Asymptotic Values and Expansions for the Correlation Between Different Measures of Spread. Anirban DasGupta. Purdue University, West Lafayette, IN

Transcription

1 Aymptotic Value and Expanion for the Correlation Between Different Meaure of Spread Anirban DaGupta Purdue Univerity, Wet Lafayette, IN L.R. Haff UCSD, La Jolla, CA May 31, 2003 ABSTRACT For iid ample from a normal, uniform, or exponential ditribution, we give exact formula for the correlation between the ample variance and the ample range for all fixed n. Thee exact formula are then ued to obtain aymptotic expanion for the correlation. It i een that the correlation converge to zero at the rate log n 1 n in the normal cae, the n rate in the (log n)2 uniform cae, and the n rate in the exponential cae. In two of the three cae, we obtain higher order expanion for the correlation. We then obtain the joint aymptotic ditribution of the interquartile range and the tandard deviation for any ditribution with a finite fourth moment. Thi i ued to obtain the nonzero limit of the correlation between them for ome important ditribution a well a ome potentially ueful practical diagnotic baed on the interquartile range and the tandard deviation. It i een that the correlation i higher for thin tailed and maller for thick tailed ditribution. We alo how graphic for the Cauchy ditribution. The graphic exhibit intereting phenomena. Other numeric illutrate the theoretical reult. 1

2 1 Introduction The ample tandard deviation, range, and the interquartile range are variouly ued a meaure of pread in the ditribution of a random variable. Range baed meaure are till common in proce control tudie, while meaure baed on the interquartile range are ometime ued a robut meaure of pread. It would eem natural to expect that the three meaure of pread hare ome reaonable poitive dependence property for mot type of population, and perhap for all ample ize n. The purpoe of the preent article i to invetigate the interrelation between them in greater depth than ha been done before, mathematically a well a numerically. For example, we invetigate the correlation between the ample range (W ) or the interquartile range and the ample tandard deviation, both for fixed ample and aymptotically. We alo invetigate the joint aymptotic behavior of the tandard deviation and the interquartile range in the ene of their joint ditribution, and ue thee aymptotic to evaluate the exact limiting value of their correlation for a number of important ditribution. A a matter of mathematical tractability, it i much eaier to analyze the correlation between 2 and W, both for fixed ample and aymptotically. In the next three ection, we preent exact formulae for the correlation between 2 and W, for every fixed ample ize n, when the underlying population i a normal, or an Exponential, or a uniform. They are common ditribution, and manifet a ymmetric population with no tail (uniform), a ymmetric population with thin tail (normal), and a kewed population widely ued in practice. Another reaon for pecifically working with thee three cae 2

3 i that they eem to be the only three tandard ditribution for which an exact formula for the correlation can be given for fixed ample ize. Uing the fixed ample formula, we then derive aymptotic expanion for the correlation. The firt term in the expanion give u the rate of convergence of the correlation to zero in each cae. For intance, we prove that the correlation converge to zero at the rate 1 n in the uniform cae, at the rate log n (log n)2 n in the normal cae, and at the rate n in the exponential cae. Thee derivation involve a great deal of calculation, much of which, however, ha been condened for the ake of brevity. Next, by ue of the Bahadur repreentation of ample quantile, we work out the aymptotic bivariate normal ditribution for interquartile range and tandard deviation for any ditribution with four moment. We apply it to obtain the limit of the correlation between them for five important ditribution. The general reult can be ued to obtain the limiting correlation for any ditribution with four moment. We alo ue thi general reult to form ome quick diagnotic baed on the ratio of the interquartile range and tandard deviation.thee may be of ome practical ue. The article end with graphic baed on imulation for the catterplot of againt W and IQR for the Cauchy ditribution. The graphic how ome intereting outlier and clutering phenomena. We hope that the aymptotic calculation and the graphic preented here give ome inight to a practicing tatitician a well a to an applied probabilit. We alo hope the aymptotic expanion are of ome independent technical interet. There i coniderable literature on a related claical problem, namely the ditribution of W ; ee, for example, Plackett(1947), David, Hartley and Pearon(1954), and Thomon(1955). We have not ad- 3

