Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

8.5.∗Minimax and Classification Procedures 511

choose the distribution indexed byθ′′; that is, we classify (x, y)ascomingfromthe
distribution indexed byθ′′. Otherwise, choose the distribution indexed byθ′;that
is, we classify (x, y) as coming from the distribution indexed byθ′. Some discussion
on the choice ofkfollows in the next remark.

Remark 8.5.1(On the Choice ofk).Consider the following probabilities:

π′ = P[(X, Y) is drawn from the distribution with pdff(x, y;θ′)]

π′′ = P[(X, Y) is drawn from the distribution with pdff(x, y;θ′′)].

Note thatπ′+π′′= 1. Then it can be shown that the optimal classification rule
is determined by takingk=π′′/π′; see, for instance, Seber (1984). Hence, if we
have prior information on how likely the item is drawn from the distribution with
parameterθ′, then we can obtain the classification rule. In practice, it is common
for each distribution to be equilikely, in which case,π′=π′′=1/2 and, hence,
k=1.

Example 8.5.2.Let (x, y) be an observation of the random pair (X, Y), which has

a bivariate normal distribution with parametersμ 1 ,μ 2 ,σ^21 ,σ^22 ,andρ. In Section 3.5

that joint pdf is given by

f(x, y;μ 1 ,μ 2 ,σ 12 ,σ 22 )=

1

2 πσ 1 σ 2

√

1 −ρ^2

e−q(x,y;μ^1 ,μ^2 )/^2 ,

for−∞<x<∞and−∞<y<∞,whereσ 1 > 0 ,σ 2 > 0 ,− 1 <ρ<1, and

q(x, y;μ 1 ,μ 2 )=

1

1 −ρ^2

[(

x−μ 1

σ 1

) 2

− 2 ρ

(

x−μ 1

σ 1

)(

y−μ 2

σ 2

)

+

(

y−μ 2

σ 2

) 2 ]

.

Assume thatσ 12 ,σ 22 ,andρare known but that we do not know whether the respective

means of (X, Y)are(μ′ 1 ,μ′ 2 )or(μ′′ 1 ,μ′′ 2 ). The inequality

f(x, y;μ′ 1 ,μ′ 2 ,σ^21 ,σ^22 ,ρ)

f(x, y;μ′′ 1 ,μ′′ 2 ,σ^21 ,σ^22 ,ρ)

≤k

is equivalent to

1

2 [q(x, y;μ

′′

1 ,μ

′′

2 )−q(x, y;μ

′

1 ,μ

′

2 )]≤logk.

Moreover, it is clear that the difference in the left-hand member of this inequality
does not contain terms involvingx^2 ,xy,andy^2. In particular, this inequality is the
same as

1

1 −ρ^2

{[

μ′ 1 −μ′′ 1

σ 12

−

ρ(μ′ 2 −μ′′ 2 )

σ 1 σ 2

]

x+

[

μ′ 2 −μ′′ 2

σ^22

−

ρ(μ′ 1 −μ′′ 1 )

σ 1 σ 2

]

y

}

≤logk+^12 [q(0,0;μ′ 1 ,μ′ 2 )−q(0,0;μ′′ 1 ,μ′′ 2 )],

or, for brevity,
ax+by≤c. (8.5.3)