This question is a continuation of this question: Derivation of Box-Cox and Yeo-Johnson Log-Likelihood Functions.

In order to derive the maximum lambda value in log-likelihood objective function for both Box-Cox and Yeo-Johnson power transformation, the Scikit-Learn PowerTransformer implementation uses Brent method (optimization method without derivatives), which should only apply to unimodal function, that is, unique local maximum is also global maximum. However, can these two objective functions be proved to be unimodal function?

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    $\begingroup$ what is the question? $\endgroup$
    – utobi
    Nov 27, 2022 at 14:44
  • $\begingroup$ The question is whether the log-likelihood function is really a unimodal function? If it is, whether there is any existing proof on that? If not, is there any anti-example? If Scipy implementation is correct, since it use Brent Optimization, then that implies this is a unimodal function. $\endgroup$
    – Audison
    Nov 28, 2022 at 0:17


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