Anhang A

Herleitung des multinomialen Logit-Modells aus der Gumbel-Verteilung

Für die Gumbel~~Verteilung gelten folgende Eigenschaften

l. Der Modus ist 17·

2. Der Mittelwert ist 17 + 1/ p, mit 1 als Euler'sche Konstante.

3. Die Varianz beträgt 1r2 /6p,2 •

4. Wenn c gumbelverteilt ist mit den Parametern (17, f.t), W und a > 0 seien Skalare, dann ist auch m + W gumbelverteilt mit den Parametern (a17 + W,f.t/a), d. h. die Eigenschaften der Gumbel-Verteilung sind invariant gegenüber linearen Transfor­mationen.

5. Wenn c1 und c2 unabhängig gumbelverteilt sind mit den Parametern (171,f./,) bzw. (172 , f.t), dann ist c• = ( 1 - ~:2 logistisch verteilt mit

(A.l)

6. Wenn ~: 1 und (2 unabhängig gumbelverteilt sind mit den Parametern (171,1-") und (172, f.t), dann ist max(t1, t 2 ) auch gumbelverteilt und zwar mit den Parametern (1/p,ln(exp(J.L17d + exp(f.t172)),Jt).

7. Aus der Eigenschaft 6 folgt, dass wenn t1, ~: 2 , ... , t'J J unabhängige gumbelverteil~ te Zufallsvariablen mit den Parametern (171, f.t), (172 , f.t), ... , (17J. f.t) sind, dass auch max(t1, t2, ... , tJ) gurnbelverteilt ist mit den Parametern (1/ f.t, ln ~f=1 exp(f.t17i), f.t).

Gleichung (4.3.8) kann aus den Eigenschaften der Gumbel-Verteilung wie folgt hergeleitet werden. Für c gumbelverteilt gilt

F(() ~ cxp(- cxp( -J.L(t -17))), f.t > 0 (A.2)

und die Dichtefunktion f lautet

(A.3)

186 Herleitung des multinomialen Logit-Modells

mit 1) als Lokations-und 1~ als positivem Skalenparameter (Hen-Akiva & Lerman, 1985).

Aus den zuvor dargestellten Eigenschaften der Gumbcl-·Vert.eilung kann nun die Glei-chung (4.3.8) hergeleitet werden. Zur Vereinfachung gelte TJ 0. Außerdem werden o. B. d. A. die Alternativen so angeordnet., dass i - 1 gilt. Nun folgt

Prn(1) = Pr[V!n + f]n 2': . rpax (V;n I C1n)J. J c 2, ... ,./n

(A.1)

Es sei (A.5)

Da t 1n gumbelverteilt ist mit den Parametern (0, /.!), ist auch fjn + V;n gumbel­verteilt und zwar mit den Parametern (V,/.l) gemäß Eigenschaft 1. Aus der Eigen­schaft 7 der Gurnbel-Verteilung folgt, dass U~ gumbelverteilt ist mit den Parametern ((1/ /.l) ln I;j;"2 exp(/.l\l;n), 1-l)· Unter Verwendung der Eigenschaft 4 gilt für U~ ~·- v,: t t;,

(A.6)

und t~ ist gumbelverteilt mit den Parametern (0,/.l)· Nun kann aus Gleichung (A.4) ge­folgert werden

(A.7)

Sowohl t 1n als auch t~ sind gumbelverteilt mit den Parametern (0, /.l)· V,;' t c~ ist gurnbel­verteilt mit den Parametern (V,;',/~) und Vin + t 1n ist gnrnbelverteilt mit. (Vin,ft). Gemäß Eigenschaft 5 gilt nun für Gleichung (A. 7)

1 Pr n ( 1) ~· -1-~--e-xp-(:-/.l..,.-(v.-,;---~~,-ln-))

exp(/.lVIn) + exp(/.lV,7)

exp(f~ Vin)

exp(pYin) t exp(ln I;j" 2 exp(ft\l;n))

exp(f~Vln)

(A.S)

Damit ist die Gültigkeit der Gleichung (4.3.8) gezeigt. Für einen anderen Weg der Her­leitung siehe auch Cramer (1991).