4 dreed that problem here. 2 Uniform Ditribution Uing a well known property of the order tatitic of an iid uniform ample, we derive below an explicit formula for ρ n = Corr( 2,W) for every fixed ample ize n. For brevity of preentation, certain intene algebraic detail have been omitted. Theorem 1. LetX 1,X 2,...,X n be iid U[a, b]. Then for each n 2, ρ n = 2 5n(n+2) (n+3) (1) 2n+3 Proof Without lo of generality, we may aume that a =0andb =1. We derive expreion for Cov( 2,W),Var( 2 ), and Var(W ) in the following tep. We will evaluate the correlation between n i=1 (X i X) 2 and W,which i the ame a ρ n. Cov( 2,W): Step 1. It i well known that for U[0, 1] data, W ha the denity n(n 1)w (n 2) (1 w), 0 <w<1. It follow immediately that E(W )= n 1,and n+1 Var(W )= 2(n 1). Step 2. Alo, obviouly, (n+1) 2 (n+2) E( n i=1 (X i X) 2 )= n The ret of the tep analyze the term E(W n i=1 (X i X) 2 ). Step 2. Toward thi end, we ue the fact that conditional on X (n),x (1), the remaining order tatitic X (2),...,X (n 1) are ditributed a the order tatitic of a ample of ize n-2 from U[X (1),X (n) ]. 4

5 Thu, denoting the order tatitic of an iid ample of ize n-2 from U[0, 1] a U (1),...,U (n 2),wehave: E(W n i=1 (X i X) 2 ) = EE(W n i=1 (X i X) 2 X (1),X (n) ) = EE(W [(X (1) X) 2 +(X (n) X) 2 + n 1 i=2 (X i X) 2 ] X (1),X (n) ) = E(W 3 )E( 1 + n 2 n n U)2 + E(W 2 )E( n 1 n [U (i) n 2 n U])2 ). n 2 n U)2 + E(W 2 )E( n 2 i=1 ( 1 n = E(W 3 ) E[(1 + (n 2)U) 2 +(n 1 (n 2)U) 2 + n 2+n 2 n 2 n 2 i=1 (U (i) U + 2 n U)2 2n n 2 i=1 (U (i) U + 2 U)] (2) n n(n 1) Step 3. From (2), uing the fact that E(W 3 )=,E(U) = 1, (n+2)(n+3) 2 and Var(U) = 1, after a few line of algebra, it follow that : 12(n 2) Cov( n i=1 (X i X) 2,W) = E( n i=1 (X i X) 2 W ) E n i=1 (X i X) 2 E(W ) = n 1 3(n+1)(n+3) (3) Step 4. Now, finally, we have to evaluate Var( n i=1 (X i X) 2 ). Firt note the identity ( n i=1 (X i X) 2 ) 2 =( n i=1 X 2 i ) 2 2nX 2 n i=1 X 2 i +n 2 X 4 (4) Therefore, E[ n i=1 (X i X) 2 ] 2 = Var( n i=1 X 2 i )+(E n i=1 X 2 i ) 2 2n 2 E(X 2 1X 2 )+n 2 E(X 4 ) (5) 5

