／ TRU Mathematics 21−1 〔1985〕
ARE THE OPTIMUM SPAC INGS SYMMETRIC ？
IN THE CASE OF NORMAL DISTRIBUTION FOR SMALL SAMPLES
Kyo MIYAKAWA， Takanori TAMIYA and Kouichi KOTANI 〔Received May 10， 1985） Abstract This paper deals with the small sa∬庄）1e eStimation in the nol皿al dis’tribu− tion． It is reasonable to consider the three cases： （1〕 to 6stimate the mean μ when the standard deviatiohσis㎞oぬ，〔2〕σ when”v is㎞ぽm， and〔3）both 1」andσ・ The esti皿ation is made by the use of．the bestl linear unbiased esti− mator based on the case of selection of I（ordered samples fr㎝ndata （n≧ k〕． The optimurn spacings of the k ordered sanrples togetber with the coefficients to be used in computing the estimates bfμ andσ are obtai皿ed in a number of tables fbr k＝2〔1）10 atid n＝．k（1〕30． It is found that the sy㎜6tries of the optimum spacings are bro1（en．for various values of n and’k in CasC （1〕 and （3）， and for odd l（except k ＝ n．in CaSe （2），曲ereas they hold for even k irrespec− tively of n in Case．（2）． 1． Introduction C°nside「n°「de「ed s卿1es X〔・）≦X（2〕≦…≦X〔。）dra岨f・㎝・h・p・pu− 1・・i・nwh・・e c・nti皿・u・．P・・b・bi・ity d・n・i・y fm・ti・n〔Ptlf）ha・th。 f。m÷・（18’〕，曲ere脚dσare th…ca…一・Sca・・pa蜘・ters re・pecti・。・yl
and f（u〕is the standardized pdf． Selecting k（≦n）samples out of th㎝， we w「’te X〔・、〕≦X（・、）≦’”≦X〔・、）・曲e「e npn・・’°’・n・a「e’ntege「s and・・y・t satisfy th・f・11・wi・g・rder ．c。・diti・n・・．≦・、・n2・…＜nk≦・・．P…id・d th・v・1ues n1・・2・…・・k are gi・・n・w・m・y・…ider th・bρ・t linear．unbiased estimators 〔BLUEs ） of the parameters p and σbased on the corresponding k samples． And we may raise their efficiencies by choosing the values suitably． The set〔・1・・2・…・・k〕f・r曲i・h the eff‡・i・n・y・f・n estim・t・r att・i・・t・it・ maXimUm Va1Ue iS Called an Optin1Um SpaCing． 、 a ． ．The prob1㎝of detemining the optimuln spacings has been studied by a lot of authors． For large samples， Mosteller［111 proved the theor㎝that the limiting distribUtion of sa町）1e・（luantiles becomes a normal distribution・ On the bas：is of this theorem， Ogawa ［12】、，［13］，［15］，［16］，［17］ obtained the BLUEミ． 161162 K．MIYAKAWA， T． TAMIYA AND K．1（OTANI of the’垂≠窒≠hneters and was able to detemine the optiJnuln spacings in the cases of nomal and exponential populations． According to Ogawa’s theory， Kulldorff ［6］ and S…irnda1 ［18］． obtained the optimum spacings for extreem value and Gamna distributions respectively． Kulldorff［5］found the same Optimum spaci㎎s as Ogawa，s applying the concept of optimunl groupings to the exponential popula− tion． On the other hand， for small samples， Lloyd ［7］ found the BLUEs of the parameters based on order statistics． On the basis of them， Ishil（awa ［4］ and Miyakawa and Kotani ［8］ obtained the optimum spacings for the normal distribu− ・tion under the、condition．that the spacings are s）㎜etric． One’of the−important subj ects in the problem of the opti㎜spacings is whether the optirnurn spaci㎎s become s）㎜etric or not when the・standardized pdf is s）㎜etric． with respect to the origin． It is because the BLUEs of l．