Let , i=1, 2, ...,m be a simple regression, where xi unobservable and assumed to be independent and identically distributed with common distribution Instead, we assume that, for each xi, zi1, zi2, . . . , zin are observed. Given xi, the conditional distribution for the observed zij is assumed to be normal with mean xi and variance . Therefore, zi = (zi1, zi2, . . . , zil)' has a multivariable normal distribution with mean vector and variance-covariance matrix , where and 1=(1, 1,...,1)'. Find the MLE for and .

Let , i=1, 2, ...,m be a simple regression, where xi unobservable and assumed to be independent and identically distributed with common distribution Instead, we assume that, for each xi, zi1, zi2, . . . , zin are observed. Given xi, the conditional distribution for the observed zij is assumed to be normal with mean xi and variance . Therefore, zi = (zi1, zi2, . . . , zil)' has a multivariable normal distribution with mean vector and variance-covariance matrix , where and 1=(1, 1,...,1)'. Find the MLE for and .

Solution. By hypothesis, zi = (zi1, zi2, . . . , zil)' has a multivariable normal distribution with mean vector and variance-covariance matrix . For convenience, denote by the variance-covariance matrix .
We know that the probability density function of
zi = (zi1, zi2, . . . , zil) is of the ...

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