6 Step 5. Of thee, obviouly, Var( X 2 i )= 4n 45 and (E n i=1 X 2 i ) 2 = n2 9. And E(X 2 1X 2 ) = 1 n 2 E(X X 3 1 i 1 X i + X 2 1( i 1 X i ) 2 ) = 1 n 2 ( n ( n (n 1)2 4 )) = 15n2 +20n (6) Step 6. Finally, E(X) 4 = 1 E( n 4 i j k l X i X j X k X l + i j k Xi 2 X j X k + i j Xi 2 Xj 2 + i j Xi 3 X j + i Xi 4 ) (7) There are n(n 1)(n 2)(n 3) 4! term of the firt kind, 4! 1!1!1!1! term of the econd kind, n(n 1) 4! 2! n(n 1)(n 2) 3 4! 3! 2!1!1! 2 4! term 2! 3!1! term of the third kind, n(n 1) 2!2! of the fourth kind, and, n term of the fifth kind. Therefore, from (7), on ome tediou algebra, it follow that : E(X) 4 = 1 n 3 15n 3 +30n 2 +5n (8) Step 7. Combining the previou tep, Var( n i=1 (X i X) 2 )= 2n2 +n 3 360n, Var(W )= 2(n 1) (n+1) 2 (n+2), and, Cov( n i=1 (X i X) 2,W)= ρ n follow. n 1, from which the formula for 3(n+1)(n+3) 6

7 Example 1 Uing the exact formula of Theorem 1, the following correlation are obtained for elected value of n : Table 1 n ρ n We ee that in the uniform cae, the correlation between 2 and W i quite high till about n = 30, and the correlation eem to be the maximum when n =3. Theorem 2. ρ n admit the aymptotic expanion ρ n = 10 n n 3/2 16 2n 5/2 + O(n 7/2 ). (9) Thi aymptotic expanion follow on very careful combination of term from the exact formula in (1). We omit the derivation. Note that we are able to give a third order expanion in the uniform cae. Thi i becaue formula (1) i amenable to higher order expanion. The accuracy of the third order expanion i excellent; for example, for n =15,ρ n =.691, and the third order aymptotic expanion give the approximation Normal Ditribution The formula for ρ n in the normal cae follow from a familiar application of Bau theorem(bau(1955)). Theorem 3. LetX 1,X 2,...,X n be iid N(µ, σ 2 ). 7

8 Let a n = b n = Γ( n+3 2 ) Γ( n 1 2 ); Γ( n+2 2 ) Γ( n 1 2 ); γ n = E(W ); δ n = E(W 2 ); and λ n = E() = 2(n 3)/2 (Γ( n 2 ))2 π(n 1)(n 2)! (10) Then, ρ n = γn( 2 2an (n 1) 3/2 1) λn ( 4bn (n 1) 2 1)(δn γ2 n). (11) Proof Without lo of generality, we may take µ =0,σ =1. Step 1. By Bau theorem(bau(1955)), W and are independent, and hence, Cov(W, 2 )=E(W 2 ) E(W )E( 2 )=EE( 2 E(W )) E(W )E( 2 )= E( 3 ) E(W ) E(W E() )E(2 2 )=γ n ( 2a n (n 1) 3/2 λ n 1), by a direct calculation of E( 3 ) uing the chiquare(n 1) denity. Step 2. Next, Var( 2 ) = E( 4 ) 1 = 4bn 1, again by a direct (n 1) 2 calculation of E( 4 )fromthechiquare(n 1) denity. Alo, from definition of δ n and γ n,var(w)=δ n γn. 2 Step 3. Uing the expreion for Cov(W, 2 ),Var(W), and Var( 2 ), the formula for ρ n follow on algebra. Example 2 Although a general cloed form formula for γ n and δ n in term of elementary function i impoible, they can be computed for any 8

9 given n (cloed form formula for mall n are well known; ee David(1970), or Arnold, Balakrihnan and Nagaraja(1992)). The following table give the numerical value of the correlation ρ n for ome elected value of n. Table 2 n ρ n The Table reveal that the correlation are ubtantially higher than in the uniform cae (ee Table 1). Even at n = 100, the correlation i.6. Alo, again we ee that the maximum correlation i for a mall n, namely n = 4. The next reult give the aymptotic order of ρ n. Due to the great technical complexity in finding higher order approximation to the variance of W for normal ample, thi reult i weaker than the aymptotic expanion in Theorem 2 for the uniform cae. Theorem 4. ρ n log n 2π n (12) Remark It i rather intereting that aymptotically, the correlation between and W i of the ame order, with the exception that the contant i 2 6 π, which i about 1.559, while the contant in Theorem 4 above i π, which i about.975. Thu, and W are lightly more correlated than 2 and W. Thi i in fact manifet for fixed n a well. For example, at n =10, 20, 30, 100, the correlation between and W can be computed to be.901,.812,.773 and.606, compared to.891,.809,.772 and.602 for 2 and W (Table 2). Proof. The proof of Theorem 4 require ue of aymptotic theory for W 9

