Gertrude M. Cox SO~lli USES OF QUASI-RANGES by John T. Chu Department of Statistics University of North Carolina, Chapel Hill Special report to the Office of Naval Research of work at Chapel Hill under Contract NR 042 031, Project N7··onr284(02), for research in probability and statistics. Institute of Statistics Mimeograph Series No. 124 February 23, 1955 SQIIE USES OF QUASI-RANGES By ~ohn 1,2 T.Chu, University of North Carolina 1. Summary. Confidence intervals for, and tests of hypotheses about, the difference of two quantiles ( of the same distribution ) are obtained, usine one or two properly chosen quasi-ranges. proved. Consistency ( of the estimates ano tests ) is Applications are discussed to the standard deviation of a normal distribution and the parameter of a binomial distribution. 2. Introduction. Let a population ( not necessarily continuous ) be given with cdf ( cumulative distribution function) F(x). ~ p For a given p, 0< p < 1, any satisfying (1) F( ~ P - 0 ) < P = <;= F( ~ P ) is called a quantile of order p ( or p-quantile ) of the given distribution. Usually one of them is chosen, by one way or another, as the Let ~q be the is drawn and x q-quantile, where r and x order of magnitude). s p < q < 1. p-quantile. Suppose that a sample of size n are the r-th and s-th order statistics ( in ascending Intuitive17, it seems obvious that if n is sufficiently large, and r is small compared with n while s is not much less than n, then X - x r is likely to be relatively large, consequently the event ~q - s ~p occurs with probability close to 1. : for given 0 <; p < q < 1, and 0 <; a <; s - xr X ~ An interesting question then arises 1, what would be a nice way, assuming there exists at least one way, for choosing two order statistics x r and x s from a sample of given size n, such that (2) P( x s - xr >= ~ q - ~ p ) ~ 1 - a, where P(E) denotes the probability of the event E? One may also ask similar questions concerning ,. lSponsored by the Office of Naval Research under Contract NR 042 031. 2Presented at the 1954 Annual meeting of the Institute of Mathematical Statistics. -2p( (2 1 ) X S' -xr P( x -x Sf f <1; = q -1; P » l . a , and = -1; <x -x ) > 1 - 2a. q p = s r = These questions, of course, are rather loosely stated. For example, the meaning r f <1; = of the word " nice " is not specified. Besides, there are at least two different approaches to the problems, namely: parametric and non-parametric. in a sample of size n, Incidentally, x n .. xl .is known as the range, and xn-r+ 1 - x r , where 1 < r ~ ( n + 1 ) / 2, a quasi-range or the r-th range, e.g., ~4_7. Here, how- ever, any x s - xr , s ?_ r, will be called a quasi-range. In this paper, an attempt is made to obtain some distributionfree methods, optimal in a certain sense, of choosing r,s,r l ,and (2) and (2 1 ) hold for given p,q,a, and n. (6), Sl such that In section 3, lower bounds (4), (5), (14), and (15), are obtained for the probabilities given in (2) and (2 1 ). Corresponding to the bounds (14) and (15), two integers k and kl are defined by (19) and they can be found by using (5.7. If r, 5, r l , and 51 are defined by (8) and (9) in terms of these k and k ' , then the corresponding order statistics satisfy (2) and (2'). 