1. Consider the Bradley-Terry model with home advantage (BTHA) which is defined as v di vdi+d j X₁ di + vdj P(i beats j) = if i is at home, if j is at home, where > 0 measures home advantage (> 1) or disadvantage ( < 1), and A₁, A, are the ability parameters for competitors i and j, respectively. It can be shown that the log likelihood of A, based on results of n independent matches is K l(A, y) = H log(v) + Σw; log (A;) - ΣΣn¡j log(vλ¡ +Aj), i=1 i=1 ji K (1) where w; denotes the number of wins for i, H denotes the total number of home wins, and nij denotes the number of times that i plays at home against j.

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Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer.

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1. Consider the Bradley-Terry model with home advantage (BTHA) which is defined as
P(i beats j)
& di
vdi + Aj
Xi
Ai + vλ j
if i is at home,
if j is at home,
where > 0 measures home advantage (½ > 1) or disadvantage (½ < 1), and \¡, λj are
the ability parameters for competitors i and j, respectively.
It can be shown that the log likelihood of A, & based on results of n independent matches
is
K
l(A, y) = H log(y) + Σw; log(A;) - ΣΣn¡¡ log(VA¡ + Aj),
i=1 ji
K
(1)
where w, denotes the number of wins for i, H denotes the total number of home wins,
and nij denotes the number of times that i plays at home against j.
Transcribed Image Text:1. Consider the Bradley-Terry model with home advantage (BTHA) which is defined as P(i beats j) & di vdi + Aj Xi Ai + vλ j if i is at home, if j is at home, where > 0 measures home advantage (½ > 1) or disadvantage (½ < 1), and \¡, λj are the ability parameters for competitors i and j, respectively. It can be shown that the log likelihood of A, & based on results of n independent matches is K l(A, y) = H log(y) + Σw; log(A;) - ΣΣn¡¡ log(VA¡ + Aj), i=1 ji K (1) where w, denotes the number of wins for i, H denotes the total number of home wins, and nij denotes the number of times that i plays at home against j.
2.
The likelihood function under the Bradley-Terry model
with home advantage can be written
hij
di
ZA) -ĤIII(A)(^.^).
L(A, &) =
+ λ;
K
aij
+ j
where hij denotes the number of home wins for i against j and aij denotes the number of
away wins for i against j. Show that the log likelihood is given by the expression in (1).
i=1 ji
Transcribed Image Text:2. The likelihood function under the Bradley-Terry model with home advantage can be written hij di ZA) -ĤIII(A)(^.^). L(A, &) = + λ; K aij + j where hij denotes the number of home wins for i against j and aij denotes the number of away wins for i against j. Show that the log likelihood is given by the expression in (1). i=1 ji
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