10 for the normal cae a well a careful algebra with Stirling approximation for the variou term in ρ n. Step 1. By ue of Stirling formula, a n ( n+2 2 )3/2 (1 3 n+2 )2 ( (n+2) ), after everal line of algebra. 10 Step 2. Again, by Stirling formula, λ n =1 1 + o( 1 ), after algebra. 2n n Step 3. From thee, on a little more algebra, one get Cov( 2,W)= 2logn log n + o( ). 4n n Step 4. By tandard uniform integrability argument, Var( 2 )= 2 n +o( 1 n ). Step 5. For iid N(0, 1) ample, 2logn(W µ n ) H, for µ n =Φ 1 (1 1 ), with H denoting a ditribution with denity 2 exp( x)k n 0(2 exp( x)),k 2 0 being the appropriate Beel function (ee, e.g., Serfling(1980)). The variance of the ditribution H, by a direct integration, work out to π2. From 3 uniform integrability argument, that hold for the normal cae, it follow that Var(W ) π2. 6logn Step 6. The firt order approximation to ρ n now follow by combining Step3,4,and5. Remark. Comparing the leading term for ρ n in the uniform cae (Theorem2)toTheorem4,weeethatthecorrelationdieoutatalowerratein the normal cae. Thi i intereting. Thi aymptotic obervation i in fact clearly oberved by comparion of Table 1 and Table 2 a well. 10

11 4 Exponential Ditribution An exact formula for the correlation ρ n for the exponential cae will be derived by uing a repreentation for the order tatitic of an exponential ample in term of iid exponential variable. Theorem 5. LetX 1,X 2,...,X n be iid Exp(λ) variable. Let a n = n 1 i=1 1 i ; b n = n 1 i=1 1 i 2 ; (13) Then, ρ n = 4(nbn an)+2((n+1)an nbn+n n 1 i=1 a i n 1 i i=1 a i n+1) (14) (8n 3 14n 2 +6n)b n Proof We may aume without lo of generality that λ = 1. The derivation ue the following well known repreentation for the order tatitic of an Exp(1) ample : Let Z 1,Z 2,...,Z n be iid Exp(1) variable. Then the order tatitic X (1),X (2),...,X (n) of an iid Exp(1) ample admit the repreentation X (i) = ik=1 Z k n k+1. Step 1. Firt note the obviou fact that ρ n alo equal the correlation between i<j(x (i) X (j) ) 2 = n(n 1) 2 and W. Step 2. Uing the above repreentation for X (i), for i<j,(x (i) X (j) ) 2 = ( j k=i+1 Z k n k+1 )2 = j k=i+1 Therefore, i<j(x (i) X (j) ) 2 = i<j j k=i+1 Z 2 k (n k+1) 2 +2 j 1 k=i+1 j l=k+1 Zk 2 +2 j 1 j (n k+1) 2 i<j k=i+1 l=k+1 Z k Z l (n k+1)(n l+1) ; (15) Z k Z l (n k+1)(n l+1) = n k=2 (k 1)Z 2 k n k+1 +2 n l 1 (k 1)Z k Z l l=3 k=2, n k+1 11