'When the parent distribution meets certain continuity requirements, we are able to show ( section 3, Theorem ) that both x - x and s r x s ' - xrl ' chosen in the way just described, are consistent estimates of ~q - ~p and provide consistent tests with respect to various hypotheses and alternatives concerning ~ q - ~ p • 2 For a normal parent distribution with variance (] , if we choose q =1 - p , then defined by (17). ~ q - ~ = 200 , where {(-a) p. = p , and I ( x ) is To obtain, e.g., a 100 ( 1 - 2a ) per cent confidence interval for (], one proceeds as follows: Take any ,p (0 < p < 1 ). Let wr -= (xn-r+ 1 - x r ) / 2a, wr 1 = (xn-r '1 + - xr , ) / 2a, where I(-a) = p and r and r' are respectively the largest and smallest integers (3) which satisfy (26) with q'" 1 - p. Then p( w , < (] < r - = W r ) > 1 - 2ci. = Here, a question follows naturally: p is taken arbitrarily, but, does the value of p -3effect the efficiencies of the corresponding estimates and tests ?; and if so, how to choose p to maxlluze the efficiencies? To these questions, some answers are given in section 5. The efficiencies of w and .wrt relative to S' of (29), r are defined by (30) in the conventional way, i.e., as the ratios of the variance of 8 ' to the variances of w (j/E(w ) and w r r r ,a/ E(wr ,) which are unbiased estimates of a. However, exact values of the efficiencies of w and wrf are r hard to obtain, because the variances of w and w r usually cannot be found r without very laborious computations. r For large samples, a method of approximation is suggested for obtaining the expectations and variances of w and w , and r r' the efficiency of w ( and similarly w,) is approximated by the asymptotic r. r )/2a l , where ~f' = rnpl 7' + 1, efficiency of ( x ,- X I ~ / n -< pI < r/n, q' 1 - pt, and ~(-at) == pl. v wrl are about == .65. ~ - vI == rnqJ 7+ 1,(r -- Maximum efficiencies of w - 1 ) and r A method is suggested of finding, for given n and a, the values of p which maximize respectively the efficiencies of w r and Table 2 in section 5 is given for illustration. For samples of moderate sizes, we made no attempt to solve the problem of " how to find P ". There is evidence, hOl1ever, that if p is properly chosen, the efficiencies of the corresponding wr and wrl are reasonably high. We find, e.g., for sample sizes around 20 and 30, if p == .25, the efficiencies of the corresponding w and w are about .70 ( Tables 3 and 4 ). r r' In section 6, we discuss_some applications to the parameter Q of a binomial distribution f(x) = Ql tests are obtained for testing p Q ~ tives Q < P or Q > q, and known one ["7, p. 14_7, Q f ~ p. - x ( 1 _ Q )x, x q and Q For testing = 0, 1. Consistent = P with respect to the alternaQ == P , our method improves a in making unique the choice of critical region and the test consistent with respect to Q ~ p. Finally He note that the idea is by no means new of using quasi-ranges, as well as sDuilar statistics, in estimations and testing of hypothese~. In fact, much work has been done along these lines. For references, see [1_7 , (4_7, and those cited there. ·43.Consistency. Let a population be given with cd! F(x). O' < P < q < 1 , the corresponding quantUes of orders are uniquely defined. ~ Let xl x ~ 2 ••• ~ x ponding to a random sample of given size If Lemma 1. (4) =1 L - P( x <; s ~ ) q 1 r p( x <; (5) L =1 - p( X s > l::'q ) - p( x f "" p( <; = , (6) L + L - 1 p( x <; = occurrence of the events L <; = - x _ sl Proof. p( x > ~ p( x - = f = r <;= r' X t s <; <; f <; = ~ t q ~p , and t q , are integers, then • xr = > ~ - ~ ) q p ) + p( x <; ~ ) = U ; p( x <; = s r l::' ) ""p <; =' p( x ) + p( X x - x t - <; S <; n = = ~: ) q and Then, <; Sl > t = q s r p p be the order statistics corres- n n. <; Suppose that for given <; r p _ x: sIr' t > ~ ) <; l::" _ = ""q l::" ""p ) 1 ):: U ; p <; J U ... U <N 1. - ~ ) > rJ = q p = s r = Let p( A, B ) denote the probability of simultaneous A and B. Clearly p( x s - x r > ~ = q p = > ~ , x <; ~ ) > p( x > ~ ) ... PC x <; ~ ) - 1. Therefore we have s=q r=p'" .s=q r=p p( x - r > ~ - ~ ) . Likewise we obtain the other inequalities. s r= q p For samples drawn from a binomial distribution, say Remark. with pdf ( prohability densit-,r fl.ll1ction) i f we define ~p =0 and ~q:: f(x) = pI - x ( 1 _ p )x , x ". 