Question

Let the independent normal random variables
Y1,Y2, . . . ,Yn have the respective distributions
N(μ, γ 2x2i
), i = 1, 2, . . . , n, where x1, x2, . . . , xn are known
but not all the same and no one of which is equal to
zero. Find the maximum likelihood estimators for μ
and γ 2.

Answer 1

The probability density function of a normal random variable is,

The likelihood function of x1, x2, . . . , xn is,

The log-likelihood function of x1, x2, . . . , xn is,

Form maximum likelihood,

---(1)

and

---(2)

From (1),

Thus, the maximum likelihood estimate of μ is,

From (2),

Thus, the maximum likelihood estimate of γ 2 is,

where