t andσ have beautifUIl and natural shapes such as linear co皿正）inations of quas i−cen− ters and quasi−ranges respectively when the spacings are symmetric． ・Speaking to the nomal distribution for large sa㎎1es， the opt迦um spacings were shown to be s）㎜etric except for the cases of estimatingσ whenμ is ㎞own and k is odd 〔Ogawa ［16］，［17］，Higuchi ［3］ and Miyalくawa and Kotani［9］ 〕． But we don’t know whether the same thing can be said to be true for small samples． The purpose of this paper is to investigate the symnetries of the optiJTIuM spacings for small samples of the normal distribution， and to put to practical use giving the coefficients of the BLUEs corresponding to the optiJnutn spac− ings． The section 20f this paper will deal with the BLUEs of the parameters based on the k ordered sa呵）les selected fr㎝the n Samples for small sal1iples・ Section 3 will be concemed with the efficiencies and the relative efficien− cies of the BLUEs． Section 4 will be devoted to the dualities and the sy㎜｝e− tries of the spacings in the case of symmetric distributions． Section 5 will give the numerical results about the nomal distribution and compare them with the ones for large samples． And the last section will state various defini− tions of optim皿spacings and take a new look at them fr㎝the pQint of view of the point prediction theory． 2． The best linear unbiased estimators As stated’n sect’°n 1・we c°ns’de「k°「de「ed samPles X〔・、）・X（・、〕・’”・
蹴∵慧1〕㌘1惣：罵㌘《監
regarded as k out of n ordered samples taken fr㎝the population with the standardized pdf f〔u） ぬich is parameter−free． We use the vecter notationsARE IHE． OPTIMUM SPACINGS SYMMETRI C ？ U’＝（U（・、）・U〔・、〕・’”・U〔・k〕〕and ・X’＝（X scriPt is used to denote the’t士ahsposition． t「ix°f旦’a「e den°ted by y’；（un、・un、・’”・un 〔・、）・X（・、〕・’一・’・X〔・k〕）・ whe「e・the supe「− The mean“vecter and variance ma一 、〕and W＝（Wn、・j〕・ K11＝1’Ω2・ K22＝旦’迦・ K12＝ystu−・ ・・K、、K22−・、22， where l denotes a k−dimensiona1 colunm vector of ones and 9 is trix of W． We discuss the following three cases． Case i Estimation of p whenσ＝σo 〔σ0 1s given） Case 2 Estimation ofσwhen l」＝μo 〔1」0 1s given） Case 3 Joint estimation of u andσ ハ Let i：lO，60 and θ〆』．〔fi，θ） be the BLUEs denote the variances of口o，∂o and the “ A V〔θ〕 respectively． rameters and their variances （ or Case 1 60 ＝ a’X−aoσo， 2 v〔fi・）＝ili、・ ガ
・here ・’・Kl、エΩand…lli・
Case 2 60 ＝ b’X−boμo， 2 V〔θ・〕＝Ki、・ ・here V・S、口d b・−lli・
ハ Case 3 θ’＝ 〔｛｝，6〕＝ 〔c’X，d’X〕， ＾ 2 V〔θ〕＝㌃IK22K・2・ ［K、2K、、l l，il・e・e 9’・ k（K221’・一・、2・’・〕・n・旦’・去（・、、・’・一・、2王’・）・ And we’垂浮 〔2．1〕 （2．2〕 〔2．3〕 〔2．4） the inverse ma一 in Case 1，2 and 3 respectively． And we A varla血ce matrix of θ トy V〔APO〕，V〔∂o） and According to the theory of Lloyd ［7］， the BLUEs of the pa− variance Inatrix ） are as follows． 〔2．5） 〔2．6〕 〔2．7〕 〔2．8） （2．9〕 （2．10〕 〔2．11） （2．12） （2．13） 3． The efficiencies and the relative efficiencies First we consider Fisher’s amounts of infomation〔or information matrix163
