SOME NOTES ON THE LOGISTIC DISTRIBUTION

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1 Quatitativ Mthods Iquirs SOME NOES ON HE OGISIC DISRIBUION Aladru ISAIC-MANIU PhD, Uivrsity Prossor, Dpartmt o Statistics Ecoomtrics Uivrsity o Ecoomic Studis, Bucharst, Romaia (CoAuthor o th books: Proictara statistica a primtlor (6, Eciclopdia calitatii (5, Dictioar d statistica grala (3, Statistica ptru maagmtul aacrilor (, Mtoda Wibull (983 al.isaic-maiu@csi.as.ro, Wb pag: Viorl Gh. VODĂ PhD, Sior Scitiic Rsarchr, Gh. MIHOC - C. IACOB Istitut o Mathmatical Statistics ad Applid Mathmatics o th Romaia Acadmy, Bucharst, Romaia Co-author o th books: Proictara statistica a primtlor (6, Dictioar d statistica grala (3, Maualul calitatii (997 vo_voda@yahoo.com Abstract: I this papr w study som proprtis o th dsity uctio ( + ( ; ( + th last ordr statistic rom a rducd logistic populatio ( / ( +, R, R, > which may b obtaid rom th distributio o trucatd variat is also discussd. h poit ad itrval stimatios or ar providd also. h so-calld ( P,γ - typ statistical tolracs ar costructd ad a commt o th hazard rat is do also. h last paragraph is dvotd to tstig procdurs o th paramtr ivolvd.. Its Ky words: logistic distributio; Burr-Hatk amily; last ordr statistic; trucatio; ME; (P, γ - typ tolracs. Itroductio As it is wll-kow, th Blgia scitist Pirr raçois VERHUS ( proposd i 838 a dmographic growth cur which was calld latr as th logistic uctio: A A Y or Y ( a+ bx a+ bx B + B + whr, a,b >, a,b R, big th Eulr s umbr (,788. h usual orm usd ow i coomtric studis is th ollowig : 3

2 Quatitativ Mthods Iquirs A Y, a, B, C >, CX B + ( (or historical dtails, s Iosiscu t al, 985 [8, pags 8-8]., Vrhulst s uctio is obtaid rom a dirtial quatio o th typ y u( y whr y y(, u(, R, by takig u( y y y. y hror, w hav: dy d ( y y or dy y d (3 ( y which provids th rducd orm o th logistic uctio: y, R + I w cosidr y( as a cumulativ distributio uctio (cd o a giv radom X : y Pr ob X, th dirtial quatio: variabl ( { } d d ( g(, (, R whr g( is a arbitrary positiv uctio ( R, provids - or svral choics o g( - various cd(s. h orm (4 grats th wll-kow BURR-HAKE amily o distributios - s Burr (94 [] ad Hatk, (949 [6]. Johso, Kotz ad Balkrisha (994 [4, pag 54] lists twlv such cd(s, dotd rom I to XII, th scod o big just th Vrhulst distributio uctio (4 i o taks g(. Notic that th most usd cd rom ths c twlv cd(s is XII-th o: ( +,, c,k > k (s Rodriguz, 977 [6] or Vodă, 98 [8]. I this papr, w shall ivstigat maily, th dsity uctio o th last ordr statistics rom th logistic orm (4.. Prlimiaris X as a cd ad i X( X( X( k X( + X ( is a ordrd sampl o X, th, th distributio o th last ordr statistics X ( is giv as: I X is a cotiuous radom variabl with ( ( ( ( { X } Pr ob{ all X } ( Pr ob (6 X (s David, 97, [3 pag 38]. I our cas, i X ( ( / ( + ( / ( + R, N \ { } ( X k, th X (4 (5. It is mor itrstig to chag by - a positiv ral paramtr ad th w obtai th cd: X : ( ;, R, > (7 + ( 33