0, 1, 1 , then the lower bounds in Lemma 1 are actually attained. Lemma 2. For all integers k and kl 0;; k, k I ~ n - 1, ; and c = q - p , choose r r where t =~ ( n - k ) p / ( 1 - c =~ ( n - , k ) p / ( 1 • c [a.? denotes the integral part of a. ).7 + 1, )_7 + 1, non-decreasing and non-increasing functions of c 2 be given such that 0 <; 01 <; c <; c 2 Sl =r +k ; =rl + k' , Then the corresponding RHS ( right I hand sides) of (14) and. (15), 10lfer bounds for and s <; L and L , are respectively k 1. and If k1.Further, let c k = [n0 .7 2 and k l 1 -5- [nc1 •..7 ' then (10) Lim L ~ 00 = Lim LI n--> ..,. :t. 00 t: I On the other hand, if k :: [" nc1 _7' and k :: nc 2_7 parent cdf F(x) is continuous at X= l; and x = ~ p (11) Lim U :: n-> 00 q , and, if, in addition, the , then I Lim U:: O. n~ 00 t t Proof. It can be seen that 1 ~ r, r ~ n and 1_~ s, s ;; n , e. g., s ~ n because s < ( n - k ) p / ( 1 - c) .. k + 1 < n + 1. Hence I L and L, as well as the RES of (14) and (15), are well defined functions of k I and k. Now (12) p( x s < t' ~q ) = 1- B ( s - 1, F( n t' - ~q 0 ) ) = < 1- Bn (s - 1,q). where n( r, p ) = (13) e B r ~ i = ° is, for fixed n and r, a decreasine function of p, 0 < p < 1. (14) Hence Bn (s - 1 , q ) - Bn (r - 1 , P ) ; I t t L > - B (s - 1 , q ) + B ( r - 1 , p ) (15) n n = Now r is a non-increasing function of k. But i f k is increased by 1 L> :: decreased at most by 1. . Hence quent1y so is the RES of (14). s , r is is a non-decreasing function of k , conse- In a silnilar way, we show that the RHS of (15) is t a non-increasing function of k. It is well knm~n, ["2, p. 200_/ and ~3, p. 193_7 that, as n tends to 00, (16) B ( r, p ) n uniformly in r I( n tends to is fixed, and --> , 0 :; r :;: n, where x:: ( r - np ) (17) As I( x) 1.- np ( 1 - p ) _7 - 1/2 2 x) :: and - ( 2n ) -1/2 e -t /2 dt • it can be shmin that if k:: ["nc2_7 and s are defineu by (8), 00, r 0, , where c2 > c and < 1 -6(18) for ( r - 1 - np ) / n ( s - I-nq ) / n = ( c -> _ 00, n-1/2 ( r _ 1 _ np) 2 ... ( c - c 1 + o( n- ) =b 2 b 1 - q ) / ( 1 - c ) > O. - c ) ( Lim L:: 1. n-> 00 ... o( n-1) ) p / ( 1 - c) where -> n-1/2 ( s - 1 - nq) =( 1 - c 2 and ) p / ( 1 - c) + c 2 - q (16), and (18), we obtain Combining (14), Likewise we prove the rest of (10) and (11). Lemma 3. k (k t = the ) Corresponding to given n and a, 0 < a < 1, let least ( ~reatest ) integer between t s are defined accordingly ~ 1 - by (8) and (9), the RHS of (14) and (15) . ( From Lennna 2, such and k For fixed k Pi and t and n - 1 0 , such that, when r, s, r , and ly large ). 00, - a. exist for any a, 0 < a < 1 , i f n is sufficient' i =- 1, 2, where qi PI < P < P2 and ql < q < q2, define (20) = {npi.7 ri Assume that l~ge F(x) + 1 is continuous at x = (nqi_7 + _1 = ~ P and x = ~ q • si , • Then, for sufficiently n, < s = 2, (21) t where r, s, r , and s' are defined by (8) and (9) with Proof. k ~ ~nc2_7 for any fixed Choose c 2 By ~ nP1 + 1 ~ r 1 and s ~ n [ n k I given by (19). is sufficiently large. r ~ n ( 1 - c c, then 2 ( 1 - c 2 ) p / ( 1 - c ) + c2 Similarly following (11), we have and (10), if k is the integer defined by (19), then c 2 > c , provided that sufficiently close to k r ~ r2 ) P / ( 1 - c ) _7' ~ nq2 ~ s2. and s ;;; sl. The fol10lTin[~ lemma is a known fact (2, p. 369_7. We state it without a proof. Lemma F(x) and pdf defined and of x =~ p f(x). 