SR (notations by means of the ratios of to the variance of each of the other three respective estimators. with respect to Fig. 2(A) shows the increasing trend of . It also reveals the negative relationsample size ship of the convergence speed of to . It can be observed that despite the negative effect of , all four curves stand above and approach 98% when . Fig. 2(B) 96% for , , and for and . compares We only compare , and in the null case due to except for . It can the lack of theoretical results of for , sugbe easily seen that gesting the advantage of OSCC over GC and SR. Moreover, , [20] and it follows that for [5] as . In other words, and can never approach 100% no matter how large is. On the can be made as close to 100% as possible by other hand,
Table I lists the observed mean values and from the Monte Carlo simulations for , , , respectively. to is It can be seen that 1) the convergence speed of consistently, 3) for most cases, quite fast, 2) is even smaller than that of , and the biasedness of 4) despite few exceptions, the biasedness of is negative with small and positive with large . and In Table II, we present the simulation results of together with theoretical values from (49). Unlike , the convergence speeds of is rather slow. However, it and is less appears that the difference between and negligible for . Therefore, noticeable for by (49) for . it would be safe to approximate can be considered asymptotic in A sample size of practice. Tables III and IV contain, respectively, the observed and theand oretical values of skewness and kurtosis with respect to . It can be seen that although their convergence speeds to (50) and behave compaand (51) are much slower, rably with increase of and . In other words, we do not conand sider that there are signi?cant differences between as far as convergence rates of their skewness and kurtosis are concerned. C. Comparative Results of Relative Ef?ciency Study As mentioned in Section III-D, both and are eligible as estimators of the population correlation . Besides, it has been

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Fig. 2. Relative ef?ciency of w . (A) relationship of (B) comparison of RE , RE and RE when  = 0 TABLE V OBSERVED AND THEORETICAL VALUES OF E (z
RE
versus
n and .
) AND E (z )
Fig. 3. Empirical distributions of w , z , r , and z from populations having correlation 0 and 0.8 when the sample size n = 30. (A) Histograms of w . (B) Histograms of z . (C) Histograms of r . (D) Histograms of z . It can be observed that the two distributions of w are far from normal for n = 30. On the contrary, both the two distributions of z are roughly normal with nearly equal variances. Moreover, the distributions of w and z are very similar to their respective counterparts even when n is as small as 30.
TABLE VI COMPARISON RESULTS OF V (z
) AND V (z )
Fig. 3 shows the property of variance-stabilization of Fisher’s transform. In Fig. 3(A) are plots of the histograms of ( ) from populations with correlations 0 and 0.8; Fig. 3(B) . For comparison, shows the corresponding histograms of and are also presented in the histograms with respect to in Fig. 3(C) and (D), respectively. The two distributions of Fig. 3(A) are drastically distinct in both their modal heights and forms — the one being symmetrical, the other highly skewed. On the other hand, in Fig. 3(B) the two distributions do not differ greatly in height and are approximately the same in form. Beand are very similar to their sides, the distributions of respective counterparts even when is as small as 30. This justo in ti?es to some extent the fast convergence rate of terms of distribution. VI. CONCLUDING REMARKS
choosing suf?ciently large. This is ensured by the asymptotic equivalence established in Section III. D. Distribution of Fisher’s -Transform In Table V, we tabulate the expectations of the transformed values , and the asymptotic theoretical excorresponding to , 50, 100. We pression is consistently greater than for all can see that . Furthermore, the biasedness of is more noticeable as large. However, the difference between and can be considered nonsigni?cant for . In Table VI, we summarize the observed variance values and from the experimental data. For comparison, the are also provided in the last row. theoretical values of is consistently larger than , It can be seen that 1) is approximately independent of for , and 3) 2) the difference between and are less noticeable for and negligible for .
In this paper, we have investigated the properties of the order statistics correlation coef?cient proposed previously by the present authors. Theoretical derivations and simulation results suggest that the new coef?cient is asymptotically equivalent to the Pearson’s product moment correlation coef?cient in the sense of distribution as well as moments in the normal cases. The new coef?cient also has close relationship with the other two correlation coef?cients, namely, Spearman’s rho and Gini correlation. The advantages of the order statistics correlation coef?cient over Pearson’s and other correlation coef?cients have been discussed in [7] and [8]. The results in this paper further justify that the order statistics correlation coef?cient can be used as an alternative to Pearson’s coef?cient in correlation analysis. APPENDIX I PROOF OF LEMMA 1 Proof: It is well known that the normality holds under linear transformations of normal random variables [13]. There-