3 Quatitativ Mthods Iquirs I a rliability ram w prr to hav [,+ dsity o X amly: X : ( ; ( ( + +, R, > ( ; ad hc it is worth to trucat th as ollows: ( ;, ( ;,, > (9 or ( ( + ;, ( + I th trucatio poit is, th: / - ( ( + +,, > - ( ;, ( ( + +,, > 3. Estimatio problms (8 ( ( irst, lt us otic that v or th rducd orm (4 th mthod o momts is diicult to b applid. I w cosidr th orm ( cosidr th itgral: k m k ( t ( t dt ( which provids th quatio (th drivativ o (: k ( ( m k (3 ad th k - th momt is giv as: E k k ( X ( d (4 It is asy to s ow that E( X m( m( lis i th approimatio o m (. with m ( ad th problms h ME - maimum liklihood stimatio - mthod givs straightorward rsults i th cas o (7. W hav th liklihood uctio ( ( ( + i,, ; p i + (5 whr,, is a radom sampl o X. Atr som simpl algbra, w obtai th ME or / as: ˆ i l( + h cas o th trucatd variabl ( provids succssivly: (,, ; ( ( ( + i p i + (7 (6 34

4 Quatitativ Mthods Iquirs i ( + l i ( + l( + l l l (8 i ( l l l + l + (9 l l i ( + l ( h liklihood quatio ( is o th ollowig typ: A B ϕ ( u C ( u u whr A, B, C > ad it is clar that it has a solutio scic lim ( u + ad limϕ( u C ϕ. u + u + Numrical mthods ar dd to approimat (u. W shall stat ow th ollowig. Propositio. I X is a logistic radom variabl with cd giv by (7, th th variabl Y l( + is potially distributd ad th cosqucs ar: (i th ME or / is ubiasd ad with miimum variac; (ii th distributio o ˆ is a Gamma o; (iii th statistic U / ˆ has a Chi - squar ( χ distributio with dgrs o rdom. Proo. W hav immdiatly: z ( z Pr ob{ l( + z} Pr ob{ < l( } z l( ( + d ( akig ito accout that ( b z ( d b ( z ϕ[ b( z ] a ( z ϕ[ a( z ] ϕ (3 a( z w obtai rom (: 35

5 Quatitativ Mthods Iquirs ( z p( z with z, > (4 Sic ˆ ˆ E ad Var (5 th proprty (i is provd. Now, sic ach variabl is potially distributd, th its sum is Gamma distributd (s Johso t al, 994 [4, pag 337 ad pag 494]. h dsity o ˆ is thror ˆ ϕ ;, Γ ˆ ( p ˆ whr >, >, N \ {} ad Γ( ( +! ˆ (6 is th Gamma uctio; hr w hav Γ (th proprty (ii is also provd. o dmostrat (iii, w may writ th charactristic uctio o U, ad w hav: ϕ U itu j ( t E( E pi( t l( + λt ( it j which is just th charactristic uctio o a Chi-squar variabl with dgrs o rdom (s or istac Wilks, 96 [ pag 86]. Basd o this proprty, o ca costruct coidc itrvals o miimum lgth ( or. Namly, w hav to dtrmi two limits i ad sup such that Pr ob{ < ˆ i / < sup} α (8 whr < α <, α - giv, with th proprty (9 ( sup i Y i ( + Y l. j (7 miimum, whr I accordac with at ad Kltt`s rsults (959, [7] th ollowig systm has to b solvd: sup sup sup u / i ( ( u du α Γ ; p (3 i i h solutios ( i, sup ar oudd by trig at - Kltt`s tabls i th cll corrspodig to dgrs o rdom (s also Isaic-Maiu ad Vodă, 989 []. 36

6 Quatitativ Mthods Iquirs 4. (P, γ - typ tolrac limits h litratur dvotd to th problm o statistical (or atural tolrac is vry wid. I 98, Miloš Jílk (Pragu compild a larg bibliography o this subjct mattr (s Jílk [] ad i 988 th sam author (s Jílk [3] providd a soud moograph with applicatios (chaptr, pags o [3] o ths tolracs. Origi o this cocpt - which gos back to W. A. Shwhart (89-967, Samul S. Wilks (96-964, Abraham Wald (9-95, Hrbrt Robbis ( ad som othr (historical aspcts may b oud i [7]. h mathmatical ormulatio o th problm is th ollowig: i X is a cotiuous radom variabl did o D R ad havig a dsity uctio ( ;, th w hav to costruct two statistics ad U (lowr ad uppr such that at last a proportio (P o this populatio {X} will b oud btw ad U, ad this must happ with a giv γ < γ <, that is. probability ( Pr ob u whr P, γ < U ( ; d P γ (3 < ar prviously chos. hs lmts ϕ(,, Ψ,, whr { i} ar sampl valus o X ar calld (,γ i ad P - typ tolrac limits. I a rliability cott o is itrstd i a lowr tolrac limits sic X : D [,+ ad w d that at most a proportio ( - P o th populatio to li btw ad. I our logistic cas, w shall writ: statistic Y Pr ob + ( ; d P γ (3 wo ways to dduc will b prstd. a h cas o larg sampls I this situatio, w may stat that th ˆ (33 whr ˆ is th ME o /, is approimatly ormally distributd with E [Y] ad Var [Y]. h rlatioship (3 may b writt as. Pr ob or { ( + P} γ (34 37

