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I have a data set representing a random vector $\mathbf{X}=(X_1,\ldots, X_p)'$. Define $Z=\alpha' X $, where $\alpha \in \mathbb{R}^p$ and $\alpha'\mathbf{1}_p =1$.

I would like to find the $\alpha$ which maximises the likelihood of $Z$ falling in a given interval $(a,b)$, that is: $$ \max_\alpha\; \mathbb{P}\left[\alpha' \mathbf{X} \in (a,b)\right] $$

The task is relatively simple if the $X_i$ variables are assumed normal i.i.d.. Can you tell me some methods, and related R packages, for a more general case?

Edit

To make the problem solvable, perhaps numerically, I assume $\mathbf{X}$ follows a skewed-$t$ distribution.

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    $\begingroup$ This may be stated too generally to permit any effective answers: based on what you have given, your objective function could be literally any function of $\alpha$ that takes on values between $0$ and $1$. Perhaps you could make it more specific by providing more information about the multivariate distribution of $X$? $\endgroup$
    – whuber ♦
    Commented Nov 5, 2017 at 17:31
  • $\begingroup$ @whuber: I am now assuming a skewed-$t$ distribution. $\endgroup$
    – antonio
    Commented Nov 5, 2017 at 20:57
  • $\begingroup$ Could you make your edit more precise? IID skewed-$t$? ??? $\endgroup$
    – kjetil b halvorsen ♦
    Commented Aug 3, 2019 at 22:18

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