164
K．MIYAKAWAプT． TAMIYA・AN］D K： 1（σrANI 〕 derived from the original whole sainple， and denote them「in Case 1， 2 and 3 by I o（μ），Io（σ〕and．1（e） ’respectively． SirppOse． tha；f〔u）is differentiable a1− most everywhere and satisfies． the following condition， 1im uf’〔u） ＝ 1im uf’〔u） ＝ 0．． u→ ＋oo u→』−co Then we have （see Ogawa ［16］ 〕…e…（・〕・σ1、E圏2・ ．． 〔3・1）
Case 2 Case 3 ・・〔・・?E圏2−…
1（θ〕＝2
σ2・團2
−・o警〕
一E ・〔U｝’〕2−・ ● （3．2〕 （3．3） FQr exa叩1e， in the case．of the ponnal distributi㎝・Case 1ヒ16〔V）・．n ． ’ （3・4〕
2 ． σOC。，e2 ・。〔。）・≧ ． 』 〔3．』5）
’ 2 σCase
@3・ω＝言｛川 ・ （3・6）q
N（》wit is we11㎞om that the efficiencyηof an estmator in the tWo one−parameter cases is defined by the ratio of the imrerse of the amomt of infbrmation tq the variance of the estimator． Therefore， we have Case 1 n（fio〕＝ 1 ＾ ， Io〔1．1）V（Po） （3．7）C。，e2 ，〔；。〕・ 1＾． （3．8）
Io（σ）V（σo） ． ． t／ In the two−parameter case， we define the efficl〔mcy of j oint estimators by th・rati・・f th・・quare・f the area・f th6・11ipse d・termi・・d by th・ ・qgati・・ バ ベ （θ一θ）’1（θ）（θ一θ〕＝4 ・ to the square of the area of the’ellipse“・of concentration of the esti皿ato「s・ Thus． we have ’ ‘ ． ・ARE THE OPTIMUM SPACINGS SYMMETRI C ？
Case 3 ・◎・ 1＾， （3．9〕
ll〔θ）lIV〔θ）lwhere the s＞mbo1川denotes a deteminant〔see Og鎚［14］〕． Substituting
the fb㎜1as（3．1〕，（3．2〕孤d〔3．3〕for these efficiencies， we haveCase・，〔fi。〕． K・・’， （，．、。）
祀巳〕2Case・，（G。）． K22， 〔，．、、）
等’〕2−・・ …｛C・・e3 n⑨・ △ ． （3．12〕
n2［・闇2司U￡〕2−・｝一・・〔Uii2〕・・］ Fr㎝（3．4），（3．5〕and（3．6〕，in the case of the normal distribution， we have…e・，（fi。〕・K…・，1 （・．・3）
Case・，（ε。）．K22，． 〔，．、4）
2nC・・e3 ・（9）・△． 〔3．・5）
2n2 The formulas given above are analogous to the ones foT large samples due to Ogawa ［12］，［13］ alld ［16］． 、 e Now， if we want to㎞㎝how much infonnation we lose in the est血ation of the pa「aJTI・ter・by th・u・e・f th・BL皿・b・・ed・n th・k・rd・・ed・amples select・d from the n data instead of the BLUEs based on the original whole sample， we ㎜st c°nside「the「e1・tive effi・i・n・yη。 d・fi・・d・・f・11・w・・1・th・tw・・ne− parameter cases，ηr of the former is defined as the ratio of the variance of the latter to the variance of the former・ In the two−parameter case， it is defined as the ratio of the square of the area of the ellipse of concentration of the latter to that of the former． Hence， we have へ…e・・。〔fi・）・V〔μ・F’））， 〔・．・6〕
V〔1．t。〕 パ…e・・。（〈σ0）・V〔σ・S’〕〕， （・．・7）
V〔σo〕 165166
K．MIYAI（AWA， T． TAMIYA AND K． KOTANI Case 3 ハ 〈 IV〔θ〔n）〕1 ηr〔θ〕＝ Iv（6〕「 ， 〔3．18〕 where fi o〔n），∂o（n〕 and θ〔n〕 are BLUEs based on the original whole sample and ム are given by putting k＝n in（i o，δo and旦of Case 1，2． and 3 respectively． Then， it follows that…e・，。〔旬一K且， （・．・9）
K11（n〕 ・…e・，。〔・。〕・K・・， ・・．…
K22（n〕．…e…◎・△：）・ 〔3・21・