Anhang B

Die Schätzung der Loyalitätsvariablen gemäß Fader et al. 1992

Die Methode von Fader et al. (1992) zur Schätzung des Loyalitätsparameters basiert auf einer Taylorreihenentwicklung. Die r--te erklärende Variable Xintn die nichtlinear abhängig ist von einem Parameter, hier .X, wird zur Vereinfachung x(t, .X) genannt. Für sie wird am Startpunkt .X0 eine Taylorreihe entwickelt mit

( .X) = ( .X ) 8x(t, .X) I (.X- .X ) ~ anx(t, .X) I (.X- .Xo)n X t, X t, 0 + ß,X 0 + ~ ß,Xn n!

..\o n=2 ..\o

(B.l)

Unter der Annahme, dass x(t, .X) glatt sei (ihre Ableitungen bzgl. .X seien beschränkt) in einem Intervall, dass sowohl .X0 als auch die Maximum-Likelihood-Schätzung von .X ent­halte, gehen die Terme zweiter und höherer Ordnung aus Gleichung (B.l) gegen Null, wenn .X0 gegen seine Maximum-Likelihood-Schätzung strebt. Damit lässt sich Gleichung (R.J) unter Verwendung von x'(t, .X)= 8x(t, .X)/8-X vereinfachen zu

x(t, .X) -::e x(t, .Xo) + x'(t, .Xo)(.X- .Xo). (B.2)

In Gleichung (B.2) gilt die Gleichheit bei der Konvergenz von .X0 zu .X. Unter Einbeziehung des Koeffizienten ß der erklärenden Variable x(t, .X) aus dem multinomialen Logit-Modell wird Gleichung (B.2) zu

ßx(t, .X) -::e ßx(t, .Xo) + ß(.X- .Xo)x'(t, .Xo). (B.3)

Aus Gleichung (B.3) wird ersichtlich, dass der Term ßx(t, .X) durch die Terme x'(t, .X0 )

und x(t, .X0 ) in der Nutzenfunktion des Logit-Modells repräsentiert werden kann. Die resultierenden Schätzungen können auch mit ß' bzw. ß bezeichnet werden. Damit wird Gleichung (B.3) zu

ßx(t, .X) -::e ßx(t, .Xo) + ß'x'(t, .Xo). (B.4)

Aus dem Vergleich von Gleichung (B.3) und Gleichung (B.4) folgt

ß' -::e ß(.X- .Xo) (B.5)

bzw. .X -::e .Xo + ß' / ß. (B.6)

188 Die Schät;;;tmg der Loyalitätsvariablen

Zur Bestimmung der Gewiehtungsparameter .\aus der Gleichung ( 5.:l.l) wird zunächst die erste Ableitung D LOY benötigt. Die Gleichung ( 5.3.1) kann umgeschrieben werden zu

t~l

LOY.nt = (1 - .\) L A1 Yint~l~1· [.]

(H.7)

Die Ableitung der Gleichung (B. 7) nach .\ lautet

DLOY.nt = .\DLOlint~I I LOlint~I- Yint~l· (B.8)

Zur lnitialisierung wird LOY;n1 - l/ J gesetzt mit .J als Anzahl der Marken, und es gilt DLOY.n1 = 0.

Unter Verwendung der Taylorapproximat.ion für x(t, .\) gelt.en für den nichtlinearen Schätzlogarithmus die folgenden Schritte

l. Wähle Startwerte für .\ als .\0 .

2. Berechne x(t, .\0 ) und x'(t, .\0 ) am Punkt .\0 für alle Beobacht.ungcn zum Zeitpunkt t.

3. Beziehe x(t, .\0 ) und x'(t, .\0 ) mit allen anderen erklärenden Variablen in das Logit~ Modell mit ein und bestimme die Koeffizienten durch eine gewöhnliche Schätzung.

4. Aktualisiere .\0 gemäß Gleichung (1:3.5) mit .\0 <--- (.\0 + ß' /ß).

5. Gehe zurück zu Schritt 2 und führe die Iteration durch bis .\0 konvergiert bzw. der Koeffizient von x' nicht mehr unterscheidbar ist von Null.

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