Moors, J.J.A.

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On the absolute moments of a normally distributed random variable

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    Tilburg UniversityOn the absolute moments of a normally distributed random variableMoors, J.J.A.Publication date:1973Link to publication in Tilburg University Research PortalCitation for published version (APA):Moors, J. J. A. (1973). On the absolute moments of a normally distributed random variable. (EIT Researchmemorandum / Tilburg Institute of Economics; Vol. 39). Unknown Publisher.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portalTake down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 12. Nov. 2023CBM ~. ~.,,, . ~. . ~R7626197339 EIT-398eetemmin4~~CHRIFTENBLiREr~U~ BIBI.IOTF~.E1C~ KAiHr~LIL:KEHOGP..SCHOOLTIVBURaJ. J. A. MoorsNr.On the absolute momentsof a normally distributedrandom variableIVIIIII IIIN IINII NI ~III NIII NIN IINI ~I IIIIResearch memorandumTILBURG INSTITUTE OF ECONOMICSDEPARTMENT OF ECONOMETRICSBIBLInTHEE~ KATH~~ ""ON THE ABSOLUTE MOMENTS OF A NORMALLY DISTRIBUTEDRANDOM VARIABLE.byJ.J.A. MoorsKatholieke Hogeschool Tílburg, The Netherlandst ~ 1~ ~7~I ~ ~ i r iMarch 1973- ~ -Summary.Estimation of shortage and excess of supply on the h~usingmarket leads to the evaluation of the absolute moments of anormally distributed variable. A formula for these momentsis given in section 2; since for the first two central momentsthe function H(x) -~(x) - x~(-x) appears to be of considerableinterest, its nroperties are investigated in section 3. Thefunction K(x) - H(x) H(-x), that appears in the formula forthe second central moment, is the subject of section 4. Theconsequences of the derívedproperties for the estimation ofhousing shortage and excess are discussed. After multiplicationby a constant, K(x) can be interpreted as a density which isnearly standard normal. A formula for the moments correspondingwith this density is presented.- 2 -l. INTRODUCTI~N.In view ofthe stil existing housing shortage in The Netherlands,an optimal use of the housing stock is necessary. This can beachíeved by trying to allocate households to dwellings insuch a way that each houshold occupies a house that suíts itbest. Of course, this ís possible only if the number of house-holds wanting a dwelling of a certaín type (related to rentand number of rooms, for example) is known, as well as theaumber cf availabl2 dwellir.gs of that typc. Comparing demar.3and supply will reveal in what categories of dwellíngs ashortage or an excess of supply exísts, which is of courseof considerable importance for building programs. Denotingsupply and demand by S and D respectively, we have for theexcess E of supply:E - max(S-D,0)and for the shortage F:F - max(D-S,0)A related concept is the potential shift P, defined as thenumber of households able to find suitable dwellings on [heexisting housing market. This important quantity indicatesto what extent the housing needs could be solved if no newbuilding activities were started.The following relations are immediately clear:P - min(S,D) - }(StD) - ~IS-DIE - S-P; F - D-P (~,~)To estimate these quantities a survey is needed in which theactual as well as the favoured housing situation is asked for.Denoting the sample totals of the above quantities by the- 3 -corresponding minuscules, the most obvious estimator in a simplerandom sample with sampling fraction f is p~f,wherep - min(s,d) - }(std) - ~~s-dlSince s~f and d~f are unbíased estimators of S and D respecti-vely, [he bías B of the estimator p~f equalsB - E{P~f} - P - ~IE{sfd}I - ~ E~sfd~as follows from formulae (1.1) and (1.2).Now,if the sample is large enough (and the sampling fractionsmall enough), (s-d)~f will be nearly normally distributedand the problem is reduced to fínding the first absolutemoment of a normally distributed variable u. Similarly,calculating the variance of p~f involves evaluation of thevariance of lul. See for details [1].It is therefOre necessary to investigate the absolute momentsof a normally distributed variable, which is the subject ofthe present paper. The major part is denotes to the study ofthe function H(x) defined byH(x) - ~(x) - x~(-x)which almost completely determines the first two centralmoments of lul. In equation (1.4) ~(x) and ~(x) denoterespectively the density and the distribution function of astandard normal variable x ti N(0,1).Of course, the results presented here may be of interestwhenever a variate following a normal distribution occurs,thesign of which is irrelevant. Moreover, the function H(x) arisesin other contexts as well.It can be shown for example, that the number of times avehicle with given speed is overtaken per kílometer, can beexpressed under certain conditions by the function H(x) [2].