bv_cvxbook_extra_exercises

73 minimum total covering ball volume we consider a

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Unformatted text preview: es tj and feature values xj , for j = 1, . . . , N . In other words, we observe that the event with features xj occurred at time tj . You can assume that the baseline hazard rate λ0 is known. Show that maximum likelihood estimation of the parameter w is a convex optimization problem. Remarks. Regularization is typically included in Cox proportional hazards fitting; for example, adding ℓ1 regularization yields a sparse model, which selects the features to be used. The basic Cox proportional hazards model described here is readily extended to include discrete times of the event, censored measurements (which means that we only observe T to be in an interval), and the effects of features that can vary with time. 6.13 Maximum likelihood estimation for an affinely transformed distribution. Let z be a random variable on Rn with density pz (u) = exp −φ( u 2 ), where φ : R → R is convex and increasing. Examples of such distributions include the standard normal N (0, σ 2 I ), with φ(u) = (u)2 + α, and the multivari+ able Laplacian distr...
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