7 Quatitativ Mthods Iquirs { l( P l( + } γ Pr ob (35 ( + l Pr ob γ l( P which may b rarragd as ollows: ˆ Pr ob / / ( + l l / ( P (w did rplac i (36 by its ME. γ (36 (37 h right-had sid o th iquality i (37 is just th γ - quatil o N (, distributio - lt it b or l ( + ( l l u γ - ad hc w may writ: ˆ u + + ( γ P ˆ + u γ l P (38 (39 Sic < <, th w could us th approimatio o l (+, < < giv i Abramowitz ad Stgu (964, [] to id a polyomial quatio i : 5 ( + a + a + + a5 + ε( 5 whr th rror ( ( l (4 ε is ε < ad a a.s.o. O may rstrict th approimatio to th roughst o, amly: ( a l + (4 ad hc may b dducd asily by takig logarithms i (39. W did call this mthod to id statistical tolracs as ormalizig o (s Isaic- Maiu ad Vodă, 98, [9], ad 993, []. b h gral cas. Wh w hav a arbitrary sampl siz, w may r-writ (35 as ollows: Pr ob ˆ ME l P l ( + i i ( + γ l (4 38

8 Quatitativ Mthods Iquirs Sic ˆ / ME is Chi-squar ( χ distributd with dgrs o rdom, th righ-had sid i th iquality o (4 is just th γ - quatil o th ( χ distributio - lt it b ( χ ;γ. hror, a similar quatio with (39 is obtaid: i ( l l( + l + χ P ; γ i (43 5. A discussio o th hazard rat I X is a cotiuous radom variabl rprstig th tim-to-ailur o a crtai dvic, th th hazard (or ailur rat associatd to X is giv as: ( ( h ( whr ( is th cd o X ad ( ( (44 is th corrspodig dsity. h variabl X is assumd to b positiv. I our cas w hav to work with th trucatd variabl. Suppos that our logistic is,+. hror: trucatd at, that is [, ( ; ( ; whr K /( ( u; du K [ ( ; ( ] ad ( ; /. I this situatio, w hav: (45 or ( + ( + ( ( ; h ; (46 ( ; ( ( ( + ; - ( + + h (47 h may b prormd dotig y (w hav < y < ad cosqutly, w gt. K y + ( ( + y h y ; (48 K + y or h study o ( ; ( 39

9 Quatitativ Mthods Iquirs h K y y, < y (49 ( ; [( + y K]( + y < A itrstig situatio ariss wh w cosidr th rducd orm o th logistic ( / ( +, R ad writ ormally: d ( d ( h ( (5 I this cas, th hazard rat coicids with th distributio uctio ad th bhaviour o h( is ow obvious. 6. stig a simpl statistical hypothsis Sic oly o paramtr is ivolvd i ( ;, w may mak us o th act that i ( + U l ˆ (5 M i is Chi-squar distributd with dgrs o rdom ad hc to tst: h : vrsus h : ( < (5 th critical valu will b χ ( α - quatil o th χ distributio with dgrs o α ; rdom. Mor coomical is to apply SPR (Squtial Probability Ratio st o Abraham WAD ( s his wll - kow book [ 9, Chaptr 3, pags 37-54]. W shall writ straightorwardly th logarithm o liklihood ratio (r : l r whr ( i,,,, i l + (53 i ( l( + i is th squtial sampl. I accordac with Wald`s ruls [9, pag 49] w could tak th ollowig dcisios: a I i l i ( + l A l + w accpt (H ad automatically rjct th altrativ (H : (54 b I i l i ( + l B l + th rjct th ull hypothsis (H ad accpt (H (55 4