- 4 -2. THE ABSOLUTE MOMENTS OF A NORMALLY DISTRIBUTED VARIABLE.In this section an expression ís derived for the absolute mo-ments of a normally distributed random variable u ti N(u,Q).The density of u is given byf(u) - Q~exP ~-~(u-u)2~QZ~By means of the transformationx - (u-U)~Q (2.2)a standard normal variable x ti N(0,1) may be obtaíned withthe densityQ(x) - ~ exp(-}xZ) (2.3)Some wellknown properties of ~ which will be used in the sequelarelim ~(x) - 0 ;~(0) - ~; lim ~(x) - 0x~,-~ x~oo4(x) ' ~ ; ~(-x) - ~(x); ~'(x) --x~(x)The probability distribution of x ís denoted by~(x) - J xm4(Y)dy - ~ J x~exP(-~iYZ)dYwith the propertieslim ~(x) - 0 ; ~(0) - ~ ; lim ~(x) - 1xy-~ x~~(2.4)(2.5)(2.6)~(x) ~ 0; ~(-x)}~(x) - 1 ; ~'(x) - ~(x)- 5 -Integrals of the type j ~ urf(u)du will be needed, which maybe evaluated by use of transformation ( 2.2); definingb - u~owe findJ 0 urf (u)du - Q~JD "r exPl -~ (u-u) Z ~Q2] du-I-b(oxtu)2~(x)dx(2.7)~ r (~1 urf(u)du - E (r)Q~ur-~J - x~~(x)dx (2.8)0 j-0 ~ bAn expression for the integrals on the right follows fromlemma 2.1, which can easily be proved by taking derivatives.Lemma 2.1 The primitive function of x~~(x) is given by-~(x) E2m-i m:x2ii:i-0for j - 2mt1m i-m (2m):i.' 2i-1 -m (2m):-~(x) E 2 ( 2i):m:x i 2 m, ~(x) for j- 2mi-1with m - 0,1,2,...Denoting- Q~I (Qx}U)2 exp(-~x2)d(ax}u)JO~~Fj(-b) - J ~ x~~(x)dxb- 6 -it follows thatF.(-b) -im m-i m: 2i~(b) E 2 i;b for j-2m}Ií-0m i m(2m) i. 2i 1 m(2i) m-~(b) E 2'- ~'~b '- t2- (2m)~~(b) for j-2m(2.9)For the rth moment m of u is follows thatr[ r~2) r: z m r-2mmr - m~~ (r-2m):m:(a ~2) u (2.10)where [aJ denotes the largest integer smaller than or equalto a.Expressions for the absolute moments Sr of u are noweasily derived. For the even moments we have of courseS2k - m2kand for the odd moments (2.8) gives:s2k-1 - I~lul2k-If(u)du - 2i~u2k-lf(u)du- m2k-1J ~ 102k-I 2k-j-1- 2 E (2k-1)O~U F.(-b) - mj-0 j ] 2k-1(2.11)(2.12)where F.(-b) and m2k-1 are .given by (2.9) and (2.10) respecti-Jvely. For the first two absolute moments we find in this waySl - U-2u~(-b)t2a~(b)Q2 - uztQzIf the function H(b) ís defined as(2.13)(2.14)- 7 -H(b) - ~(b)-b~(-b) (2.15)we get for the expectation and variance of lu~:Elul - R1 - u}2oH(b) (2.16)Var~u~ - ~2-R~ - 6Z-(2a)2H(b){H(b)tb}It is easily seen from the definition that H(b) has the propertyH(-b) - H(b)fb, so that Varlu~ can be written asVarlu~ - aZ-(2a)ZH(b)H(-b) (2.17)The expectations of~ul and u differ by a term in which thefunction H(b) plays a dominant part and the same holds forthe variance of lu~ and u with respect to the function H(b)H(-b).The properties of H(b) and H(b)H(-b) are investigated in thenext sections.3. PROPERTIES OF THE FUNCTION H(x) -~(x)-x~(-x).Some ímportant properties of H(x) are:lim H(x) - 0; H(0) -~: lim H(x) - 0xi-~ x~mH(x) ~ 0; H(-x)-H(x) - x; H'(x) --~(-x) JMost of these relations follow immediately from the propertiesof ~(x) and ~(x) mentioned in formulae (2.4) and (2.6). Thelimiting value of H(x) when x nears infinity is somewhat moredifficult to find; it is however a special case of lemma 3.1.Lemma 3.1 For all real numbers c relation (3.2) holds:- 8 -clim x H(x) - 0xy~Proof.Of course, the relationlim xc ~(x) - 0x-~~(3.2)holds for all real numbers c. Further we have according tode 1'Hospital'srulelím ~(-x) - lim {~(-x)}'~ - lim -4(x) - 1x~,~ 4(x)Ix x~~ 4(x)Ix x-~~ -~h(x)-~(x)IxzCombination of these resultes gives:lim xc 4(-x) - lim xc-~~(x). lim ~(-x) - 0x-~~ xy~ x-~~from which the lemma follows. ~Q~(x)IxA consequence of the lemma is that the ratio ~(-x)Ib(x)-calledMills' ratío-is approximately llx for large x. It is alwayssmaller, however, since H(x) is positive everywhere. In thefollowing sectíon also a lower bound for Mills' ratio isneeded, which is provided by lemma 3.2.Lemma 3.2 For all positive x inequality ( 3.3) holds:~(x) ~ (xt~2ln)~(-x)Proof.DefineJ(x) - ~(x)-(xt~2l~t)9(-x)The first two derivatives are thenJ'(x) - f2ln ~(x)-~(-x)(3.3)- 9 -J"(x) - (1-x~~r)4(x)Consider first the values of x with 0 ~ x ~~2. For thesevalues the second derivative of J(x) is positive and hence thefirst derivative is an increasing function. Sínce J'(0) isnegative and J'(~) posítive, J(x) has local maxima in theendpoints 0 and J~r~2.On the other hand, for x~~n~2 the second derivative isnegative and the first decreasing; since J'(x) tends to zerofor large x, the first derivative ís positive. So, J(x) isincreasing for all x~~n~2 and has a local maximum for x-~ ~.Since J(0) - J(~) - 0, J(x) is negative for all positive x. ~Relation ( 3.3) r~ay be written alternatively asH(x) ~~ ~n ~(-x) for x~ 0 (3.4)For ~Tills'ratio we fínd for positive x the double inequality:1 ~ ~ (-x) ~ 1xt 2 ~r Q(x) xOther inequalíties for this ratio are1 - ~ ~ ~(-x) ~ Ix x Q(x) x(see for example Feller [ 3] ) and1 ~ ~(-x) ~ 1xtl~x Q(x) x(3.5)due to Gordon [4]. The first part of this last relation isstronger than Feller's, but for x ~ r~r~2 weaker than (3.5).If equation (2.16) is written asElu~-Eu - 2QH(b)- 10 -relations (3.1) imply that lul is always larger than u inexpectation. For constant variance aZ the difference betweenE~u~ and Eu - u decreases as U increases in absolute value.This result is quite plausible, since the difference betweenthe two expectations is caused by theoccurrence of positiveand negative values of u. If b- UIQ moves farther away fromthe origin in either direction, positive or negative valuesdominate. So, the bías of the estimator p~f is largest whendemand and supply are about equal, but tends to zero whenIS-DI is large. .4. PROPERTIES OF THE FUNCTION k(x) - H(x)H(-x).The function k(x) is defined byk(x) - H(x)H(-x) - ~Z(x)tb~(x){~(x)-~(-x)}-bZ~(x)~(-x)(4. 1 )Some important properties are1 ' lim k(x) - 0lim k(x) - 0: k(0) - 2,~ .xi-pp x~ook(x) ~ 0: k(x) - k(-x) : xk'(x) ~ 0(4.2)All these relations but one follow ímmediately from thesimilar properties of iï(x) given ín (3.1). The last assertionconcerning the sign of the first derivatíve, is the subject oflemma 4.1.Lemma 4.1 k(x) is increasing for x ~ 0 and decreasíng for x~ 0.Proof.In view of the synmetry of k(x) the two assertions of the lemmaare equivalent. So ít is sufficient to prove that k'(x) isnegative for x~ 0. Now,- tt -k'(x) - H'(x)H(-x)-H(x)H'(-x)- -~(-x){H(x)tx}tH(x)~(x)- H(x)~(-x){á-I(-x)-xH-~(x)-2}and the expression between {} tends to 0 for x-~ 0. So, iffor x~ 0 this expression can be shown to be decreasing, orits derivative to be negative, the proof is given. Indeed,{~-1(-x)-x H-~(x)-2}' - -~(x) - ~(x)~Z(-x) H2(x)- ~(x) {HZ(x)-~2(-x)} ~ 04(-x)HZ(x)because of inequality (3.4). ~If equation (2.17) is written asVarlul - Var u - -(26)Z H(b)H(-b)relations (4.2) imply that lu~ has always a smaller variancethan u. For constant 62 the difference between them decreasesif u increases in absolute value, which is again very plausible.Figure 1 shows the functions ~(x),~(x),H(x) and k(x). Itappears that k(x) differs from ~(x) by a factor that isreasonably constant. The following values can be obtained, forexample, from tables of the normal density and distribution:x 0 tl t2 t3k(x)~~(x) 0.3989 0.3823 0.3158 0.2573So, the function g(x) defined asg(x) - k(x)~J~ k(y)dy (4.3)~- 1 2 -W~JULL- 13 -can be interpreted as a density whích resembles the density ofa standard normal varíate. This density could be used, for exampleto detect numerically the influence of small deviations fromnormality. The moments of a variable x with density f(x) canbe found wíth lemma 4.2.Lemma 4.22nThe primitive function of x k(x) is~ ~ x2n}3~(x)~(-x)tQ(x){2~(x)-]}(x2nt2- E2n-i n~x2i)t2nt3 i-0t~2(x)x(x2n} E ci[ 2n~ x2i)}d[ 2n~ ~(xJ2)IJn~ (4.4)i-0for all non-negative integers. The ci[2n)follow the recursionformulac[ 2nJ - -2 ~ c[ 2n]n-1 i-1- (it~)ci-2n-i (4.5)and d[ 2n~ ís given by- ( 2 for n - 0d[ 2nj {I2nn:-}c[ 2n~ for n~ 00(4.6)Proof.The proof can be given by differentiating formula (4.4). (Thisformula was formed with the help of (4.1).) '2ntlThe primitive function of x k(x) may be obtained in asimilar way; it reads2nt4[x2n}4~(x)~(-x)}~(x){2~(x)-1}xpn}~(x2) tf ~2(x)9ntl(x2] (4.7)- 14 -where pn}~(x2) and qn}~(x2) are polynomials in x2 of degreenfl, the coefficients of which can be found by differentiating.The most important difference with formula (4.4) is the absenceof the term in which ~(x~2) occurs; this corresponds withlemma 2.1.Especially interesting are the integrals of xlk(x) over thewhole real axis, to which only the term containing ~(xr2)contributes. So, it follows from formulae (4.4) and (4.7) forall non-negative integers n:(~I-~xlk(x)dx -r 0 for i-2nt1d[ 2n](2nt3) ~rfor i-2n(4.8)The result for odd i follows of course immediately from thesymmetry of k(x).In particular, it follows that- ~k(x)dx 2and substituting this in equation (4.3) we find thatS(x) - Zr~ k(x) (4.9)is a density.To calculate the even moments correspondíng with S(x) it isnecessary [o evaluate có2n} Recursion formula (4.5) givesfor all j ~ n:j J 2n-kn: k-1 n-1c[o2n]-c[ 2n] n (it~i)- E k: ]I (i}])-2 n:J i-j k-2 í-1w'ith the wellknown relation- 15 -knl (it]) -(2k):i-1 22k-lk;we find in particular for j- n:c[ 2n] - c[ 2n] (2n): - 2ntln:n~l (2kj2-3k-2n-]n:o n-1 22n-]n:k-2 kor, bringing terms under the sommation sign:c[ 2n] - -2ntln:0n~2n ) 2-3n} ~ ( 2k ) 2-3k~L` n k - 1Expression (4.6) can now be wrítten asd[ 2n] I nL U-whích holds under the usual conventions even for n-0.