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fore, and , which are linear transformations of normal and , must follow a bivariate normal random variables and the distribution. Recall that i.i.d. assumption, we have
and
(77) From (2) it follows that
(67) that is, . and are asymptotically normal It can be shown that is also normally distributed as [13]. Therefore, large. From Wishart’s formulae [21], we have (68) (69) and (70) Given (68)–(70) and by the Delta method, we have (71) We have shown in Lemma 1 that large. Then we have For brevity, let is independent of
(78) Applying some basic identities together with (68), (70), and (78), we have
(79) . It has been known long ago that [13]. Hence
(80) as
and (72) Since, by de?nition, , then (73) Noticing that arrive at and , we ?nally Substituting (81) and (82) into (80) leads to (74) and (75) thus completes the proof. thus establishing the lemma. APPENDIX III PROOF OF LEMMA 3 Proof: Write be the cdf of and On the other hand, from (1) it follows that
(81)
(82)
(83)
APPENDIX II PROOF OF LEMMA 2 Proof: From (1) in Lemma 1 we have (76)
and . Let . It follows [10] that, as (84)

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It can be easily shown that Writing into (84) yield
, with being the pdf of [10]. and substituting the expression of
Substituting (91) into (89) leads to (92) Write from the L’Hospital’s Rule that . Then it follows directly
(85) We can prove that the function is nondecreasing over by verifying the nonnegativity of its derivative, as
(93) Applying the L’Hospital’s Rule once more, we have
where
(86) is a compact notation of . Write . Now we show that when is ?x, . Noticing that , we have, for odd
(94) Then, for being suf?ciently large (95) whence the result. APPENDIX IV PROOF OF LEMMA 4
(87) and for even
Proof: The relationship between moments and cumulants gives [22]
where
(96) denotes the cumulants. It has been shown that [23] as large (97)
(88) It is easy to verify that both and are left Riemann sums over . Therefore, and are of underestimations of since is nondecreasing over , that is, for any ?x . Write . We have
Hence, it is suf?cient to study the orders of the summations of the remaining two terms, which are of order [23]. It follows that , , and (say) [13]. Substituting these identities into , we have (96) and ignoring the terms
(98) (89) Let . We ?rst prove that . Now we show that , as Applying the Cauthy–Schwarz inequality yields Recalling from Lemma 3 that have . , we then
(99) thus completes the proof. (90) REFERENCES Then we have and hence (91)
[1] R. A. Fisher, Statistical Methods, Experimental Design, and Scienti?c Inference. New York: Oxford Univ. Press, 1990. [2] R. A. Fisher, “Frequency distribution of the values of the correlation coef?cient in samples from an inde?nitely large population,” Biometrika, vol. 10, pp. 507–521, 1915.