10 Quatitativ Mthods Iquirs c I l l l B l A i + < l( + < + (56 i th, th procdur cotius by takig th t obsrvatio. Hr A ( β / α ad B β/ ( α, α, β big th classical statistical risks i th thory o hypothsis tstig (s Wilks, 96 []. h OC - uctio (th Oprativ Charactristic o th tst is giv by: h h h ( ( A /( A B whr h is th solutio o Wald`s quatio (, h, ( ; ( ; E th z l. I our cas: z l + l + ad cosqutly: ad zh ( ( h ( ( + + zh h [ ] ( ( + d E h Imposig th coditio > h(, w gt a paramtric rprstatio o th OC - uctio as h( (6 h h ASN (Avrag Sampl Numbr dd to prorm SPR is giv b E( { ( l B + [ ( ] l A} / E( z whr - i o cas: E( z l + ( hror, th squtial tst is compltly costructd. W did ot r-statd th whol thory - all dtails ar giv i Wald [9], Dio ad Massy (97, [4, pags 3-3] or i mor rct works such as thos o Govidarajulu ( [5] or Pham (6, [5]. (57 (58 (59 4

11 Quatitativ Mthods Iquirs Bibliography. Abramowitz, M. ad Stgu, Ir, A. Hadbook o Mathmatical uctios, NBS Applid Math. Sris. Nr. 55, Washigto, D. C. (Russia ditio, 979, Moscow, NAUKA, 964. Burr, I. W. Cumulativ rqucy uctios, Aals o Mathmatical Statistics, vol. 3, 94, pp David, H. A. Ordr Statistics, Joh Wily ad Sos Ic, Nw York, Dio, W. J. y Massy,. J. (Jr Itroduccio al Aalisis Estadistico (sguda dicio, rvisado por D. Sito Rios; traducido par J. P. Vilaplaa, Edicio Rvolucioaria, Istituto Cubao dl ibro a Habaa, Cuba, Govidarajulu, Z. Statistical chuiqus i Bioassay (scod, rvisd ad largd ditio, KARGER Publishrs, Basl, Switzrlad, 6. Hatk, M. A. (Sistr A crtai cumulativ probability uctio, Aals o Mathmatical Statistics, vol., 949, pp Iliscu, D. V. ad Vodă, V. Gh. Statistica si tolrat (Statistics ad olracs - i Romaia, Ed. hica, Bucharst, Iosiscu, M., Moiagu, C., rbici, Vl. ad Ursiau, E. Mica Eciclopdi d Statistică (ittl Statistical Eciclopdia- i Romaia, Editura Stiitiica si Eciclopdica, Bucharst, Isaic-Maiu, Al. ad Vodă, V. Gh. olrac limits or th Wibull distributio usig ME or th scal paramtr, Bull. Math. Soc. Sci. Math d la R.S. d Roumai (Bucarst, om 4(7, o. 3, 98, pp Isaic-Maiu, Al. ad Vodă, V. Gh. Estimara durati mdii d viata a produslor pri itrval d îcrdr d lugim miima (Estimatio o product avrag li via miimum lgth coidc itrvals - i Romaia, Stud. Crc.Calc.. Eco. Cib. Eco. Aul XXIII,o., 989, pp Isaic-Maiu, Al. ad Vodă, V. Gh. O a spcial cas o Sdrakia class o distributios applicatios, Eco. Comp. Eco. Cyb. Stud. Rs. (Bucharst, Vol. XXVII, o. - 4, 993, pp Jilk, M. A bibliography o statistical tolrac rgios, Mathmatisch Opratios orschug ud Statistik, Sr. Statistics (Brli, vol., 98, pp Jilk, M. Statistick olraci Mz. (Statistical olrac imits - i (czch, SN/KI, Praha, Johso, N.., Kotz, S. ad Balakrisha, N. Cotiuous Uivariat Distributios, Vol. (Scod Editio, Joh Wily ad Sos Ic., Nw York, Pham, H. (ditor Sprigr Hadbook o Egirig Statistics, Sprigr Vrlag, Brli, 6 6. Rodriguz, R. N. A guid to th Burr yp XII distributios, Biomtrika, vol. 64, o., 977, pp at, R.. ad Kltt, G. W. Optimum coidc itrvals or th variac o a ormal populatio, J. Amr. Statist. Assoc., vol. 54, 959, pp Vodă, V. Gh. (98: Burr distributio rvisitd, Rv. Roum. Math. Purs t Appl., om XXVII, o. 8,98, pp Wald, A. Squtial Aalysis. Dovr Publicatios Ic. Nw York (Studt ditio: uabridgd rpublicatio o th origial (947 - Joh Wily ad Sos Ic., 973. Wilks, S. S. Mathmatical Statistics (Scod Editio, Joh Wily ad Sos Ic., Nw York, 96 Codiicatio o rrcs: [] Abramowitz, M. ad Stgu, Ir, A. Hadbook o Mathmatical uctios, NBS Applid Math. Sris. Nr. 55, Washigto, D. C. (Russia ditio, 979, Moscow, NAUKA, 964 [] Burr, I. W. Cumulativ rqucy uctios, Aals o Mathmatical Statistics, vol. 3, 94, pp. 5-3 [3] David, H. A. Ordr Statistics, Joh Wily ad Sos Ic, Nw York, 97 4