(4.10)(4.11)The even moments of a variate x with density g(x) are given by~ x2ng(x)dx - 3n:2n-1 r(2n)2-3n} ~ (2k)2-3k~J-~ 2nt3 I (4.12)n k-0 kfor non-negative n. This gives for the first three even moments:uo - l; u2 - 9~10; U4 - 69I28 'Note that for large n the factor between [] tends to r2sincem ~E (2k)2-3k - E (-~)k(-~) - (1-~)-~ - ,~2.kz0 k k-0 kFinally, an easier method for calcuting có2n] and d[2n] isby use of the recursion formulae- 2nn: (2n)2-3n} ~ (2k)2-3k~n -~ kc~2(ntl)] - 2(ntl)c[ 2n] } (2n):(4.13)0 0 22n-ln:- 16 -d[ 2(nfl)] - 2(n}1)d~ 2n] - (2n):0 22nn;(4.14)which can be derived immediately from ( 4.10) and (4.11).M.v.d.B.- 17 -REFERENCES.[1] Statistische en econometrische onderzoekingen no. 14(forthcoming).[2] Centraal Bureau voor de Statistiek, "Aantal inhaal-manouvres bij een vereenvoudigd verkeersmodel",report no. 1497-68-B.S.A. (Ag 06-67-027);[3] Feller, W., "An introduction to probability theory andits applications I", New York, Wiley;[4J Gordon, R.D., "Values of Mill's ratio of area to boundingordinate and of the normal probability integral forlarge values of the argument", Annals of ~tathematicalStatistics XII (1941), 364-366.PREVIOU3 NUMBERS:EIT 1 l. Krlens') . .. .... Het verdelen van eteekproeven over aubpopu-laties biJ accountantscontroies.EJT 2 1. P. C. Kleynen ') . . . . .ER 3 S. R. Chowdhury and W. Vandaele')ER 4 Prof. dn. 1. Kriens . .ER 5 Prof. dr. C. F. 8eheHer') .EIT 8 S. R. Chowdhury ~. .. .EIT 7 P. A. Verhsyen ') . . .EIT 8 R. M. l. Heute enWalterA. Vandasle') . .E1T 9 3. R. Chowdhury ~. . .E!T 10 A. l. van Rseken ') .Een toepaeeing van „importance sampiing".A bayesian anaiyais of heteroecedasticity inregresslon models.. De bosllskunde en haar toepasafngen.. Wlnstkapitelisatie veraus dividendkapitalieatiebiJ het waarderen van aandelen.. A bayesian approach In multiple regresslonenelysls with Inequafity constralnte.. Investeron en onzekerheid.. Problemen rond nlet-iineaire regreasle.. Bayesian anelysis In Ilnear regresslon withdifferent priors.. The effect of truncation tn stetistlcel compu-tatton.EIT i t W. H. V~ndaels andS. R. Chowdhury') .EIT 12 J. de Blok') . .EIT 13 WelterA. Vandaele')ER 14 J. Plaamane ') . .EIT 15 D. Nesleman ') . .EIT 18 H. N. Weddepohl .ER 17EIT 18 J. Plasmans') .A revtsed method of scortng.Reclemo-uitgaven in Nederlend.M0dsc0, a computer programm for the revisedmethod of scoring.Alternative production models.(Some emplrical relevance for poatwar BelgianEconomy)Multiple regression and eerially correlatedarrors.. Vector repreaentation of majority voting.The general linear seemingiy unrelaied regres-sion problem.I. Modeis and Inference.ER 19 1. Plaamans and R. Van Straslen ~. The general ilnear seemingly unrelated regres-slon problem.II. Feasible etatiaticai estimatlon and an appii-cation.qI ~N~V~~~~I ~éY ~ ~Í~W IVIIh IEIT 20 Pleier H. M, Ru~re . . . . . A procedure for an economy wlth collectlveEIT 21 D. Neeleman ') . .tora.EIT Z2 R. M.1. Heute . . .EIT 23 D. Neeleman ') . . .ER 24 R. Stobbsringh ~ .goods oniy.. . . An alternative derivation of the k-clasa esttma-. . . Parameter eatimation In the exponential dtstri-butlon, confidence intervals and a monte carlostudy for some goodness of flt tests.. . . The claselcal multiverlate regresslon modelwith singular covarlance matrix.. . . Tho derivat?on of the optimal Karhunen-Loèvecoordinate functions.EIT 25 Th. van de Klundert') . .EIT 28 Th. van de Klundert') . .EIT 27 R. M.1. Heuts ') . . .. . Produktle, kapitaal en Interest.. . Labour values and intemational trade; a refor-mulation of the theory of A. Emmanuei.. . Schattingen van parameters In de gamma-verdeling en een onderzoek naar de kwaliteltvan een drletal schattingsmethoden met behulpvan Monte Cario-methoden.ER 28 A. van Schaik') . . . . . A note on the reproduction of fixed capital Inwvo-good techniquea.EIT 29 H.N. Weddepohl') . . . . . Vector representation of majority voting; arevised paper.EIT 30 H. N. Weddepohl . . . . . Duality and Equilibrlu:n.EIT 31 R. M. l. Heuts and W. H. Vandaels ') Wumerical rasults of quasi-newton methods forunconatrained functlon minimization.EIT 32 Pleter H. M. Ruya . . . . . On the extatence of an equtllbrtum for aneconomy with publlc goods only.E{T 33 , . . . . . . . . . Het rekencentrum bi~ het hoger onderwiJs.E!T 34 R. M. l. Heute and P.1. Rens . . A numerical comparison among some algo-rithms for unconstrafned non-Iinear functtonmin(mization.EIT 35 1. Kriens . . . Systematic inventory management with a com-puter.EIT 38 Pieter H. M. Ruys . . . . . On convex, cone-interior processes.EiT 37 1. Plasmans Adjustment cost modela for the demand ofinvestment.EIT 38 H. N. Weddepohl . . . . . Dual sets and dual correspondences and theirapplication to equilibrium theory.EIT 1973e) not available

