If the random variable Y denotes an individual’s income, Pare to’s law claims that P ( Y ≥ y ) = ( k y ) θ , where k is the entire population’s minimum income. It follows that F Y ( y ) = 1 − ( k y ) θ , and, by differentiation, f Y ( y ; θ ) = θ k θ ( 1 y ) θ + 1 , y ≥ k ; θ ≥ 1 Assume k is known. Find the maximum likelihood estimator forθ if income information has been collected on a random sample of 25 individuals.
If the random variable Y denotes an individual’s income, Pare to’s law claims that P ( Y ≥ y ) = ( k y ) θ , where k is the entire population’s minimum income. It follows that F Y ( y ) = 1 − ( k y ) θ , and, by differentiation, f Y ( y ; θ ) = θ k θ ( 1 y ) θ + 1 , y ≥ k ; θ ≥ 1 Assume k is known. Find the maximum likelihood estimator forθ if income information has been collected on a random sample of 25 individuals.
Solution Summary: The author calculates the maximum likelihood estimator for theta , if income information has been collected on a random sample of 25 individuals.
If the random variable
Y
denotes an individual’s income, Pare to’s law claims that
P
(
Y
≥
y
)
=
(
k
y
)
θ
, where
k
is the entire population’s minimum income. It follows that
F
Y
(
y
)
=
1
−
(
k
y
)
θ
, and, by differentiation,
f
Y
(
y
;
θ
)
=
θ
k
θ
(
1
y
)
θ
+
1
,
y
≥
k
;
θ
≥
1
Assume
k
is known. Find the maximum likelihood estimator forθ if income information has been collected on a random sample of 25 individuals.
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