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[3] R. A. Fisher, “On the ’probable error’ of a coef?cient of correlation deduced from a small sample,” Metron, vol. 1, pp. 3–32, 1921. [4] E. Fieller, H. Hartley, and E. Pearson, “Test for rank correlation coef?cients. I,” Biometrika, vol. 44, pp. 470–481, 1957. [5] M. Kendall and J. D. Gibbons, Rank Correlation Methods, 5th ed. New York: Oxford Univ. Press, 1990. [6] D. D. Mari and S. Kotz, Correlation and Dependence. London, U.K.: Imperial College Press, 2001. [7] W. Xu, C. Chang, Y. S. Hung, S. K. Kwan, and P. C. W. Fung, “Order statistic correlation coef?cient and its application to association measurement of biosignals,” in Proc. ICASSP’06 , 2006, vol. 2, pp. II-1068–II-1071. [8] W. Xu, C. Chang, Y. S. Hung, S. K. Kwan, and P. C. W. Fung, “Order statistics correlation coef?cient as a novel association measurement with applications to biosignal analysis,” IEEE Trans. Signal Process., vol. 55, no. 12, pp. 5552–5563, Dec. 2007. [9] J. D. Gibbons and S. Chakraborti, Nonparametric Statistical Inference, 3rd ed. New York: Marcel Dekker, 1992. [10] H. David and H. Nagaraja, Order Statistics, 3rd ed. Hoboken, NJ: Wiley-Interscience, 2003. [11] N. Balakrishnan and C. R. Rao, Order Statistics: Applications. New York: Elsevier, 1998, vol. 17, Handbook of Statistics. [12] N. Balakrishnan and C. R. Rao, Order Statistics: Theory & Methods. New York: Elsevier, 1998, vol. 16, Handbook of Statistics. [13] A. Stuart, J. K. Ord, and S. F. Arnold, Kendall’s Advanced Theory of Statistics: Volume 1 Distribution Theory, 6th ed. London, U.K.: Edward Arnold, 1994. [14] D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Classical and New Inequalities in Analysis. Dordrecht, The Netherlands: Kluwer Academic, 1993. [15] O. E. Barndorff-Nielsen and D. R. Cox, Asymptotic Techniques For Use in Statistics. London, U.K.: Chapman & Hall, 1989. [16] A. W. Vaart, Asymptotic Statistics. Cambridge, U.K.: Cambridge Univ. Press, 1998. [17] H. Hotelling, “New light on the correlation coef?cient and its transforms,” J. R. Statist. Soc. Ser. B (Methodological), vol. 15, pp. 193–232, 1953. [18] A. Stuart and J. K. Ord, Kendall’s Advanced Theory of Statistics: Volume 2 Classical Inference and Relationship, 5th ed. London, U.K.: Edward Arnold, 1991. [19] E. Schechtman and S. Yitzhaki, “A measure of association base on Gini’s mean difference,” Commun. Statist.-Theor. Meth., vol. 16, no. 1, pp. 207–231, 1987. [20] W. Xu, C. Chang, and Y. S. Hung, “The asymptotic distribution of sample Gini correlation in normal cases,” J. Multivariate Anal., 2008, submitted for publication. [21] J. Wishart, “The generalised product moment distribution in samples from a normal multivariate population,” Biometrika, vol. 20A, pp. 32–52, 1928. [22] R. A. Fisher, “Moments and product moments of sampling distributions,” Proc. London Math. Soc., vol. 30, pp. 199–238, 1929. [23] F. David and N. Johnson, “Statistical treatment of censored data. Part I: Fundamental formulae,” Biometrika, vol. 41, pp. 228–240, 1954. Weichao Xu (M’06) received the B.Eng. and M.Eng. degrees in electrical engineering from the University of Science and Technology of China, Hefei, China, in 1993 and 1996, respectively, and the Ph.D. degree in biomedical engineering from the University of Hong Kong, Hong Kong, in 2002. Since 2003, he has been a Research Associate with the Department of Electrical and Electronic Engineering, University of Hong Kong. His research interests include the areas of mathematical statistics, pattern recognition, digital signal processing, and applications.
Chunqi Chang (M’06) received the B.Sc. and M.Sc. degrees in electronic engineering from the University of Science and Technology of China, Hefei, China, in 1992 and 1995, respectively, and the Ph.D. degree in biomedical engineering from the University of Hong Kong, Hong Kong, in 2001. He is currently a Research Fellow with the Department of Electrical and Electronic Engineering, University of Hong Kong. His main research interests include statistical signal processing theories and methods and their applications to biomedical engineering and computational molecular biology.
Y. S. Hung (M’88–SM’02) received the B.Sc. (Eng.) degree in electrical engineering and the B.Sc. degree in mathematics from the University of Hong Kong, Hong Kong, and the M.Phil. and Ph.D. degrees from the University of Cambridge, Cambridge, U.K. He was a Research Associate with the University of Cambridge and a Lecturer with the University of Surrey, Surrey, U.K. In 1989, he joined the University of Hong Kong, where he is currently an Associate Professor and the Head of the department. His research interests include robust control systems theory, robotics, computer vision, and biomedical engineering. Dr. Hung was a recipient of the Best Teaching Award in 1991 from the Hong Kong University Students’ Union. He is a chartered engineer and a fellow of IET and HKIE.
Peter Chin Wan Fung received the B.Sc. (Phys.) degree, the B.Sc. degree (special hons) in radio astronomy, and the Ph.D. degree in radio astronomy from the University of Tasmania, Hobart, Australia. In 1999, he became the ?rst Chair Professor of Medical Physics with the Department of Medicine, University of Hong Kong, Hong Kong. He is also an Honorary Professor in the Department of Psychiatry and the Department of Electrical and Electronic Engineering, University of Hong Kong, where he works on multidisciplinary projects. He has been with the University of Montreal, Montreal, QC, Canada, and Stanford University, Stanford, CA. During the early 1970s, he joined the University of Hong Kong and became the Personal (Chair) Professor of Physics in 1984 and the Director of the Centre for Materials Science in 1992. He has published around 280 articles in international reviewed journals. His current research interests include the areas of biophysics, medicine, and signal processing.