12 by rearranging the order of ummation in the iterated um. Step 3. Likewie, obviouly, W = n j=2 Z j n j+1. Step 4. By an eay calculation, Cov( n (k 1)Zk 2, n Z j k=2 n k+1 j=2 n j+1 )=4 n 1 i=1 n i = i 2 4(nb n a n ). Step 5. On the other hand, Cov( n l 1 (k 1)Z k Z l l=3 k=2, n Z j ) n k+1 j=2 n j+1 = n l=3 l 1 k=2 k 1 Cov(Z (n k+1) 2 k Z l,z k )+ n l 1 l=3 k=2 = n 1 (k 1)(n k) k=2 + n 1 k 1 n k (n k+1) 2 k=2 n k+1 j=1 1 (16) j k 1 Cov(Z (n k+1)(n l+1) kz l,z l ) From (16), by a change of variable, Cov( n l 1 (k 1)Z k Z l l=3 k=2, n Z j ) n k+1 j=2 n j+1 = n 1 i=2 (n i)(i 1) i 2 + n 1 i=2 n i i 1 i j=1 1 j =(n +1)a n nb n + n n 1 i=1 a i i n 1 i=1 a i n + 1. (17) Step 6. Having the covariance term done, we now have to find the variance of i<j(x (i) X (j) ) 2 and of W. Of thee, trivially, Var(W )= n 1 i=1 1 = b i 2 n. To find Var( i<j(x (i) X (j) ) 2 )=n 2 Var( n i=1 (X i X) 2 ), we will ue the binomial expanion argument we ued in Theorem 1. Toward thi end, one obtain the following expreion on detailed calculation : E( n i=1 X 2 i ) 2 =4n 2 +20n; E(X 2 1X 2 )= 2(n2 +5n+6) n 2 ; 12

13 and, E(X 4 )= n3 +6n 2 +11n+6 n 3. (18) Combining all thee expreion into the binomial expanion, a in Theorem 1, one get : Var( i<j(x (i) X (j) ) 2 )=8n 3 14n 2 +6n. (19) Step 7. The formula for ρ n now follow by ubtituting the covariance and the variance expreion from Step 1-6. Example 3 Exact value of ρ n are given in the Table below for ome elected value of n. Table 3 n ρ n Remark From Table 3, we notice an intereting diverity of phenomenon from the uniform and the normal cae. For mall n, the correlation are maller in the exponential cae, and the maximum correlation appear to be attained at n = 2 itelf, while for larger n, the correlation i in between the correlation for the uniform and the normal cae. The final reult give an aymptotic expanion for ρ n with an error of the 1 order of n. Although it i a two-term expanion, due to the error being of 1 the order of n, the accuracy i poor unle n i very large. For the ake of completene, however, it i nice to have the expanion. Theorem 6. ρ n = 3 2 (log n)2 ( n +2(1+C) log n n )+O( 1 n ) (20), 13

14 where C i the Euler contant. Proof : The proof ue the aymptotic expanion, derived from the Euler ummation formula, given below : i=n 1 i 2 = 1 n + 1 2n n 3 + O(n 4 ), ni=1 1 i = C +logn+ 1 2n 1 12n 2 +O(n 4 ). (21) Step 1. By uing (21), nb n a n = O(n), on ome algebra. Step 2. Alo by uing (21), (n +1)a n nb n n +1=nlog n + O(n), on algebra. Step 3. Similarly, n 1 i=1 a i = n log n + O(n). Step 4. The final term n 1 i=1 a i i i the hardet one. We analyze thi term by uing the ummation by part formula and (21): n 1 i=1 a i i = n 1 i=1 2C log n + O(1)) = a i+1 + a 2 i n =( Clog n (log n)2 + C log n + O(1). 2 (log n)2 2 + O(1)) + ((log n) 2 + Thee four tep take care of the covariance term in ρ n. Step 5. A regard the variance term in ρ n, 8n 3 14n 2 +6n =2 2n 3/2 (1+ O( 1 )), and b n n = π2 (1 + O( 1 )). 6 n Step 6. Subtituting thee approximation into the exact formula (14) for ρ n, the aymptotic expanion in (20) follow after ome algebra. 14

