Median as a Weighted Arithmetic Mean of All Sample Observations

Transcription

1 Meda as a Weghted Arthmetc Mea of All Sample Observatos SK Mshra Dept. of Ecoomcs NEHU, Shllog (Ida). Itroducto: Iumerably may textbooks Statstcs explctly meto that oe of the weakesses (or propertes) of meda (a well kow measure of cetral tedecy) s that t s ot computed by corporatg all sample observatos. That s so because f the sample x = ( x, x,..., x ), where the varate values are ordered such that x x... x the meda( x) = ( xk + x+ k ) / ; k = t(( + ) / ). Here t(.) s the teger value of (.). For example t( (+)/ < ) =. Ths formula, although queer ad expressed a lttle roudabout way, apples uformly whe s odd or eve. Evdetly, meda( x) s ot obtaed by corporatg all the values of x, ad so the alleged weakess of the meda as a measure of cetral tedecy.. The Meda Mmzes the Absolute Norm of Devatos: It s a commoplace kowledge Statstcs that the statstc x (the arthmetc mea of x) mmzes the (squared) Eucldea orm of devatos of the varate values from tself or explctly stated, t mmzes S = x c sce S attas ts mmum whe c = x. To obta ths result, oe may = mmze S (the Eucldea orm per se) also. O the other had the meda mmzes the Absolute orm of devatos of the varate from tself, expressed as whch = M = x c yelds c = meda( x). I a geeral framework, we obta arthmetc mea or meda by p mmzg the geeral Mkowsk orm x c for p= or p= respectvely. Ths = vew of the arthmetc mea ad the meda gves them the meag of beg the measures of cetral tedecy. 3. Idetermacy of Meda whe the Number of Values the Sample s Eve: Whe the sample x = ( x, x,..., x ), the umber of observatos,, s odd, the value of meda( x) = ( xk + x+ k ) / ; k = t(( + ) / ) s determate; xk = x + k mmzes the absolute orm, M. However, whe s a eve umber, x k ad x + k are (very ofte) dfferet. As a matter of fact, ay umber z for whch the relatoshp ( xk z x + k ) holds, mmzes the absolute orm of devatos. Thus, the meda s determate. It has bee customary, therefore, that absece of ay other relevat formato, oe uses the prcple of suffcet reaso ad obtas meda( x) = ( xk + x + k ) /. However, t remas a truth that ay umber z for whch the relatoshp ( xk z x + k ) holds, s the value of the meda as much as z = ( xk + x + k ) /. / p

2 4. Meda as a Weghted Arthmetc Mea of Sample Observatos: If x = ( x, x,..., x ) are ordered such that x x... x, t s possble to express meda as a weghted arthmetc mea x jwj wjwhere j = w + j j= j= =.5 for j = t(( + ) / ) else w j = for j t(( + ) / ). However, ths s trval. Now we preset a o-trval alteratve algorthm to obta meda( x ). I order to use ths algorthm t s ot ecessary that the values of x be arraged a ascedg (or descedg) order, that s x x... x codto s relaxed. The steps the algorthm are as follows: () Set w = =,,...,. Obvously, = = w =. xw = () Fd v =, the weghted arthmetc mea of ( x, x,..., x ). w () Fd ew w = f d = x v ε ( ε > s a small umber, say.), d else w =.or ay such small umber; =,,...,. xw = (v) Fd v = w = usg the weghts obtaed () above. (v) If v v τ (where τ s a very small umber, say,. or so, cotrollg the accuracy of result) the v s replaced by v (that s, v s reamed as v ) ad go to step (); else (v) Meda s v ad w = ( w, w,..., w ) are the weghts assocated wth ( x, x,..., x ). Stop. Ths algorthm yelds o-trval weghts w = ( w, w,..., w ). It yelds the meda detcal to that obtaed by the covetoal formula f s odd. If s eve, t gves a umber z : ( xk z x + k ), whch s meda as metoed secto. 5. Some Mote Carlo Expermets: We have coducted some Mote Carlo expermets to study the performace of the alteratve method (weghted arthmetc mea represetato) vsà-vs the covetoal method of obtag meda. Three sample szes (of =, ad 5) have bee cosdered. Samples have bee draw from fve dstrbutos (Normal, Beta, Beta, Gamma ad Uform). I each case, expermets have bee carred out. A success of the alteratve estmator s there f t obtas meda detcal to that obtaed by the covetoal method case s odd ad obtas meda = z : ( xk z x + k ) case s eve. The summary of results s preseted table.