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[YOUR COMPANY NAME] [Your Company Slogan] [YOUR ADDRESS] [YOUR ADDRESS 2] [YOUR CITY], [YOUR STATE/PROVINCE] [YOUR ZIP/POSTAL CODE] [YOUR COUNTRY] Phone: [YOUR PHONE NUMBER] Fax: [YOUR FAX NUMBER] Purchase Order The following number must appear on all related correspondence, shipping papers, and invoices: P.O. NUMBER: 100 To: Name Company Address City, State ZIP Phone Ship To: [NAME], [TITLE] [YOUR COMPANY NAME] [YOUR ADDRESS] [YOUR ADDRESS 2] [YOUR CITY], [YOUR STATE/PROVINCE] [YOUR PHONE NUMBER] 1. Please send two copies of your invoice. 2. Enter this order in accordance with the prices, terms, delivery method, and specifications listed above. 3. Please notify us immediately if you are unable to ship as specified. Authorized by Date

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purchaseorder模板

采购订单PURCHASE ORDER

For M/S : 华瑞益通 Name:Date: 本订单合同由买方和卖方共同签署,双方同意按本合同各项条款,买方购买且卖方出售合同规定的货物并提供相应服务。 The Purchase Order is made by and between the Buyer and the Seller, whereby the Buyer agrees to buy and the Seller agrees to sell the Goods and provide services as covered and described hereunder. SECTION 1 - 供货范围 SCOPE OF SUPPLY: You have discussed all technical requirements for this project and confirmed the compliance to the same. 卖方已经了解此项目的所有技术要求并确认完全满足所有要求： 序号Item 货物描述 Description / Scope of Supply 单位 Unit 数量 Qty 单价 Unit Price 总价 Total Prices According to (MR编号及设备名称) 1 2 3 4 5 6 Start-up and Commissioning Spare Included 7Inspection , testing and Factory Acceptance Certificate(FAT) Included 8Design, Drawing, Documentation and Manual(Required Quantity) Included 9Export Packaging Included 10Inland Transportation Included 11Commodity Inspection(if needed) Included 总价（至买方指定港口）