・h・・eKii（・）・・d△〔・）are Kii ・nd△f・r k＝n re・pectiv・1y・It i・ ea・ily・h・wn that the efficiency and the relative efficiency for the same BLUE satisfy the relation： o≦n≦nr≦1・ Especially in the case of、the noτmal distribution， it follows that Case 1 Case 2 nr（fio） nr〔60〕＝ K ll n ， K 22 K22（n〕 ， ・・〔A ﾆ）＝ u、i〔。）・ 〔3．22） Case 3 （3．23〕 〔3．24〕 砲ere we u・e th・r・1・ti。・・K11（・〕＝n ・nd K12〔・〕：°d・・t・L1・yd［7］・lt び ， should be noted that in this case bothμo〔n） andμ〔n〕 in Case l and 3 respec− tively reduce to the sample means， and the efficiency and the relative effi一 ム ム ciency of the BLUE p e in Case l are equivalent， that is， n〔μo） ＝ nr〔Vo〕． 4．The dualities and symmetries of the spacings F・r ・ny・gi・・n・paci・g・1・・2・…・nk・・ne c・n・1w・ys c・田truct it・dua1 ・paci・g・f・・S・…・・1・where n；−n・1−nk−i．1〔i＝1・2・…・k）・A・elf−dual spacing・ i・・…窒・ni（i・1・2・…・k〕・r ni・］・k．i．1＝n＋1 i・ca11・d・・）m・et「ic spacing・ W・ ・a・ily・bt・i・th・f・11・wing re1・ti・n・wh・n f〔u）is s＞㎜・tri・・L・t nf・167
ARE THE OPTIMUM SPACINGS SYmaT［RIC ？ n》・…・ni．b・th・・桓・1・pacing f・r a giv・n・p・とing n Kss〔・1・・2・…・・k） K12〔n1・n2・…・nk〕 △〔・p・2・∵’・・k） ＝KSS〔nl・n；・’・’・喋） ；−j12〔n至・n覆・…・喋）・ ＝△〔nf・n》・…・nl〕・ 1，n2，°°6，nk， then we have 〔s＝1，2〕， Us ing these relations， we can prove easily the following． ⑳ The・r㎝S・PP…th・t th・ BLUEsαnd磁・ffi・tenci・・tn a・yne・etri・dis− f」ribution ave gZτ戊ργ2α8 r2．5ノ⊃r2．8／3r2．ZZ） and r3．ZO）3r3．ZZノ，r3．Z2J resρeetive− z〃・Th・緬θ8ρ・・w・ヱ・・2・…・％・・脇・duaZ・ρ・・z・g ・1・nS・…」嘆』吻 s・励θefficiencyαnd reZatiVe effieien・y． And拗B五〃Es・f the輪Z spα・W nS・n』・…・・2鍾θ勿・・、α・f・zz・w・・ ＾ k 伽θヱ1・t°㍉三、a・X・・i．、．、ノ≠α゜σ゜『・ ． 「4”1） ＾ kCa＄e 2σ゜＝−i三、b♂・』、ノ鋤μ゜・ 「4’2ノ
・ ＾ ヒ＾＾ k k °αsθ3♀＝「“・σノ＝㌦三、°iX・ni．、．ゾ、三、醐・え．、．プ・「4・3ノ tohere 一 ．τ ノ 1 ≠ 鰺k ．η r＝x
（n＋ヱーn．ノ ￠ r乞＝ヱ⊃23●・●，k）． Next we consider the symmetry of the spacing that we will investigate in the p「esent・pape「・tS・um・th・t f（・〕is s）㎜・t・i・and th・・paci・g・1・・2・…・nk is s）mmetric・ Then it can be easily shown that the corresponding BLUEs ofμ and ° a「e耐tten as 1’nea「c°mb’nat’°ns°f quas’−cente「s（X（…−n、）＋X〔・、））／2 and quas’一「anges X〔…−n、）−X〔・、〕「espect’vely・Such f°「ns°f these est’mat°「・ are very desirable， takihg account of the fact that the quasi−centers and quas i−ranges have often been used in the estimation of the mean and standard deviation respectively 〔Dixon ［ll，Harter ［2］ and Mosteller ［11］ ）． Accord− ingly， it is worth investigating whether the optimum spacings are s）rmmetric． But it is too obvious that there i’s only one spacing that is s）mmetric and oP− timulll， whenever k＝n． 5． The opti皿um spacings for the nolmal distribution The detailed tables with respect to the m㎝ents of order statistics from the nonna1．population are published ［20］． Feeding them into a computer， the168
