( µ /σ)ζ/(ζ+1) µ /σ ( µ /σ)ζ/(ζ 1)

A eective CI for the mea with samples of size 1 ad Melaie Wall James Boe ad Richard Tweedie 1 Abstract It is couterituitive that with a sample of oly oe value from a ormal distributio oe ca costruct a ite codece iterval of ay size for the mea. It goes just as much agaist stadard teachig that from a sample of size two such a CI might be shorter tha that based o the t statistic. We ree a earlier versio of this rst result ad use it to prove the secod. For samples of three ad larger we show that the t-based iterval caot be improved usig this approach. A \partly-bayes" approach idicates that i some istaces with such small samples a quite reasoable codece iterval ca be costructed. keywords locatio estimatio Studet's t ivariat estimators codece itervals 1 Itroductio It has bee kow but ot well kow sice the 1960's that from a sigle observatio x from a ormal distributio with ukow mea ad ukow stadard deviatio it is possible to create a codece iterval (CI) for with ite legth [1 8 9 10]. This remarkable result seems to completely cotradict the stadard statistical ituitio that at least two observatios are ecessary i order to have some idea about variability. Noetheless it is a special case of a eve more surprisig result [ 4 11] that for ay (0; 1) there exists a ite 100(1 )% CI for the mea of ay uimodal distributio of the form x jxj: (1) Based o icreasigly strog assumptios about the uderlyig distributio (uimodal uimodalsymmetric or ormal) the value of decreases; ad for each of the three classes the relevat value of is solely a fuctio of the size 1 ad thus is appropriate for all distributios i that class. I this paper we discuss the case where the uderlyig distributio is ormal. We rst ree the results i [4] usig the methods of [] ad give a formula for that results i shorter codece itervals tha those i [4]. We the show that it is possible to use (1) with samples of size >1 ad that give a sample of size = from a ormal distributio this CI ca yield better results i some cases tha the usual Studet's t iterval. For > however the usual Studet's t iterval is always better. We the discuss itervals of the form x jx Aj () where A is some xed predetermied costat. These have the same coverage properties as do those i (1) but if (usig a prior assessmet of the likely locatio of ) we are able to pick A appropriately we will get shorter itervals from this approach. 1 Divisio of Biostatistics School of Public Health Uiversity of Miesota Mieapolis MN 55455. 1

We coclude with a discussio of the ituitive reasos behid these results ad i particular we discuss the apparet cotradictio with the kow optimality of the CI based o Studet's t. A codece iterval for = 1 Let X be a ormal radom variable with mea ad stadard deviatio. To derive (1) we wat to d a value such that P (X jxj X + jxj) 1 ; o matter what the values of ad. For such aiterval the coverage probability for >1is P (X jxj X + jxj) = 1 P (jx j jxj) X = 1 P X + = 1 1 +1 where is the stadard ormal CDF. Note that this depeds oly o the ratio. I order to set the miimum coverage probability equal to (1 ) eeds to be chose so that sup 1 +1 = : (3) We shall show that is possible to solve umerically for the value of that satises (3). Table 1 gives the exact values of associated with several values of assessed usig (7) below. Table 1: Values of for costructig 100(1 )% CIs for usig (1).0.15.10.05.01.4 3.3 4.84 9.68 48.39 So for example a 90% codece iterval for is give by x 4:84jxj. Note that this is arrower tha the rage x 5:84jxj give by the approximatio (4) from [4]. Prior to derivig the results i Table 1 we preset two simple closed-form approximatios for usig Figure 1 which displays the regio where the codece iterval fails to cover. Edelma [4] derives as a approximatio for the value ecessary to esure the rectagle formed with the dotted lies i Figure 1 has area less tha or equal to. This gives (1) +1 (4) where is the stadard ormal desity. Sice this area is always greater tha the area uder the ormal desity eclosed by the same upper ad lower limit this value for is typically rather coservative.

0.1 0. 0.3 0.4 ( µ /σ)ζ/(ζ+1) µ /σ ( µ /σ)ζ/(ζ 1) Figure 1: Regio uder the stadard ormal desity where the codece iterval (1) fails to cover. Rectagle formed by the dotted lie relates to (4) ad trapezoid formed by the bold lie relates to (5). Blachma ad Machol [] derive a approximatio usig the value of eeded so that the trapezoid formed by takig the rst order approximatio to the ormal desity at the poit (bold lie i Figure 1) has area less tha or equal to. This gives (1) : (5) The formula (5) is remarkably close to the exact umerical solutio to (3) for. Numerical methods idicate that for all <:3 it is withi.005 of the exact solutio. We ow idicate how to derive the exact value of which solves (3). For each y deote by h y the \least favorable value" which maximizes the area ( y y ) ( i. ) I the Appedix y1 y+1 followig the reasoig i [] we show that this least favorable value is = 1 1 s y y y y log 1 : (6) Thus the which solves (3) is the same as the which solves!! 1 +1 = ; (7) ad (7) ca easily be solved umerically usig ay software that ca calculate quatiles of the ormal distributio. The results are show for commo values of i Table 1. 3