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PurchaseOrder模板

苏丹 XXX 项目：PROVISION OF PCC SERVICE S FOR PROJECT 项目合同编号 POJECT CONTRACT NO: P etro-Energy/ (2) 交货期 DELIVERY SCHEDULE All goods of the PO shall be delivered completely to the Carrier nomin ated by the Buyer no later tha n xxxxxxxxxxx. (3) 交货事项 DELIVERY POINT SECTION 3- 文件 REQUIRED DOCUMENTATION: (1) 1套正本增值税专用发票， 增值税专用发票的内容必须与报关单相一致。卖方必须在收到 买方发出的开票通知后 7天内将此正本增值税专用发票提交买方经办人。 如果由于卖方不 能及时提交增值税专用发票或出具的发票不符合要求， 由此产生的损失由卖方承担， 买方 有权从货款中直接扣除。 One set of original Value Added Tax (VAT) Invoice(s) submitted by the Seller. The content of VAT Invoice(s) must be in accordanee with the Customs Declaration. The Seller shall submit the VA T In voices to the Buyer dur ing seve n (7) days after receiv ing the Buyer ' s Notice about the In voice issu ing. All the losses arise n from late submissi on or in correct ness of the VAT In voices shall be borne by the Seller. The Buyer has the right to deduct the corresp ondent losses directly from the payme nt un der the Con tract; (2) 3份正本和3份副本装箱单(中英文对照)，装箱单要求详细注明订单号、 件号、货物名称、 规格尺寸、毛重、净重、包装类型。 3 origi nals of Pack ing List and 3 copies in both Chin ese and En glish Ian guage, issued by the Manufacturer or the Seller with indication of PO No., package number, the name of Goods, specificati on. Package dime nsions and weight (net and gross), List of Items with qua ntity and tag Number t, type of pack ing; (3) 制造商签署的3份中英文对照的质量证书原件，3份正本检验报告原件。 Three (3) origi nal Quality Certificates and T est and In specti on Reports in both Chin ese and En glish Ian guage issued by the Manu facturer; (4) 1份由当地商检局出具的出境货物换证凭条 (单)。(如需商检，商检费用已包括在总价里) One original Inspection Certificate issued by local Import and Export Commodity In specti on Bureau of the P .R.Chi na; (5) 工厂验收测试(FAT )报告正本1份。 Certificate of factory accepta nee test. (6) 1份由买方出具的运输释放通知 Shipp ing Release Note issued by the buyer. (7) 1份由买方指定承运人签发的收货单原件。 One origi nal and two copies of Cargo Receipt issued by the Buyer appo in ted Freight Forwarder; (8) 卖方与合同签订后一周内提交报批图纸。 MR 、补遗等相关文件中规定的最终文件 ，包括图 纸、数据表、技术手册等相关文件的 9份硬拷贝和6份CD 软拷贝。 Other technical documents, including but not limited to the following, drawing, data sheets and tech ni cal manual in 9 hard copies and 6 CDs; 9套硬拷贝和6套软拷贝（CDs ）的最终文件邮寄到买 方办公室。 项目合同编号 POJECT CONTRACT NO: P etro-Energy/ 合同货物必须在 XXXXXXXXXX 前交至买方指定的承运人处。 如果分批运输，则分批付 卖方应在两周内用特快专递方式将

奢侈品 世界十大品牌

简单汇总十大品牌： LV（louis viutton）路易威登(法) GIVENCHY纪梵希（法） HERMES爱马仕（法）CHANEL香奈儿（法） PRADA（MIU MIU）普拉达（意）VERSACE范思哲（意）GUCCI古奇（意大利）DOLCE GABBANNA（D&G）杜嘉班纳（意）BURBERRY巴宝莉（英）CARTIER卡地亚（瑞士）Ω OMEGA 欧米茄（瑞士）ROLEX劳力士（瑞士 十大奢侈品十大服装 唐纳?卡兰、Louis Vuitton、Chanel、范思哲、Dior、古驰、瓦伦蒂诺?加拉瓦尼、PRADA、GUESS、乔治?阿玛尼 珠宝 卡地亚、蒂芬尼、ENZO、Oxette、宝诗龙、Bulgari、御木本、Graff、Georgjensen、波米雷特 皮具 Louis Vuitton、Chanel、Dior、古驰、瓦伦蒂诺?加拉瓦尼、PRADA、乔治?阿玛尼、登喜路、芬迪、COACH 顶级名表 https://www.wendangku.net/doc/cc9111485.html,nger、积家、伯爵、江诗丹顿、劳力士、卡地亚、爱彼、万国、宝玑、百达翡丽 汽车 法拉利、Porsche、奔驰、宝马、莲花、宾利、凯迪拉克、菲亚特、奥迪、劳斯莱斯 豪宅 “三湖”别墅、曼德勒农场、“向往东方”的海滩、"燃点"的海滩、“拉?阿密提”、“迪奥”宫殿、纽约曼哈顿区的一座大厦顶楼三层、佛罗里达棕榈滩的“观光别墅” 化妆品 娇兰、兰蔻、娇韵诗、伊丽莎白?雅顿、奥伦纳素、雅诗兰黛、倩碧、资生堂、Dior、Chanel

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