    Tilburg UniversityOn the absolute moments of a normally distributed random variableMoors, J.J.A.Publication date:1973Link to publication in Tilburg University Research PortalCitation for published version (APA):Moors, J. J. A. (1973). On the absolute moments of a normally distributed random variable. (EIT Researchmemorandum / Tilburg Institute of Economics; Vol. 39). Unknown Publisher.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portalTake down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 12. May. 2021CBM ~. ~.,,, . ~. . ~R7626197339 EIT-398eetemmin4~~CHRIFTENBLiREr~U~ BIBI.IOTF~.E1C~ KAiHr~LIL:KEHOGP..SCHOOLTIVBURaJ. J. A. MoorsNr.On the absolute momentsof a normally distributedrandom variableIVIIIII IIIN IINII NI ~III NIII NIN IINI ~I IIIIResearch memorandumTILBURG INSTITUTE OF ECONOMICSDEPARTMENT OF ECONOMETRICSBIBLInTHEE~ KATH~~ ""ON THE ABSOLUTE MOMENTS OF A NORMALLY DISTRIBUTEDRANDOM VARIABLE.byJ.J.A. MoorsKatholieke Hogeschool Tílburg, The Netherlandst ~ 1~ ~7~I ~ ~ i r iMarch 1973- ~ -Summary.Estimation of shortage and excess of supply on the h~usingmarket leads to the evaluation of the absolute moments of anormally distributed variable. A formula for these momentsis given in section 2; since for the first two central momentsthe function H(x) -~(x) - x~(-x) appears to be of considerableinterest, its nroperties are investigated in section 3. Thefunction K(x) - H(x) H(-x), that appears in the formula forthe second central moment, is the subject of section 4. Theconsequences of the derívedproperties for the estimation ofhousing shortage and excess are discussed. After multiplicationby a constant, K(x) can be interpreted as a density which isnearly standard normal. A formula for the moments correspondingwith this density is presented.- 2 -l. INTRODUCTI~N.In view ofthe stil existing housing shortage in The Netherlands,an optimal use of the housing stock is necessary. This can beachíeved by trying to allocate households to dwellings insuch a way that each houshold occupies a house that suíts itbest. Of course, this ís possible only if the number of house-holds wanting a dwelling of a certaín type (related to rentand number of rooms, for example) is known, as well as theaumber cf availabl2 dwellir.gs of that typc. Comparing demar.3and supply will reveal in what categories of dwellíngs ashortage or an excess of supply exísts, which is of courseof considerable importance for building programs. Denotingsupply and demand by S and D respectively, we have for theexcess E of supply:E - max(S-D,0)and for the shortage F:F - max(D-S,0)A related concept is the potential shift P, defined as thenumber of households able to find suitable dwellings on [heexisting housing market. This important quantity indicatesto what extent the housing needs could be solved if no newbuilding activities were started.The following relations are immediately clear:P - min(S,D) - }(StD) - ~IS-DIE - S-P; F - D-P (~,~)To estimate these quantities a survey is needed in which theactual as well as the favoured housing situation is asked for.Denoting the sample totals of the above quantities by the- 3 -corresponding minuscules, the most obvious estimator in a simplerandom sample with sampling fraction f is p~f,wherep - min(s,d) - }(std) - ~~s-dlSince s~f and d~f are unbíased estimators of S and D respecti-vely, [he bías B of the estimator p~f equalsB - E{P~f} - P - ~IE{sfd}I - ~ E~sfd~as follows from formulae (1.1) and (1.2).Now,if the sample is large enough (and the sampling fractionsmall enough), (s-d)~f will be nearly normally distributedand the problem is reduced to fínding the first absolutemoment of a normally distributed variable u. Similarly,calculating the variance of p~f involves evaluation of thevariance of lul. See for details [1].It is therefOre necessary to investigate the absolute momentsof a normally distributed variable, which is the subject ofthe present paper. The major part is denotes to the study ofthe function H(x) defined byH(x) - ~(x) - x~(-x)which almost completely determines the first two centralmoments of lul. In equation (1.4) ~(x) and ~(x) denoterespectively the density and the distribution function of astandard normal variable x ti N(0,1).Of course, the results presented here may be of interestwhenever a variate following a normal distribution occurs,thesign of which is irrelevant. Moreover, the function H(x) arisesin other contexts as well.It can be shown for example, that the number of times avehicle with given speed is overtaken per kílometer, can beexpressed under certain conditions by the function H(x) [2].