MLE in a logistic regression model Assuming that the design matrix is of full rank, in non-degenerate cases of the logistic regression model, does the maximum likelihood estimator always exists and is always unique?
It would be really helpful if someone could provide an example or a resource link.

Yes. Logistic regression model belongs to the conditional exponential family, so per Pitman-Koopman-Darmois theorem log-likelihood functions for exponential families should have unique optimal value. For a specific proof of your case where the design matrix of the logistic regression is of full rank, see reference. The basic idea is to prove the log-likelihood function of LR is convex via its Hessian similar to the single variable case via second order derivative.