3 Codece itervals for > 1 The codece iterval (1) was origially derived with the itetio of beig used whe there was oly oe observatio from a distributio. Of course (1) ca also be used to form codece itervals for whe we have more tha oe observatio. If x 1 ;x ;:::x are iid observatios from N(; ) the x = P x i=1 i= is distributed N(; ) ad therefore based o the results of the last sectio we ca form a 100(1 )% codece iterval for by takig x jxj (8) where is from Table 1. Note that the value of does ot deped o so that these itervals do ot get shorter as more data is collected. The use of (8) whe >1may be of practical use if it ca provide shorter codece itervals tha the usual codece iterval for formed with the sample stadard deviatio s ad the Studet's t-distributio i.e. s x t 1 p ; (9) where t 1 is the quatile associated with = from the Studet's t distributio with 1 degrees of freedom. To compare the margi of error of (8) versus (9) we compare E(j Xj) S toe(t1 p ). We derive a expressio for E(j Xj) i the Appedix for completeess ad a expressio for E(S) ca be foud i [3]. These lead to E(j Xj) =E(j Xj) = "!! 1 p r + # e (10) ad E(t 1 " S E(S). p )=t 1 p = t 1 p 1 r # 1 : (11) To see that it is plausible that the margi of error for (8) might be smaller tha that for (9) whe = compare the quatiles of Studet's t distributio give i Table to the values for give i Table 1 for some values of jxj ad s. We see that t 1 > for all although t < for all. Table : Quatile values of Studet's t-distributio with oe degree of freedom t 1.0.15.10.05.01 t 1 3.08 4.17 6.31 1.71 63.66 t 1.89.8.9 4.30 9.9 A careful compariso of E(j Xj) ad E(S) p shows that whe =there is ideed a regio of the parameter space where (10) is smaller tha (11). Figure shows a perspective plot of E(j Xj) ad E(t S 1 p ) as fuctios of ad for the case whe = ad = :05. I order to compare these surfaces we show their dierece over the parameter space i Figure 3 (left plot). The regio of iterest is that where the E(j Xj) S <E(t1 p ). The plot o the right of Figure 3 represets this regio i.e. where the expected margi of error of (9) is larger tha that of (8). We ca see that 4

0 50 100 150 00 50 0 50 100 150 00 50 30 30 5 0 15 σ 10 5 5 µ 0 5 10 5 0 15 10 σ 5 5 µ 0 5 10 Figure : Give = the height i each of these plots represets the expected margi of error for a 95% CI at a give ad. O the left is E(j Xj) ad o the right ise(t1 S p ) whe = ad is less tha approximately 1 (8) leads to a shorter (i expectatio) codece iterval tha (9). The amout of improvemet i accuracy icreases as sigma icreases. O the other had if >> 1 the average margi of error of (8) ca become much worse tha that of (9). To see this cosider what happes to the margi of error i Figure 3 (left plot) whe is held costat ad icreases. Outperformig the CI (9) usig iid ormal observatios might seem like a impossible task give the well kow ad widely taught optimality properties of (9). The key fact to recall is that the CI (9) is the Uiformly Most Accurate Equivariat codece iterval for the parameter ([7] p. 39). Note that the radom variable X jxj used i formig the codece iterval (8) is ot a pivotal quatity sice it depeds o both ad. Thus the CI x jxj is ot i the equivariat class of codece itervals ad so it is ideed possible that for some poits i the parameter space of ad (8) ca be more accurate tha (9). All of the above refers to the situatio =. Whe > the average margi of error for (8) ca be show to be uiformly larger tha the average margi of error for (9) ad so this o-equivariat method is ot practical for larger samples where the stadard methods are better. The followig propositio demostrates this result for the most useful rage of. Propositio: For > ad (0;:0] E(j Xj) >E(t1 S p ) (1) Proof: Note that mi E(j Xj) occurs whe =0. This ca be see by examiig the derivative of E(j Xj) with respect to ad observig that it oly ca equal zero whe =0. Thus E(j Xj) > r p >:797 p : 5

100 50 0 50 diff 30 5 0 15 10 5 σ 5 µ 0 5 10 σ 0 5 10 15 0 5 30 40 30 0 10 0 10 5 0 5 10 µ Figure 3: The height of the plot o the left is the dierece E(t 1 S p ) E(j Xj) based o a 95% CI with =ad the plot o the right is a cotour plot of the dierece show o the left but oly for the regio where (8) is more accurate tha (9) Now.h q 1 i 1 < 1 for all ad so for 3 E(t 1 S p ) <t 1 p t p : q (1) Hece we ca coclude (1) if we ca show that t <:797 for all (0;:0]. Now t = ([6] p. 11). If we use the umerically q demostrated fact that (5) is withi.005 from q the true (1) for (0;:0] it suces to show :797 ( (1) (1) :005). The quatity is mootoically icreasig sice the the rst derivative w.r.t is always positive whe (0;:0]. t Thus from Table 1 ad we coclude that max = 1:89 (0;:0] <:797 as required. :4 4 A partially Bayesia approach Cosider ow the more geeral codece iterval of the form x jx Aj (13) where A is some xed predetermied costat. I our developmet so far we have assumed A =0. Sice A is xed the coverage probability of (13) for is the same as that developed i Sectio ad thus the values for give i Table 1 will still apply. O the other had the choice of A will clearly aect the margi of error of the codece iterval. For the example we posed above whe = ad = :05 the regio i the parameter space where (13) would provide o average a smaller margi of error tha (9) is where jaj is less tha approximately 1. 6