15 5 Interquartile Range and Standard Deviation The interquartile range i another well accepted meaure of pread. A uch, the correlation between it and the ample tandard deviation i alo of intrinic interet. Unlike the correlation between and W, the correlation between and the interquartile range doe not converge to zero a n. In thi ection, we firt derive the joint aymptotic ditribution between and the interquartile range for any population with four moment, and ue it to obtain the limiting correlation between and the interquartile range for a number of important ditribution. We ee ome intereting effect of the thickne of the tail of the population on the correlation between and the interquartile range. 5.1 Joint Aymptotic of Interquartile Range and Standard Deviation Theorem 7. Let X 1,X 2,...,X n be iid obervation from a CDF F on the real line with a finite fourth moment and a poitive denity function f(x). Let 0 <p 1 <p 2 < 1, and let Q = Q p1,p 2 = X ([np2 ]) X ([np1 ]). For any 0 <p<1, let ξ p = F 1 (p). Then, n[(q, ) (ξ p2 follow : ξ p1,σ)] N(0,AΣA ), with A, Σdefineda A =((a ij )), Σ=((σ ij )), where a 11 = a 12 =0,a 13 =1,a 21 = µ σ,a 22 = 1 2σ,a 23 = 0; (22) 15

16 σ 11 = σ 2,σ 12 = E(X 3 ) µe(x 2 ),σ 13 = µ( p 1 p 2 ),σ f(ξ p1 ) f(ξ p2 ) 22 = E(X 4 ) (E(X 2 )) 2,σ 23 = ξp1 xf(x)dx f(ξ p1 ) ξp1 x2 f(x)dx f(ξ p1 ) ξp2 xf(x)dx f(ξ p2 ) ξp2 x2 f(x)dx f(ξ p2 ) E(X 2 )( p 1 f(ξ p1 ) p 2 f(ξ p2 ) ),σ 33 = p 2(1 p 2 ) f 2 (ξ p2 ) + p 1(1 p 1 ) f 2 (ξ p1 ) 2p 1(1 p 2 ) f(ξ p1 )f(ξ p2 ). (23). Proof The main tool needed in deriving the joint aymptotic ditribution of (Q, ) ithebahadur repreentation : X ([np2 ]) X ([np1 ]) = Z + o p ( 1 n ), where Z i = ξ p2 + p 2 I Xi ξp 2 f(ξ p2 ξ ) p1 p 1 I Xi ξp 1 f(ξ p1 ; ee, e.g., Serfling(1980). ) Step 1. By the multivariate central limit theorem, on the obviou centering and normalization by n, (X,X 2, Z) N(0, Σ). Step 2. Conider the tranformation h(u, v, z) =(z, v u 2 ). The gradient matrix of h ha firt row = (0, 0, 1) and the econd row = ( u 1 v u 2, 2, 0). v u 2 Step 3. The tated joint aymptotic ditribution for (Q, ) now follow from an application of the delta theorem and the Bahadur repreentation tated above. We omit the intermediate calculation. An important conequence of Theorem 7 i the following reult : Corollary 1 Under the hypothee of Theorem 7, n( Q ξp 2 ξp 1 ) σ N(0,c AΣA c), where c =( 1, ξp 1 ξp 2 ). σ σ 2 The corollary follow on another application of the delta theorem to the reult in Theorem 7. Corollary 2 Let ρ n = Corr(IQR,). Then a) lim n ρ n =.6062 if F = Normal; 16

17 b) lim n ρ n =.8373 if F = Uniform; c) lim n ρ n =.3982 if F = Exponential; d) lim n ρ n =.4174 if F = Double Exponential; e) lim n ρ n =.3191 if F = t(5). Corollary 2 follow on doing the requiite calculation from the matrix AΣA in Theorem 7. Thu, note that the correlation between IQR and doe not die out a n, unlike the correlation between W and. It i alo intereting to ee the much higher correlation between IQR and for the Uniform cae, and that ucceively heavier tail lead to a progreively maller correlation. The Table below provide the value of the correlation for ome elected finite value of n. Example 4 Table 4 n Normal ρ n Uniform ρ n Exponential ρ n