4. Let a continuous population be eiven with cdf Suppose that for 0 < p < q < 1 , ~ p and 1;. q are uniquely t f (x), the derivative, exists and is continuous in some neighborhoods and x =~ • q If I-L = r- np- 7 + 1 and \l" r- nq- 7 + 1, ( we assume that -7np ancJ nq are not integers ) , anc1 statistics in a sample of size x and !J. n, then as x are the corresponding order \I n --> 00, - x X \I ally normal distrihution with mean ~q - ~p !J. has an asymptotic- and variance 2p ( 1 .. q ) (22) f(~p) f(~q) As a consequence of the previous lemmas, we have Theorem. and a pdf satisfy the continuity conditions stated in Lemma k and , let by (8) and (9). a sample of size I If x x .. x s r n , then both Lim (x n -> 00 and X S1 l , and be defined Sl r-th etc. order statistics in etc. are respectively the r cdf 4. For given nand k be the integers defined by (19), and r, s, r ~q" ~p in the sense that of Let a continuous population be given whose .. xrl are consistent estimates Lim (x .. x ) s .. x r )::: n-->ooSI r' = ~q - ~p in probability. Proof. PI and p( x q2' and 51 are properly chosen and - x > sr.. ~ 6' ) ~ e. 4. Following Lemmas 3 and 4, for given 6, e > 0 , i f n is sufficiently large, then ~ • ~ + 6 ) < p( x .. x > ~ .. ~ + 6 ) < p( x .. x > ~ - ~ q p ::: s2 rI q P ::: s2 r1 q2 PI In a similar way we easily complete the proof. Inference. In section 3 we proved among other things the existence of k and kt ~ defined by (19), for sufficiently large k and k' , there is no difficulty either. be evaluated by means of ~5..7, (23) I where P (r, s ) = k As is well known, binomial J J 0 Therefore for a given k' , defined by (19) and the corresponding by (8) and (9). cdfs can through the relationship 0 and To actually find such B ( r - I , p ) = I - I ( r , n .. r + 1 ) , n p 00 p r r x - I ( I .. x ) s - ldx x - I ( I - x ) s - ldxl is the incomplete beta function. find n. Of course, for a small a and n, we can easily r, s , r a, in order that such k l , and and s' defined kl exist, -8~ar_g;p.!.;, n hos to be . ', .. ;.... ~ ~. Confidence ;.n,tervals-. . ..;'. _. When "~'1~'f,'" cribed, x s '" xr bounds for ( :rSf ... and r sf - X;:r' ~q'" ~p xr ! , X rnd arechosen in the way previously des- Sl are- respectively confidence upper and lower with the same confidence coefficient ]. .. cdf and pdf, then both x estimates of and s - X'r ) is a confidence interval with confidence coefficient 1 - 2a. 4, If, in addition, the continuity conditions, dtated in Lemma the parent em , ~q - ~p .. x s and x r - x. s r r I are satisfied by are consistent • B. Tests of hypotheses. Let (24) Then the tests, using as critical regions: ,• -x <d • % s r ' X's - xr < d or X r - X r • > d , are respectively: 1. of significance levels a, a, 2a ; and 2. consistent with s , respect to the alternatives ~ q .. ~ p < d; ~ q .. ~ p > d ; and ~ q .. ~ p provided that the continuity conditions in Lemma 4 are satisfied. I d , A test, for testing a given hypothesis H0 , is said to be consistent with respect to a certain . alternative, if its power, when the alternative is true, tends to 1 as sample size tends to infinity. C. A special case: q =1 - p 0 If, in par'c,icular, q = 1 -p, then, fC1110wing (8) and (9), r = L" ( n .. k for which (25) ) / 2_7 + 1, s = r + k , etc.. If we use only those k and k' (n.. k )/2 and (n - k r ) /2 are not integers, then s ~ n .. r + 1 , and Sl =n .. r l + 1 • From (14) and (15), it follows that L > 1 .. 