Question

Consider the maximum likelihood estimation problem with the linear measurement model y_i = a_i^T x + v_i, i = 1,...,m The vector x ? R^n is a vector of unknown parameters, y_i are the measurement values, and v_i are independent and identically distributed measurement errors. In this problem we make the assumption that the normalized probability density function of the errors is given (normalized to have zero mean and unit variance), but not their mean and variance. In other words, the density of the measurement errors v_i is p(z) = 1/? f((z - ?) / ?) where f is a given, normalized density. The parameters are the mean ? and standard deviation ? of the distribution p, and are not known. The maximum likelihood estimates of x, ?, ? are the maximizers of the log-likelihood function ?_{i=1}^m log p(y_i - a_i^T x) = -m log ? + ?_{i=1}^m log f((y_i - a_i^T x - ?) / ?) where y is the observed value. Show that if f is log-concave, then the maximum likelihood estimates of x, ?, ? can be determined by solving a convex optimization problem.

          Consider the maximum likelihood estimation problem with the linear measurement model

y_i = a_i^T x + v_i, i = 1,...,m

The vector x ? R^n is a vector of unknown parameters, y_i are the measurement values, and v_i are independent and identically distributed measurement errors. In this problem we make the assumption that the normalized probability density function of the errors is given (normalized to have zero mean and unit variance), but not their mean and variance. In other words, the density of the measurement errors v_i is

p(z) = 1/? f((z - ?) / ?)

where f is a given, normalized density. The parameters are the mean ? and standard deviation ? of the distribution p, and are not known. The maximum likelihood estimates of x, ?, ? are the maximizers of the log-likelihood function

?_{i=1}^m log p(y_i - a_i^T x) = -m log ? + ?_{i=1}^m log f((y_i - a_i^T x - ?) / ?)

where y is the observed value. Show that if f is log-concave, then the maximum likelihood estimates of x, ?, ? can be determined by solving a convex optimization problem.

Consider the maximum likelihood estimation problem with the linear measurement model

yi = ai^T x + vi, i = 1,...,m

The vector x ? R^n is a vector of unknown parameters, yi are the measurement values, and vi are independent and identically distributed measurement errors. In this problem we make the assumption that the normalized probability density function of the errors is given (normalized to have zero mean and unit variance), but not their mean and variance. In other words, the density of the measurement errors vi is

p(z) = 1/? f((z - ?) / ?)

where f is a given, normalized density. The parameters are the mean ? and standard deviation ? of the distribution p, and are not known. The maximum likelihood estimates of x, ?, ? are the maximizers of the log-likelihood function

?i=1^m log p(yi - ai^T x) = -m log ? + ?i=1^m log f((yi - ai^T x - ?) / ?)

where y is the observed value. Show that if f is log-concave, then the maximum likelihood estimates of x, ?, ? can be determined by solving a convex optimization problem.

Added by Andrea F.

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