K．MIYAKAWA， T． TAMIYA AND K〔KOTANI authors detennined the optimnn spacings fOr k≦10 and．n＝k（1）30． H㎝everジ the tables became so manyうwe decidedηot to present in this paper． The per− s㎝swho are interested in seeing these tables may address a letter to the au− thors． ’ The numerical results are romded to five deci皿al Place． The final rowS denoted by n＝。。 in the tables i血dieate the values’fbr large sal町）1es that are given first by Ogawa ［12］，［13］，【16］ alld are supplemented after．that by M‡yakawa and Kotani ［9］． If the opti皿1m spacing fOr some set （1（，n〕 is．as）㎜e− tric， we marl（the left−most of the correspOnding row of the tables with★・ The．dual spacing fbr such an as）㎜etriごopti皿】m spacing， which attains to the same maxtm efficiency as this， is not listed in the tables owing to li皿ited space・ It is fbund that in Case 2 the opti蜘puM spacings are s）㎜etric fbr even k irrespectively of n， and asy㎜1etric fOr odd］（regardless of n except k＝n・ Hence， speaking to the s）㎜etries of the optimum spacings，1arge samples or small sanrples， they are all the salne so far as Case 2 is concemed． However， it apPears that the symnetries of the．optimu皿spacings are broken for various values of n and k in Case l and 3． Therefore， in these two cases， we can con− clude that the symmetries in question are．broken fbr small sa珂Ples， whereas they hold fbr large samples． It is also found that the asymptotically BI．UEs with the asymptotically opti皿um spacings given by Ogawa ［16］ are useless as the apProxi皿ate estimators of the BLUEs with the optimum spacings， when the sqlTEF）1e size n is at least less than several tens． However， the values of the ・・Y・rP・・tica・・y・p・i・nmP・paci・g・n：・n；・…・nlk i・・h・…t・・w・・f・h・t・b… are usefUI to estimate roughly the． neighborhood of the opt㎞m spacings for small sampユes． 6． ．Some comnents First， we wish to touch upon the relations among’the asymptotic efficien− cies，「the．asymptotically relative efficiencies and the Ogawa’s relative effi− ciencies［16］． It is reas㎝司）1e that the as）叩ptotic efficiencies and the as− yml）totically relative efficiencies in the加o one−parameter cases are defined as the ratios of the inverses of the amomts of infdrmation derived from the distribution of the original n sa明）1es and the variances of the BLUES based On the n samples respectively to the variances of the BI、UEs based on the k sample quantiles selected from the n san唖）1es．， taking account of the definitions of the efficiencies and the relative efficiencies in small samples． The Ogawa，s relative・efficiencies are defined as the ratios of． the alnomts of infomatiσnARE THE OPTIMUM SPACING］S SYMMETRI C ？ deriv・d f・。m th・1i・niti・g di・t・ibuti・n・f・．th・k・ampl・qu・ntile。 t。 th。se d。． 「ived f・㎝th・di・t・ibuti・n・f・11 th・n・瓠Ples． H・w・v・r， f・r large sa。。1。， of the normal distribution， the BLU雀…s based on the k sanrple quantiles are ef− ficient in a sense that the variances of these BLUEs are equal to the inverses of the amounts of infOrmation derived from the limiting distributions stated above． Therefore， replacing the latter with the fomer in the Ogawa’s rela− tive efficiencies， it tums out that the Ogawa．