2I ( n .. r + 1 , r) and . = q 1 L ~ 2Iq< n .. r t + 1 , r' ) - 1. For a given n, we use [5_7 to find the largest r and the smallest integer r' for which • -9.. I q ( n - r + 1 , r ) ~ 0./2 (26) , Iq ( n f Then the corresponding L and Lf ~ 1 - a. • For example: i f n = ,30 in this way. largest r ~~~~75 - ~:25 ) ~ .95 • ~ .95 • ~8 - ~ ) t and q :: satisfying the first inequality in (26). P( x 28 - Xj r + 1 r I l ) > 1 - 0./2 • Table .3 in section 5 is obtained = .05 , a I - .75 , then .3 From (25), s is the = 28. . '!hus - Likewise, p( ~7 - x l 4 ~ ~. 75 - ~ .25 ~ For large n, a fairly good approximation for Bn(r,p) case, the largest integer 1.( ( r 7 mere !.(x) is given by (17) • In this f for which and smallest integer holds are + 1/2 - np )/ rnp( 1 - P ) the largest integer r r is 1/2 ) (26) r and smallest r l satisfying r r r < 1/2 + np - aa- np ( 1 - P ) ..71/ 2 , r' >= 1/2 + np ... aa- np( 1 - P ) - 71/~ 2' where !( -aa ) ... a/2. Table 2 in section 5 is computed _ in this way. We see that, (27) ¥ with the exception of a few cases, r' - 2np ... 1 - r • 5. The standard deviation of a normal distribution. Suppose that the parent distribution is normal with mean and variance (J 2 Then, for each p, 0 < p < 1 , and q • (28) where a =p ~(-a) = ( ~q =1 ~ - p , - ~p )/ 2a , and I(x) is given by (17). Obviously, as was pointed out in section 2 , confidence intervals for, and tests of hypotheses about a, can be readily obtained from those concerning ~ q - ~ p • A new problem arises, however, as to how to choose p to maximize the efficiencies of the corresponding wr and wrt in (,3), since they, for given n and a, are determined by p. We shall now consider this problem. A f~~iliar statistic used to estimate and test n _ 1 )1/2 r( n...;..l ) 2 ~ S , r( n/2 ) n 2 n 2 S = 1: (xi - i ) / ( n - 1 ) and i = 1: (J is f S = ( where i ciency of w r =1 i xi/n • The relative effi- =1 (and similarly w ' ) with respect to Sf is defined to be r ·10- ()o) The variance of [2, p. 484.7. I 100 var S t E1: - var [w (J!E(w ) 7 • r is easy to compute and for large n St The variances of wI' and r wI''' is about for arbitrary n, (J 2 I 2n and I' , are, on the contrary, difficult to obtain, ( Recently general formulas r' have 213 7 ), consequently so is the value of e sample size is large, and ( r . 1 ) I n and rln are fairly close to been derived by Ruben 1rJhen the ;-6, P. E(Wr ) , each other, we suggest the follOWing method of approximation for obtaining var w ,etc.. r In section 3, Lemma and ~,= ~np'_7 1 and + VI is asymptotically normal. 4, 0 < pI < ql < 1 , it was stated that if = [nq' ..7 + 1 , then the distribution of x For large samples, the mean and V I - X IJ. I variance of the asymp- totic distribution may be used as approximations to the true mean and variance of x , - X I . Now if p' and v IJ. q I "" 1 - p' , then x ... x, the denominator of ()o) (x , .. x )/2a l !J.' ~(-a') ... pl. where at2 e is ,follouing ~(a') If var S' Lermna 4, 2 ¢(x) ... (2n)-1/2 e- x /2 , / pI ( 1 _ 2pl ) , where nwnerical investigations, e.g., w I' and at pI .... 