3 3 Dstrb. Uform Gamma Beta Beta Normal Table. Performace of the Alteratve Method to obta Meda Sample Arthmetc Meda Meda Iclato Sze = Mea (Covetoal) (Alteratve) to Mea Success Rate (%) Yes Yes Yes Yes We fd that whe s odd, rrespectve of the dstrbuto or the sample sze both the methods yeld detcal results. Whe the dstrbuto s skewed (.e. there s a sgfcat dvergece betwee meda ad mea) ad s eve, the alteratve meda s slghtly pulled by the mea (ts clato s towards the mea). Ths appears justfed because t s expected that the values lyg betwee x k ad x + k (for x = ( x, x,..., x ) : x x... x ; k = t(( + ) / ) ) must be more desely dstrbuted the sde of the mea. The covetoal method, however, cosders them uformly dstrbuted wat of formato. The alteratve method appears to explot the formato cotaed the sample. 6. Aalyss of Iclato of Computed Medas to Mea Value: We have see that whe s a eve umber, the values of meda estmated by the two methods dffer ad the oe estmated by the alteratve (weghted arthmetc mea) method appears to be pulled towards the mea value, x. The, a questo arses : s the meda estmated by the alteratve method based (towards the mea)? To vestgate to ths questo, we geerate some a values ( our expermet 8) of v such that ( xk v x + k ), ad v follows the dstrbuto detcal to that of x. We do t aga ad aga for a large umber of tmes ( our expermet,,). We cout as to how may tmes the v < the meda values obtaed by the two competg methods. The probablty of v =computed medas s very small ( our expermet we ever ecoutered equalty). Table- clearly shows that case of Gamma ad Beta dstrbutos both medas are pulled by mea, though the meda obtaed by the alteratve method s more cled to mea. The pull s stroger case of the Gamma dstrbuto, sce t s more skewed tha the Beta dstrbuto. I case of ormal dstrbuto we fd the opposte tedecy (push). I case of uform dstrbuto o pull or push force s observed, whle case of Beta dstrbuto a mxed observato s there.

4 4 Table. Iclato of the Competg Methods to the Mea Value Dstrb. Sample a (o. of Meda Iclato Meda Iclato Sze = v values (Covetoal) to Mea (Alteratve) to Mea geerated) Uform No No No.588 No Gamma Yes Yes Yes Yes Beta 8.54 No.578 No No Yes Beta Yes Yes Yes Yes Normal Yes Yes Yes Yes + pull; + + stroger pull; - push; - - stroger push 7. Relatve Effcecy ad Cosstecy of the Competg Methods: Now suppose we geerate a large ( our expermet 5) umber of varate values followg a specfed dstrbuto. Let us call the collecto of these values U or the Uverse. We may obta the Meda(U) = µ, say. Ths value may ot be the true meda of the dstrbuto (or f U were very large), but t s lkely to be very close to that. Dstrb. Uform Gamma Beta Beta Normal Table 3. Effcecy of the Competg Methods to obta Meda Sample True Computed Sze = Meda Meda Norm Computed Meda Norm U(5) ( m ) (ref. m ) ( m ) (ref m )

5 5 From U we may draw some ( our case, 5 ad 9) radom values, say x = ( x, x,..., x ), compute medas ( m ad m ) by the two competg methods (respectvely) aga ad aga. I our case, tral=, wth replacemet. I each draw, the computed medas wll dffer from µ. From ths, we may obta the orms for each meda. These orms would suggest whch meda s most frequetly closer to µ. Symbolcally, tral p p, t ; = t = for tradtoal Normt = m µ t = for alteratve We have used the absolute orm (p = the formula defg orm). The results of the expermets are gve table 3. We observe that for Uform, Normal ad Beta dstrbutos orm s smaller tha orm. For Gamma ad Beta dstrbutos the opposte s true. We also observe that the orms are smaller for larger values of, dcatg to cosstecy. 8. Asymmetry of Dstrbuto ad Effcecy of the Competg Methods: It s well kow that the Gamma dstrbuto s severely skewed for small shape parameters, but wth the creasg value of that parameter, the dstrbuto teds to become symmetrc. Table 4. Asymmetry of Dstrbuto ad Effcecy of the Competg Methods Dstrbuto Sample True Computed Gamma Sze = Meda Meda Norm Computed Meda Norm (shape parameter) U(5) ( m ) (ref. m ) ( m ) (ref m ) Gamma(.5) Gamma(.) Gamma(.) Gamma(4.) Gamma(8.) Gamma(6.) Gamma(5.)

6 6 Table 4 shows the relatve orms for the competg methods due to creasg values of the shape parameter of the Gamma varate. We observe that orm becomes uformly smaller (tha orm ) whle the shape parameter reaches 6. Ths expermet reforces our cocluso that the alteratve method of obtag meda s better tha the covetoal method whle the dstrbuto s less asymmetrc. 9. Cocluso: Ths study establshes that meda may be expressed as a weghted arthmetc mea of all sample observatos. If the covetoal formula does ot corporate all sample values, t s the property of the specfc method of computato ad ot of meda per se, as ofte alleged to t. If our expermets covey somethg, the we may also state that for relatvely more symmetrc dstrbutos the alteratve formula (weghted mea) performs better tha the covetoal method. But for heavly asymmetrc dstrbutos the covetoal method of computg meda performs better, although both the methods yeld based estmates. The alteratve algorthm of computato s easly exteded to other meda type estmators - such as Least Absolute Devato (LAD) estmator of the regresso model y = X β + u - as show by Far (974) ad Schlossmacher (973). Refereces Far, RC (974). O the Robust Estmato of Ecoometrc Models, Aals of Ecoomc ad Socal Measuremet, 3 ( ). Schlossmacher, EJ (973). A Iteratve Techque for Absolute Devatos Curve Fttg, Joural of the Amerca Statstcal Assocato, 68 ( ).