s Telative efficiencies are not the asy町）totically relative efficiencies but the asymptotic efficiencies． The ・㎝・thing・an・b・ ・t・t・d・b・・t th・加・−par・m・ter case． B・t・・peci・11y． @in Case lof the nomal distribution， the Ogawa’s relative ef壬iciencies are also the の ム as）㎎）totlcally relative efficiencies， since the BLUE μo〔n） is efficient in a パsense that V〔μo〔n〕）・＝1／lo（P）． Next we wish to mention the definitions of the optimum spacings． There is no question aboUt the definitions of the opti皿um spacings in the two one− parameter cas．es． Hσwever， in the tiYo−parameter case， it is rathqr doubtful whether the spacings、，maximizing the efficiencies of the j oint esti皿ators 〔 in other words， minimizing their areas of the ellips6s of concentration） can be leterally optimum． If there is a spacing whose ellipse of concentration is included by the ones of any other spacings， Surely it can be said to． be opti− mum． But unfortunately there is no such a spacing in genera1． Therefore we can not help cons idering suitable criteria corresponding to・the circumstances under consideration ill order to decide what spacings can be said to be optimum ． For example， we can consider t｝ie following spacings to b60ptimum as the criteria．except the one that we a（lopted． ・ ‘ 「 A （1°） 〔2°〕 Especially the fomer is cal prediction． point predictor on a future sample Y f the k samples Aspacing minimizing the varlance ofμ （ In this case， we皿ake the accUracy in the estimation ofμ impor− tant， and neglect the one in the esti皿ation ofσ． 〕 Aspacing minimizing the variance of 6 （This case is contrary to （10〕． 〕 ユmportant ln relation to the theory of statisti− On this we wish to offer a few words． Suppose we wi1］． find a or the situation wh6re we have observed X’＝〔X〔・、〕・X〔・、〕・’”・X〔・、〕〕Wh’ch a「e def’n・d’n sect’・n 1 and indep・nd・nt・f Y・A・d・upP・se w・mea・ure th・g。・dn・・r・f皿鋤iased predic− tor by its variance og prediction err6r． Then to find an unbiased predictor mln1皿1zlng lts variance of predictiOn error is reduCed to・find an unbiased es＿ timatqr of the meaJlμminimizing its variance， wh⇔n Y an（11are’statistically independent 〔Takeuchi ［19］）．’Accordingly， the best linear unbiased predic一
169
170 K．MIYAKAWA， T．「TAMIYA AND K．・KOTANI A tor〔．BLUP．）of Y・is equal toゼhe BLUE U of U which ls giVen in Case 30f sec− tion’2． Therefore if we・find spacings mini皿izing the variances of｛1 （ that is ・K22／△〕．・・d・rdi・g t・the criteri・n・f（1°）「， we c・n・m・k・use・f th㎝・・th・ opt．imum spacings of the BLUP’．of Y in the．theo．ry of point prediction． The de− tailed results「dn this will be t）Ublished’receritly ’ bソthe preseht authors ［10］． ACKNOWLEDG［噸TS ’ The authors would like．to e）q）ress tiheir gratitude to Miss Narabayashi， Mr． Kaneko， M士． Kimura， Mr． Shinoda and M主． Mfurakarni for 亡heif valuable as− sistance on the numerical calculations． ・ ［1］ ［21 ［3］ ［4］ ［5］ ［6］ ［7］ ［8］ ［9］ ］］
