14.3The Exponential Distribution in Life Testing 585
and so, by independence, the joint probability density ofXij,j=1,...,ris
fXi 1 ,...,Xir(x 1 ,...,xr)=
∏r
j= 1
1
θ
e−xj/θ
Also, the probability that the othern−rof theX’s are all greater thanxris, again using
independence,
P{Xj>xrforj=i 1 ori 2 ...orir}=(e−xr/θ)n−r
Hence, we see that the likelihood of the observed data — call it L(x 1 ,...,xr,
i 1 ,...,ir) — is, forx 1 ≤x 2 ≤ ··· ≤xr,
L(x 1 ,...,xr,i 1 ,...,ir) (14.3.2)
=fXi 1 ,Xi 2 ,...Xir(x 1 ,...,xr)P{Xj>xr,j=i 1 ,...,ir}
=
1
θ
e−x^1 /θ···
1
θ
e−xr/θ(e−xr/θ)n−r
=
1
θr
exp
−
∑r
i= 1
xi
θ
−
(n−r)xr
θ
REMARK
The likelihood in Equation 14.3.2 not only specifies that the firstrfailures occur at
timesx 1 ≤x 2 ≤ ··· ≤xrbut also that theritems to fail were, in order,i 1 ,i 2 ,...,ir.
If we only desired the density function of the firstrfailure times, then since there are
n(n−1)···(n−(r−1))=n!/(n−r)!possible (ordered) choices of the firstritems to
fail, it follows that the joint density is, forx 1 ≤x 2 ≤ ··· ≤xr,
f(x 1 ,x 2 ,...,xr)=
n!
(n−r)!
1
θr
exp
−
∑r
i= 1
xi
θ
−
(n−r)
θ
xr
To obtain the maximum likelihood estimator ofθ, we take the logarithm of both sides
of Equation 14.3.2. This yields
logL(x 1 ,...,xr,i 1 ,...,ir)=−rlogθ−
∑r
i= 1
xi
θ
−
(n−r)xr
θ