07 {4.7 , approximately. is maximized i f p a:, this value of to is replaced by ,i/2n, which, in fact, is the asymptotic efficiency of .65 and and x 1 = X I • Therefore am approximation n-r+ v is the variance of the asymptotic distribution of then an approximation to (31) 1 ) / n < p r < I' /n (I' - IJ. I' v' are such that qI (x I - X , ) v !J. revealed that (31) has Therefore, for large is so chosen that p r/n n is about 2a l / a Previous • ms.."Cimum value , the efficiency of .01n • For given can be found approximately by equating to .01n n the RHS of the first inequality in (27) and then solving a quadratic equation in p In a similar way we can find the optimal p with respect to wrl • Obviously, values of p which maximize the efficiencies larger than Houever, as of the corresponding .01, and those maxir.lize the efficiencies sample size increases, both r/n and W I' 's are of the 'tvrl's , smaller. rl/n tend to p (Lemma 3 ). -11Therefore eventually the efficiencies of w and wrt are maxilrlized if P'" .01. r In the following, Table 1 gives, for different values of p, the asymptotic efficiencies . (31) of (x\I .. x_~ )/ 2a \I ... (nq.7 + 1 , q ... 1 .. P imum value J = p. and ::I(-a) .65 occurs at P'" .01. J where ~ = ~np - 7 + 1 J As was mentioned before, the max- In the neighborhood of P" .01 , the variation is small of the corresponding efficiency. In Table 2 are given, fer different choices of n and a, the values of p, and corresponding to each p, and r t the subscripts r of w r and w r They are not found, however, by t. the method of solving a quadratic equation mentioned in the previous paragraph. Instead, four different values of p are chosen , then for each p, the corresponding r ani r t are found by (21). For most combinations of n and a given in that table, at least one of the four with an efficiency fairly close to p( ( x 64 .. x ) / ~.56~;;; (] ) ~ 31 4 statistic is about .65. .65. :9S, ( p p's will provide a wr For example, if n" 500, then ....10 ) , and the efficiency of the For samples of moderate sizes, Cadwell ~1, p. 609.7 obtained recently the efficiencies e (0), for n .. 10, 20, ••• , 60 , and r :;;z 1, 2, 3 • Using his results, we see ( Tables 3 and 4 ) that if p = .25 ,then the corresponding w 's have efficiencies around •10 for n r Table 1. Aymptotic efficiency of = 20, ( x \I ..x )/ 2a IJ. P .01 .02 .03 .04 .05 .06 .01 .AE .39 .51 .58 .62 .64 .65 .65 p .08 .09 .10 .ll .12 .13 .14 .AE .65 .64 .63 .62 .61 .59 .51 30• 12 Table 2 The Largest r and smallest r' p 2a 1000 500 .10 .25 n ""'a = .04, .07, .10, and .13 = 3.50, 2.95, 2.56, and 2.25 .01 .025 .05 .005 33 48 30 51 28 53 26 55 24 57 23 58 61 80 57 84 54 87 52 89 49 92 47 94 89 112 84 117 81 120 79· 122 76 125 73 128 118 143 113 148 109 152 106 155 103 158 100 161 15 26 13 28 11 30 10 31 9 32 8 33 28 43 26 45 24 47 22 49 20 51 19 52 42 59 39 62 37 64 35 66 33 68 31 70 56 75 53 78 50 81 48 83 46 85 44 87 5 12 3 14 2 15 1 16 0 17 10 19 8 21 '7 22 6 23 5 24 4 25 15 26 13 28 12 29 10 31 9 32 8 33 21 32 18 35 17 36 15 38 14 39 13 40 2 7 1 8 0 9 0 9 0 10 0 11 4 11 3 12 2 13 1 14 0 15 0 15 7 14 5 16 4 17 3 18 2 19 2 19 9 18 7 20 6 21 5 22 4 23 3 24 200 100 n = Sample size 3 14 a = Level of significance 13 Table 3. The Largest r and Smallest r' = .25 p n~Q 2a .10 .25 = 1.35 .01 .025 .05 .005 10 1 5 20 3 8 2 9 30 5 11 4 13 3 13 3 14 2 15 2 40 7 14 6 16 5 17 4 17 4 18 3 19 50 9 17 8 19 7 20 6 21 5 22 5 23 Table 4. 2 10 1 1 Efficiencies of the Statistics (Xn _r+l - xr ) / 2a in Table 3 n~a. .25 .10 .05 .025 .01 .005 10 .85 * * * * 20 .66 .73 .73 .70 .70 * * 30 * * .70 .70 .70 .70 40 * * * * * .69 -146. The parameter of a binomia~istribution. We shall..consider a practical example. To classify lots of, say, factory products, according to the proportion e of defectives contained therein, the following way might be of It practical interest. ~ > q , a prescribed quantity, then the lot will be rejected for containing too many defectives. If p ~ e ~ q , where p < q is another given quantity, then the lot will be accepted as of average