- 4 -2. THE ABSOLUTE MOMENTS OF A NORMALLY DISTRIBUTED VARIABLE.In this section an expression ís derived for the absolute mo-ments of a normally distributed random variable u ti N(u,Q).The density of u is given byf(u) - Q~exP ~-~(u-u)2~QZ~By means of the transformationx - (u-U)~Q (2.2)a standard normal variable x ti N(0,1) may be obtaíned withthe densityQ(x) - ~ exp(-}xZ) (2.3)Some wellknown properties of ~ which will be used in the sequelarelim ~(x) - 0 ;~(0) - ~; lim ~(x) - 0x~,-~ x~oo4(x) ' ~ ; ~(-x) - ~(x); ~'(x) --x~(x)The probability distribution of x ís denoted by~(x) - J xm4(Y)dy - ~ J x~exP(-~iYZ)dYwith the propertieslim ~(x) - 0 ; ~(0) - ~ ; lim ~(x) - 1xy-~ x~~(2.4)(2.5)(2.6)~(x) ~ 0; ~(-x)}~(x) - 1 ; ~'(x) - ~(x)- 5 -Integrals of the type j ~ urf(u)du will be needed, which maybe evaluated by use of transformation ( 2.2); definingb - u~owe findJ 0 urf (u)du - Q~JD "r exPl -~ (u-u) Z ~Q2] du-I-b(oxtu)2~(x)dx(2.7)~ r (~1 urf(u)du - E (r)Q~ur-~J - x~~(x)dx (2.8)0 j-0 ~ bAn expression for the integrals on the right follows fromlemma 2.1, which can easily be proved by taking derivatives.Lemma 2.1 The primitive function of x~~(x) is given by-~(x) E2m-i m:x2ii:i-0for j - 2mt1m i-m (2m):i.' 2i-1 -m (2m):-~(x) E 2 ( 2i):m:x i 2 m, ~(x) for j- 2mi-1with m - 0,1,2,...Denoting- Q~I (Qx}U)2 exp(-~x2)d(ax}u)JO~~Fj(-b) - J ~ x~~(x)dxb- 6 -it follows thatF.(-b) -im m-i m: 2i~(b) E 2 i;b for j-2m}Ií-0m i m(2m) i. 2i 1 m(2i) m-~(b) E 2'- ~'~b '- t2- (2m)~~(b) for j-2m(2.9)For the rth moment m of u is follows thatr[ r~2) r: z m r-2mmr - m~~ (r-2m):m:(a ~2) u (2.10)where [aJ denotes the largest integer smaller than or equalto a.Expressions for the absolute moments Sr of u are noweasily derived. For the even moments we have of courseS2k - m2kand for the odd moments (2.8) gives:s2k-1 - I~lul2k-If(u)du - 2i~u2k-lf(u)du- m2k-1J ~ 102k-I 2k-j-1- 2 E (2k-1)O~U F.(-b) - mj-0 j ] 2k-1(2.11)(2.12)where F.(-b) and m2k-1 are .given by (2.9) and (2.10) respecti-Jvely. For the first two absolute moments we find in this waySl - U-2u~(-b)t2a~(b)Q2 - uztQzIf the function H(b) ís defined as(2.13)(2.14)- 7 -H(b) - ~(b)-b~(-b) (2.15)we get for the expectation and variance of lu~:Elul - R1 - u}2oH(b) (2.16)Var~u~ - ~2-R~ - 6Z-(2a)2H(b){H(b)tb}It is easily seen from the definition that H(b) has the propertyH(-b) - H(b)fb, so that Varlu~ can be written asVarlu~ - aZ-(2a)ZH(b)H(-b) (2.17)The expectations of~ul and u differ by a term in which thefunction H(b) plays a dominant part and the same holds forthe variance of lu~ and u with respect to the function H(b)H(-b).The properties of H(b) and H(b)H(-b) are investigated in thenext sections.3. PROPERTIES OF THE FUNCTION H(x) -~(x)-x~(-x).Some ímportant properties of H(x) are:lim H(x) - 0; H(0) -~: lim H(x) - 0xi-~ x~mH(x) ~ 0; H(-x)-H(x) - x; H'(x) --~(-x) JMost of these relations follow immediately from the propertiesof ~(x) and ~(x) mentioned in formulae (2.4) and (2.6). Thelimiting value of H(x) when x nears infinity is somewhat moredifficult to find; it is however a special case of lemma 3.1.Lemma 3.1 For all real numbers c relation (3.2) holds:- 8 -clim x H(x) - 0xy~Proof.Of course, the relationlim xc ~(x) - 0x-~~(3.2)holds for all real numbers c. Further we have according tode 1'Hospital'srulelím ~(-x) - lim {~(-x)}'~ - lim -4(x) - 1x~,~ 4(x)Ix x~~ 4(x)Ix x-~~ -~h(x)-~(x)IxzCombination of these resultes gives:lim xc 4(-x) - lim xc-~~(x). lim ~(-x) - 0x-~~ xy~ x-~~from which the lemma follows. ~Q~(x)IxA consequence of the lemma is that the ratio ~(-x)Ib(x)-calledMills' ratío-is approximately llx for large x. It is alwayssmaller, however, since H(x) is positive everywhere. In thefollowing sectíon also a lower bound for Mills' ratio isneeded, which is provided by lemma 3.2.Lemma 3.2 For all positive x inequality ( 3.3) holds:~(x) ~ (xt~2ln)~(-x)Proof.DefineJ(x) - ~(x)-(xt~2l~t)9(-x)The first two derivatives are thenJ'(x) - f2ln ~(x)-~(-x)(3.3)- 9 -J"(x) - (1-x~~r)4(x)Consider first the values of x with 0 ~ x ~~2. For thesevalues the second derivative of J(x) is positive and hence thefirst derivative is an increasing function. Sínce J'(0) isnegative and J'(~) posítive, J(x) has local maxima in theendpoints 0 and J~r~2.On the other hand, for x~~n~2 the second derivative isnegative and the first decreasing; since J'(x) tends to zerofor large x, the first derivative ís positive. So, J(x) isincreasing for all x~~n~2 and has a local maximum for x-~ ~.Since J(0) - J(~) - 0, J(x) is negative for all positive x. ~Relation ( 3.3) r~ay be written alternatively asH(x) ~~ ~n ~(-x) for x~ 0 (3.4)For ~Tills'ratio we fínd for positive x the double inequality:1 ~ ~ (-x) ~ 1xt 2 ~r Q(x) xOther inequalíties for this ratio are1 - ~ ~ ~(-x) ~ Ix x Q(x) x(see for example Feller [ 3] ) and1 ~ ~(-x) ~ 1xtl~x Q(x) x(3.5)due to Gordon [4]. The first