Thus whe we ca use some form of prior iformatio cocerig the mea ad stadard deviatio oly (but ot ecessarily ay other prior distributioal properties) to choose A to be withi a half a stadard deviatio of we ca provide a better 95% codece iterval with = usig x jx Aj tha usig (9). There have bee a umber of approaches to elicitig prior iformatio o momets [5]. Give the full prior distributios o ad we also ca calculate the prior probability of evets such as for some A ad thus of the prior probability that (8) will be more accurate tha (9). jaj < 1 5 Coclusio Why does this totally implausible approach work? To try ad give a ituitio for this cosider a sigle observatio of x = 10. This could come from a ormal distributio with = 0 ad = 10 or from a ormal distributio with = 10 ad = 1. However \commo sese" suggest that it ca hardly come from = 1000 ad = 0. This reasoig idicates that some pairs are highly ulikely. Thus ay codece regio i (; )-space should exclude some pairs ad should ot be iite. The rather surprisig result eve give this is that there is margially a ite CI for which is correct o matter what the ukow value of. Agai however oe might argue that if x =10 the o matter what the value of there is little probability that will be 100000 or more. Thus we are ituitively workig i a \rectagle" i (; )-space ad the margial result of that calculatio leads to the ite CI for aloe. The proof above formalizes this argumet showig that without ay appeal to prior distributios we ca calculate the actual size of the ite CI. Whe is this sort of work useful? Clearly there is a good use for this example i the classroom. We teach that oe eeds a idea of variability i order to do estimatio: it is useful to hoe this ituitio with examples such as this to make us realize that x itself tells us somethig about the variability as well as the mea. We also teach that Studet's t leads to a uiformly most accurate CI. We do ot always metio ad certaily do ot always stress that this oly applies if we restrict ourselves to equivariat CIs. It is valuable to have a o-articial example that shows the eed for the equivariace i this statemet. Whether the approach is useful i practice is dicult to judge. The authors who largely developed this approach [ 8 9 10] are from NASA ad it appears plausible that i their work small samples of 1 or really do exist. There are may other experimets where such sample sizes also apply. I these cases usig (8) may well lead to greater accuracy of estimatio. Appedix The result of our rst lemma is essetially give i [] without proof but we give it here with proof because [] is ot trivial to read. h Lemma: Let a = ( i y y ) ( ). The the value of which maximizes a is y1 y+1 = 1 1 s y y y y log 1 : 7

Proof: Dee v @a. Settig @v yv y 1 v e v equal to zero ad solvig for v gives successively: y yv y y 1 h y y i y1 y+1 " y(y +1) y 1 y(y 1) # y 1 v 0 4y 3 (y 1) log v v y 1 y 1 log y 1 1 1 y s y y log 1 1 1 y py archcot y Our secod lemma is stadard but we give it for completeess. Lemma: Give X N(; ) Proof: E(jXj) = = = E(jXj) = 1 r + Z 1 jxj 1 Z 1 0 Z 1 X 1 1 p e (X) dx p e (X) dx Z 0 1 ( + U) p e U du 1 + = = 1 + p r 1 Z Z 1 e 1 e X p 1 e (X) dx 1 ( + U) p e U du e V dv p Z 1 e V dv Refereces [1] Abbot J.H. ad Roseblatt J. (1963). Two stage estimatio with oe observatio o the rst stage. Aals of the Istitute of Statistical Mathematics 14 9-35. [] Blachma N.M. ad Machol R.E. (1987). Codece itervals based o oe or more observatios. IEEE Trasactios o Iformatio Theory IT-33 No. 3 373-38. [3] Cureto E.E. (1968). Ubiased estimatio of the stadard deviatio. The America Statisticia No. 1. 8

[4] Edelma D. (1990). A codece iterval for the ceter of a ukow uimodal distributio based o a sample size 1. The America Statisticia 44 No. 4 85-87. [5] Goldstei M. (1981). Revisig previsios: A geometric iterpretatio. J.R. Statistical Society B 43 105-130. [6] Johso N. ad Kotz S. (1970). Distributios i Statistics: Cotiuous Uivariate Distributios -. New York: Joh Wiley ad Sos. [7] Lehma E.L. (1986). Testig statistical hypotheses d ed. New York: Joh Wiley ad Sos. [8] Machol R.E. ad Roseblatt J. (1966). Codece iterval based o sigle observatio. Proceedigs of IEEE 54 1087-1088. [9] Machol R.E. (1966). Codece iterval based o sigle observatio. Proceedigs of IEEE 54 1976. [10] Machol R.E. (1967). Codece iterval based o sigle observatio. Proceedigs of IEEE 55 1. [11] Rodriguez C.C. (1998). Codece itervals from oe observatio. Public webpage: http://omega.albay.edu:8008/cot.html. 9