Klerk-Grobben, G.

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MATHEMATISCH CENTRUM 2e BOERHAAVESTRMT 49 AMSTERDAM ST ATISTISCHE AFDELING Leiding: Prof. Dr D. van Dantzig Chef van de Statistische Consultatie: Prof. Dr J. Hemelrijk 1953-49(1) An Example of FIELLER 1 s method for deter-mining confidence limits for the quotient of two me2ns by Gerdn Klerk-Grobben 4953. :it I.ntroduction. E.C.FIELLER (1944) developed a method for determining confidence limits for the ratio of the means of two nor-mally (but not necessarily independently) distributed random variables. To this end he uses statistics s 11 ,s12 and s 22 (here denoted by a 11 ,a 12 ,and a 22 ), which he calls estimates of the variances and of the covariance of these random variables. We here give a general description of his methods and show, by means of an example, that the method need not be confined to the use of estimates of the variances and covariance mentioned. 11 Notations. The random character of a variable is denoted by under-lining the symbol, thus x. "x and y are N( J , ~ ; <r11 , a-12, er 22 ) distributed", .. 1.eans: !. and y have a two dimensional normal distribution with means f and 1, variances cr; 1 and o-22 and covariance 0-12· The probability that a random variable x assumes a value x 2 a will be denoted by P[ !. ~ a J. For the variance of a random variable x we shall use the symbols <f" ! or er 2L~-1. III General formulation of the method. J We suppose that to determine confidence limits for oZ.= f' observations ,!1 , ••• ,wk are made, which enable. us to find five functions of w1 , ••• ,wk, say X, Y, .§._11 , .§,.12, and .§.. 22 , satisfying the following conditions; 1) X and Y are N( g , 12; cr'11 ., er 12 ., cr-22 ) distributed with unknown parameters (we will suppose that not · ~~ = 0 and ~ = 0 is true ) ; 2) the functions z def 1 ! _ J 1_ and 8 z ~ +V 72 2.§..11-2 ~ j .§..12 + ! 2.§..22 1 are independently distributed; 3) for some known integer f the random variable with L s2 -Z 2 I 0-z a-~= 1 2 0-11 - 2 '?.g <r:i2 + g 2 0-22' has a -;r_2 -distribution with f degrees of freedom. - 2 -From the conditions it follows that Z has Student's §.z distribution with f degrees of freedom, that is to say: from a table of Student's distribution we can determine a Thus: p[ 19ZX-EY\ _ftJ=1-t, . v~ 2~~1-2d .':!.12+ f'.':!.22' - E . or, since tt) OJ.I t.<1, pr (1 X-· g Y)2b t~ (1( 2~11-2'>z f §..42+ i2.§.22)] := 1-r.~ which ca[n b; w~itten aj follo~s: J 2 2 2 ) L l (1) P (X -te§..11)-2-(XY-\.§.12) I 2(Y -te§..22 = 0J=1-r. (supposing~ f. 0). --rz. ~ From the last formula it is clear that the values, which, substituted for J_ in (1), satisfy the inequality 1 between square brackets and form a confidence interval for o< = { , corresponding to the confidence level 1- e. • In many cases, according to condition 3) and the .definitions of <T'Z and s2 , the functions .§.11 ,§.,12 and §.,22 will be estimates of the variances and covariance, 11 , ~ 2 and <r22 , of X and Y. However, this is not necessary as will be clear from the following example. IV Confidence limits for the slope of a straight line,if both variables are subject to errors, according to A.WALD. ions: A.WALD in his paper starts with the following assumpt-1) ~ and v are N(0,0.i a-11 , o-12 ., a-22 ) distributed ( er 11 , o-12 and o-22 unknown). 2) For the variables tand 1., satisfying j =o<'n +/3 , . ( I observations (x1 ,y1), ... ,(x2m.,y2m) are made, such that X" = t i+U", -l ' -1 Z1 = "2 1 +vi' where, for each i, ui and vi are distributed like ~and~< and two pairs (~1 ,~1 ) and (~j'2.j) (if. j) (i,j = 1, ... ,2m) are independent. 3) Independent of the values of the observations the pairs (x1 ,Yi) can be renumbered and divided into two - 3 -equal groups, (x1 ,y1 ), ••. ,(xm.,ym) and (xm+1 ,Ym+1 ), .•. • • .,(x2m.,y2m)., such that all of j 1 ,. .. Jm are smaller than J m+1 ., .•• J 2m(or such that all of o/ ,,., .•• , 1 m a re smaller· than J m+'1'... p 2m). The observations xi and yi here take the place of ~1 ., .. ,.,wk in the preceding section. We now define: 1 m _ 1 2m x,, = - L~i and x2 = m L ~i' m 1 m+1 where x1 and x2 are independently distributed., according to asE._umption 3 J, and likewise y 1 ,y2,u1 ,u2 ,v1 ,v2 ., J1 , [ 2, ~ 1 and 72 2 . Taking for the five functions X,Y.,.9..11 ,.9..12 and §..22 the following expressions: X = x1 - x2 , Y = Y1 - Y2 , 2m z __ 1 ,;;;-·"·· (xi-x1)2 + [ m .9..44= m(m-1) L.~ i=1 i=m+1 a - 1 [ -12- m(m-1) 1 §..22= m(m-1) [~ 2m (yi-Y1)2 + L m+1 (x,-x2)2l, -l - J ,:f: (~i-x2)(yi-y2)l ' m+1 ~{ it is not difficult to show that the conditions of para-graph III are fulfilled and thus a confidence interval for o<...(estimate: ~=~)can be determined as is explained in that paragraph. (The proof is performed by substituting: x , = b . +u. , and -l S l -l y. = 'h • +v , . {,_ i=c< rn . + 1,,3 • ) -l l l -l > S l l we now compare the expression .9..11 with the variance it is seen that we cannot use ~11 as an estimate of Indeed: 2 21 __ l crx = er 1x1-x2j whereas; !:'._11 ~ m(~-1)[~ 1 [ m = m(m-1) ~ 2m z_ m+1 - 4 -m § - 2 2m { 2)2+ + L ( i- t 1) + z_.- ( t i -1 m+1 m 2m ( j i - 12><~1-u~ · + 2 I: ( s - t 1 )(2:3.i -u1) + 2 L 1 i m+1 as an ordinary estimate of q 11 , but the value of 5!.11 also depends on the values of Ji. Clearly the mathematical expectation of ~11 is larger than<ri,the difference in-creasing with increasing differences between the ~ 1 • References. E.C.FIELLER, A .WALD, G.KLERK-GROBBEN., H.JoPRINS., A fundamental formula in the statistics of biological assay and some applications, Quart.J.Phar., .11(1944).,pp.117-123. The fitting of straight lines if both variables are subject to error, The Annals of Math.Stat., jj_ (1940), pp.284-300. Een betrouwbaarheidsgebied voor het quotient voor de verwachtingen van twee stochastische grootheden., die een si-multane normale verdeling bezitten. Rapport S 90(M36) van de Statistische Afdeling van het Mathematisch Centrum, Amsterdam 1952. Een betrouwbaarheidsgebied voor het qu-tient van de verwachtingen van twee on-afhankelijke normaal verdeelde groothe-den met gegeven spreidingsverhouding. Rapport S 90(M36a) van de Statistische Afdeling van het Mathematisch Centrum, Amsterdam 1953. 

An example of Feller's method for determining confidence limits for the quotient of two means

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