part of this last relation isstronger than Feller's, but for x ~ r~r~2 weaker than (3.5).If equation (2.16) is written asElu~-Eu - 2QH(b)- 10 -relations (3.1) imply that lul is always larger than u inexpectation. For constant variance aZ the difference betweenE~u~ and Eu - u decreases as U increases in absolute value.This result is quite plausible, since the difference betweenthe two expectations is caused by theoccurrence of positiveand negative values of u. If b- UIQ moves farther away fromthe origin in either direction, positive or negative valuesdominate. So, the bías of the estimator p~f is largest whendemand and supply are about equal, but tends to zero whenIS-DI is large. .4. PROPERTIES OF THE FUNCTION k(x) - H(x)H(-x).The function k(x) is defined byk(x) - H(x)H(-x) - ~Z(x)tb~(x){~(x)-~(-x)}-bZ~(x)~(-x)(4. 1 )Some important properties are1 ' lim k(x) - 0lim k(x) - 0: k(0) - 2,~ .xi-pp x~ook(x) ~ 0: k(x) - k(-x) : xk'(x) ~ 0(4.2)All these relations but one follow ímmediately from thesimilar properties of iï(x) given ín (3.1). The last assertionconcerning the sign of the first derivatíve, is the subject oflemma 4.1.Lemma 4.1 k(x) is increasing for x ~ 0 and decreasíng for x~ 0.Proof.In view of the synmetry of k(x) the two assertions of the lemmaare equivalent. So ít is sufficient to prove that k'(x) isnegative for x~ 0. Now,- tt -k'(x) - H'(x)H(-x)-H(x)H'(-x)- -~(-x){H(x)tx}tH(x)~(x)- H(x)~(-x){á-I(-x)-xH-~(x)-2}and the expression between {} tends to 0 for x-~ 0. So, iffor x~ 0 this expression can be shown to be decreasing, orits derivative to be negative, the proof is given. Indeed,{~-1(-x)-x H-~(x)-2}' - -~(x) - ~(x)~Z(-x) H2(x)- ~(x) {HZ(x)-~2(-x)} ~ 04(-x)HZ(x)because of inequality (3.4). ~If equation (2.17) is written asVarlul - Var u - -(26)Z H(b)H(-b)relations (4.2) imply that lu~ has always a smaller variancethan u. For constant 62 the difference between them decreasesif u increases in absolute value, which is again very plausible.Figure 1 shows the functions ~(x),~(x),H(x) and k(x). Itappears that k(x) differs from ~(x) by a factor that isreasonably constant. The following values can be obtained, forexample, from tables of the normal density and distribution:x 0 tl t2 t3k(x)~~(x) 0.3989 0.3823 0.3158 0.2573So, the function g(x) defined asg(x) - k(x)~J~ k(y)dy (4.3)~- 1 2 -W~JULL- 13 -can be interpreted as a density whích resembles the density ofa standard normal varíate. This density could be used, for exampleto detect numerically the influence of small deviations fromnormality. The moments of a variable x with density f(x) canbe found wíth lemma 4.2.Lemma 4.22nThe primitive function of x k(x) is~ ~ x2n}3~(x)~(-x)tQ(x){2~(x)-]}(x2nt2- E2n-i n~x2i)t2nt3 i-0t~2(x)x(x2n} E ci[ 2n~ x2i)}d[ 2n~ ~(xJ2)IJn~ (4.4)i-0for all non-negative integers. The ci[2n)follow the recursionformulac[ 2nJ - -2 ~ c[ 2n]n-1 i-1- (it~)ci-2n-i (4.5)and d[ 2n~ ís given by- ( 2 for n - 0d[ 2nj {I2nn:-}c[ 2n~ for n~ 00(4.6)Proof.The proof can be given by differentiating formula (4.4). (Thisformula was formed with the help of (4.1).) '2ntlThe primitive function of x k(x) may be obtained in asimilar way; it reads2nt4[x2n}4~(x)~(-x)}~(x){2~(x)-1}xpn}~(x2) tf ~2(x)9ntl(x2] (4.7)- 14 -where pn}~(x2) and qn}~(x2) are polynomials in x2 of degreenfl, the coefficients of which can be found by differentiating.The most important difference with formula (4.4) is the absenceof the term in which ~(x~2) occurs; this corresponds withlemma 2.1.Especially interesting are the integrals of xlk(x) over thewhole real axis, to which only the term containing ~(xr2)contributes. So, it follows from formulae (4.4) and (4.7) forall non-negative integers n:(~I-~xlk(x)dx -r 0 for i-2nt1d[ 2n](2nt3) ~rfor i-2n(4.8)The result for odd i follows of course immediately from thesymmetry of k(x).In particular, it follows that- ~k(x)dx 2and substituting this in equation (4.3) we find thatS(x) - Zr~ k(x) (4.9)is a density.To calculate the even moments correspondíng with S(x) it isnecessary [o evaluate có2n} Recursion formula (4.5) givesfor all j ~ n:j J 2n-kn: k-1 n-1c[o2n]-c[ 2n] n (it~i)- E k: ]I (i}])-2 n:J i-j k-2 í-1w'ith the wellknown relation- 15 -knl (it]) -(2k):i-1 22k-lk;we find in particular for j- n:c[ 2n] - c[ 2n] (2n): - 2ntln:n~l (2kj2-3k-2n-]n:o n-1 22n-]n:k-2 kor, bringing terms under the sommation sign:c[ 2n] - -2ntln:0n~2n ) 2-3n} ~ ( 2k ) 2-3k~L` n k - 1Expression (4.6) can now be wrítten asd[ 2n] I nL U-whích holds under the usual conventions even for n-0.