18 Double Exponential ρ n t(5) ρ n The correlation i remarkably table except for the uniform cae and in the uniform cae it i table for n 15 or o, that i unle n i quite mall. We find thi tability of the correlation acro a very large range of value of n intereting and alo urpriing. 5.2 Thumb Rule Baed on IQR and Simple thumb rule for quick diagnotic can be formed by uing the reult of Corollary 1. We preent them only for normal and Exponential data here; but they can be formed for any ditribution with four moment by uing Corollary 1. Corollary 3 a) n( IQR b) n( IQR 1.349) N(0, 1.566) if F = normal; 1.099) N(0, 3.060) if F = Exponential. Corollary 3 i a direct conequence of Corollary 1 by taking p 1 = 1,p 4 2 = 3, 4 and on computing the mean and the variance of the limiting normal ditribution for IQR by uing the expreion in Corollary 1. Uing 1.5 tandard deviation around the mean value a the plauible range (choice of 1.5 i evidently omewhat ubjective) for IQR,wehavethe following thumb rule for normal and Exponential data : 18

19 Thumb Rule For iid normal data, IQR hould be in the interval [.85,1.85], [.975,1.725], [1.05,1.65], [1.08,1.6] for n = 15, 25, 40, 50 repectively. For iid Exponential data, IQR hould be in the interval [.4,1.75], [.6,1.6], [.68,1.5], [.75,1.45] for n = 15, 25, 40, 50 repectively. The overlap between the two et of interval decreae a n increae. The thumb rule for the normal cae above may be of ome practical ue. 6 The Cauchy Cae The Cauchy cae i alway an intereting one to addre becaue of it lack of moment. Thu, the correlation between and either W or the interquartile range i not defined. Still, one would certainly expect ome poitive dependence. Thi i explored in thi final ection by ue of graphic baed on imulation for three different ample ize : n = 10, 25, and 100. The v. IQR catterplot are fundamentally different; they how a maive concentration of the point cloe to the vertical axi, and a mall fraction of tray point. We believe thi i the connection of the aociation between and IQR with the thickne of the tail of the population previouly een in ection 5. But now we ee it in almot an extreme form for the Cauchy cae. The graphic how two intereting phenomena for the W v. catterplot : there i alway an outlier in the catterplot, and there i alo an intereting clutering. The clutering get omewhat blurred a n increae. However, there i an obviou poitive dependence een in each catterplot. It would be intereting to quantify thi mathematically by uing ome meaure 19

20 of dependence other than correlation. Reference Arnold,B., Balakrihnan,N. and Nagaraja, H.(1992). A Firt Coure In Order Statitic, John Wiley, New York. Bau,D.(1955). On Statitic Independent of a Complete Sufficient Statitic, Sankhya,15, David,H.A.(1970). Order Statitic, John Wiley, New York. David,H.A., Hartley,H.O. and Pearon,E.S.(1954). The Ditribution of the Ratio, in a Single Sample, of Range to Standard Deviation, Biometrika, Plackett,R.L.(1947). Limit of the Ratio of Mean Range to Standard Deviation, Biometrika, 34, Serfling,R.(1980). Approximation Theorem of Mathematical Statitic, John Wiley, New York. Thomon, G.W.(1955). Bound for the Ratio of Range to Standard Deviation, Biometrika, 42, van der Vaart, A.W.(1998). Aymptotic Statitic, Cambridge Serie in Statitical and Probabilitic Mathematic, Cambridge Univerity Pre, Cambridge 20

21 Range 500 Simulation (n = 10) IQR 500 Simulation (n = 10)

22 Range 500 Simulation (n = 25) IQR 500 Simulation (n = 25)

23 Range 500 Simulation (n = 100) IQR 500 Simulation (n = 100)