12 Quatitativ Mthods Iquirs [4] Dio, W. J. y Massy,. J. (Jr Itroduccio al Aalisis Estadistico (sguda dicio, rvisado por D. Sito Rios; traducido par J. P. Vilaplaa, Edicio Rvolucioaria, Istituto Cubao dl ibro a Habaa, Cuba, 97 [5] Govidarajulu, Z. Statistical chuiqus i Bioassay (scod, rvisd ad largd ditio, KARGER Publishrs, Basl, Switzrlad, [6] Hatk, M. A. (Sistr A crtai cumulativ probability uctio, Aals o Mathmatical Statistics, vol., 949, pp [7] Iliscu, D. V. ad Vodă, V. Gh. Statistica si tolrat (Statistics ad olracs - i Romaia, Ed. hica, Bucharst, 977 [8] Iosiscu, M., Moiagu, C., rbici, Vl. ad Ursiau, E. Mica Eciclopdi d Statistică (ittl Statistical Eciclopdia- i Romaia, Editura Stiitiica si Eciclopdica, Bucharst, 985 [9] Isaic-Maiu, Al. ad Vodă, V. Gh. olrac limits or th Wibull distributio usig ME or th scal paramtr, Bull. Math. Soc. Sci. Math d la R.S. d Roumai (Bucarst, om 4(7, o. 3, 98, pp [] Isaic-Maiu, Al. ad Vodă, V. Gh. Estimara durati mdii d viata a produslor pri itrval d îcrdr d lugim miima (Estimatio o product avrag li via miimum lgth coidc itrvals - i Romaia, Stud. Crc.Calc.. Eco. Cib. Eco. Aul XXIII,o., 989, pp [] Isaic-Maiu, Al. ad Vodă, V. Gh. O a spcial cas o Sdrakia class o distributios applicatios, Eco. Comp. Eco. Cyb. Stud. Rs. (Bucharst, Vol. XXVII, o. - 4, 993, pp [] Jilk, M. A bibliography o statistical tolrac rgios, Mathmatisch Opratios orschug ud Statistik, Sr. Statistics (Brli, vol., 98, pp [3] Jilk, M. Statistick olraci Mz. (Statistical olrac imits - i (czch, SN/KI, Praha, 988 [4] Johso, N.., Kotz, S. ad Balakrisha, N. Cotiuous Uivariat Distributios, Vol. (Scod Editio, Joh Wily ad Sos Ic., Nw York, 994 [5] Pham, H. (ditor Sprigr Hadbook o Egirig Statistics, Sprigr Vrlag, Brli, 6 [6] Rodriguz, R. N. A guid to th Burr yp XII distributios, Biomtrika, vol. 64, o., 977, pp [7] at, R.. ad Kltt, G. W. Optimum coidc itrvals or th variac o a ormal populatio, J. Amr. Statist. Assoc., vol. 54, 959, pp [8] Vodă, V. Gh. (98: Burr distributio rvisitd, Rv. Roum. Math. Purs t Appl., om XXVII, o. 8,98, pp [9] Wald, A. Squtial Aalysis. Dovr Publicatios Ic. Nw York (Studt ditio: uabridgd rpublicatio o th origial (947 - Joh Wily ad Sos Ic., 973 [] Wilks, S. S. Mathmatical Statistics (Scod Editio, Joh Wily ad Sos Ic., Nw York, 96 43