(4.10)(4.11)The even moments of a variate x with density g(x) are given by~ x2ng(x)dx - 3n:2n-1 r(2n)2-3n} ~ (2k)2-3k~J-~ 2nt3 I (4.12)n k-0 kfor non-negative n. This gives for the first three even moments:uo - l; u2 - 9~10; U4 - 69I28 'Note that for large n the factor between [] tends to r2sincem ~E (2k)2-3k - E (-~)k(-~) - (1-~)-~ - ,~2.kz0 k k-0 kFinally, an easier method for calcuting có2n] and d[2n] isby use of the recursion formulae- 2nn: (2n)2-3n} ~ (2k)2-3k~n -~ kc~2(ntl)] - 2(ntl)c[ 2n] } (2n):(4.13)0 0 22n-ln:- 16 -d[ 2(nfl)] - 2(n}1)d~ 2n] - (2n):0 22nn;(4.14)which can be derived immediately from ( 4.10) and (4.11).M.v.d.B.- 17 -REFERENCES.[1] Statistische en econometrische onderzoekingen no. 14(forthcoming).[2] Centraal Bureau voor de Statistiek, "Aantal inhaal-manouvres bij een vereenvoudigd verkeersmodel",report no. 1497-68-B.S.A. (Ag 06-67-027);[3] Feller, W., "An introduction to probability theory andits applications I", New York, Wiley;[4J Gordon, R.D., "Values of Mill's ratio of area to boundingordinate and of the normal probability integral forlarge values of the argument", Annals of ~tathematicalStatistics XII (1941), 364-366.PREVIOU3 NUMBERS:EIT 1 l. Krlens') . .. .... Het verdelen van eteekproeven over aubpopu-laties biJ accountantscontroies.EJT 2 1. P. C. Kleynen ') . . . . .ER 3 S. R. Chowdhury and W. Vandaele')ER 4 Prof. dn. 1. Kriens . .ER 5 Prof. dr. C. F. 8eheHer') .EIT 8 S. R. Chowdhury ~. .. .EIT 7 P. A. Verhsyen ') . . .EIT 8 R. M. l. Heute enWalterA. Vandasle') . .E1T 9 3. R. Chowdhury ~. . .E!T 10 A. l. van Rseken ') .Een toepaeeing van „importance sampiing".A bayesian anaiyais of heteroecedasticity inregresslon models.. De bosllskunde en haar toepasafngen.. Wlnstkapitelisatie veraus dividendkapitalieatiebiJ het waarderen van aandelen.. A bayesian approach In multiple regresslonenelysls with Inequafity constralnte.. Investeron en onzekerheid.. Problemen rond nlet-iineaire regreasle.. Bayesian anelysis In Ilnear regresslon withdifferent priors.. The effect of truncation tn stetistlcel compu-tatton.EIT i t W. H. V~ndaels andS. R. Chowdhury') .EIT 12 J. de Blok') . .EIT 13 WelterA. Vandaele')ER 14 J. Plaamane ') . .EIT 15 D. Nesleman ') . .EIT 18 H. N. Weddepohl .ER 17EIT 18 J. Plasmans') .A revtsed method of scortng.Reclemo-uitgaven in Nederlend.M0dsc0, a computer programm for the revisedmethod of scoring.Alternative production models.(Some emplrical relevance for poatwar BelgianEconomy)Multiple regression and eerially correlatedarrors.. Vector repreaentation of majority voting.The general linear seemingiy unrelaied regres-sion problem.I. Modeis and Inference.ER 19 1. Plaamans and R. Van Straslen ~. The general ilnear seemingly unrelated regres-slon problem.II. Feasible etatiaticai estimatlon and an appii-cation.qI ~N~V~~~~I ~éY ~ ~Í~W IVIIh IEIT 20 Pleier H. M, Ru~re . . . . . A procedure for an economy wlth collectlveEIT 21 D. Neeleman ') . .tora.EIT Z2 R. M.1. Heute . . .EIT 23 D. Neeleman ') . . .ER 24 R. Stobbsringh ~ .goods oniy.. . . An alternative derivation of the k-clasa esttma-. . . Parameter eatimation In the exponential dtstri-butlon, confidence intervals and a monte carlostudy for some goodness of flt tests.. . . The claselcal multiverlate regresslon modelwith singular covarlance matrix.. . . Tho derivat?on of the optimal Karhunen-Loèvecoordinate functions.EIT 25 Th. van de Klundert') . .EIT 28 Th. van de Klundert') . .EIT 27 R. M.1. Heuts ') . . .. . Produktle, kapitaal en Interest.. . Labour values and intemational trade; a refor-mulation of the theory of A. Emmanuei.. . Schattingen van parameters In de gamma-verdeling en een onderzoek naar de kwaliteltvan een drletal schattingsmethoden met behulpvan Monte Cario-methoden.ER 28 A. van Schaik') . . . . . A note on the reproduction of fixed capital Inwvo-good techniquea.EIT 29 H.N. Weddepohl') . . . . . Vector representation of majority voting; arevised paper.EIT 30 H. N. Weddepohl . . . . . Duality and Equilibrlu:n.EIT 31 R. M. l. Heuts and W. H. Vandaels ') Wumerical rasults of quasi-newton methods forunconatrained functlon minimization.EIT 32 Pleter H. M. Ruya . . . . . On the extatence of an equtllbrtum for aneconomy with publlc goods only.E{T 33 , . . . . . . . . . Het rekencentrum bi~ het hoger onderwiJs.E!T 34 R. M. l. Heute and P.1. Rens . . A numerical comparison among some algo-rithms for unconstrafned non-Iinear functtonmin(mization.EIT 35 1. Kriens . . . Systematic inventory management with a com-puter.EIT 38 Pieter H. M. Ruys . . . . . On convex, cone-interior processes.EiT 37 1. Plasmans Adjustment cost modela for the demand ofinvestment.EIT 38 H. N. Weddepohl . . . . . Dual sets and dual correspondences and theirapplication to equilibrium theory.EIT 1973e) not available

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On the absolute moments of a normally distributed random variable

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