quality. if Q < p, then the lot will be considereCl as superior ( and probably will be sold at a higher price). i.e., But Q < P ; p < While the problem, to decide between three alternatives, = Q =< q ; and Q > q , is actually a 3-c'ecision one, we may , perhaps as a preliminary approach , start with testing the hypothesis p < < q. =Q = H: (34) o Corresponding to a random observation taken from a given lot, let x =1 is found defective, and x o, 1 , where f(x) if good. . =0 and S ~ =q1 . in a sample of size Let xr xs and n, then lity of E given that H 3, a pair of integers r = • be the r-th and s-th order statistics p(x s -xr:: >~ -~ IH )=p(x =l,x =0 q pas r is true. o and s p( E I H ) , where 0 ) denotes the probabi- Following the general method given in section can be determined, (8) and (19) , for given n such that or x => 1 r Therefore the test using as critical region (35) p( x (36) xs = 0 where r x is true, we may define, following (1) , 0 I Ho ) = 1 - p( x - 0 or x = 1 I H s. r . a and a Then f(x):: Q 1 - x ( 1 _ Q )x ; if it denotes the probability of obtaining an observation x For this distribution, whenever H ~ sp =0 and s s = 0 or x r => 1 H 0 ) < a • = , are given by (35) , is of level of significance a for testing We sho1"J'ed ( section 3, Lemma 3 ) that for arbitrary but -15- I1. fixed < p and q2 > q , it r~ ... [nP-r _7 + 1 in (35). =: (nq2 _7 + 1 , then, and 6 ~ 6 ' where r and s are given 2 1 Using (16) , we easily show that the test defined by (36) is consistent when n is sufficiently large, with respect to the alternative r ~ and s2 Q< r P or g > q • The hypothesis may be considered as a limitine:: case of (34) as To test H at a pre1 scribed level of significance a, we use critical region (36) with 6 s r + k , and r = [ ( n . k ) p_7 + 1 , where (38) q -> p. is the least integer for which k Bn ( s - 1 , P ) - Bn ( r - 1 , p ) > 1 - a • =: If, in particular, P'" 1/2 , then, from (25) and (26) , s =n - r + 1 and r is the largest integer for which B ( r - 1 , 1/2 ) ~ a/2. It should be noted that n the tests just suggested are not new ["7, p. 14_7. However, for testing HI when p f 1/2 , according to the knOlTn method, one chooses a pair of r and s which satisfy (38) and uses the corresponding region (36) as critical region; and if there are more than one pair of r be made and s satisfying (38), then selection should "in accordance with practical consideration" 1:7, p.l5.7. removes this 60mewhat ambiguous way of selection, since r , s uniquely determined for given n, p , and a. is consistent with respect to the alternative See {7.7, Our method and k are Further, the test thus obtained Q f p • for applications to interval estimations testing of hypotheses of quantiles of an unknown distribution. and -16References ["1_7 Cadwell, J.H .. ,"The distribution of quasi-ranges in samples from a normal population ", Ann. Math. Stat. , Vol. 24 ( 1953 ), pp. 603 - 613 • [2_7 Cr~r, H., Mathematical Methods of Statistics, Princeton University Press, 1946 .. ["3_7 r£vy, P. , [4_7 Calcul des probabilites, Paris • Mosteller, F., " On some useful" inefficient" statistics ", Ann. Math. -stat., Vol. 11 ( 1946 ) , pp. 311 - 408 • [5_7 Pearson, K., Tables of the Incomplete Beta Function, The Biometrika Office, University College, London, 1934. [6_7 Ruben, H ., On II the moments of order statistics in samples from normal populations ", [7_7 Wilks, S.S., II Biometr!~, Order statistics It, ( 1948 ) , pp. 6 - 50. Vol. 41 ( 1954 ) , pp. 200 - 221. Bull .. A.rner. Hath. Soc. , Vol. 54
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