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Given a distribution $f(x_i,y_i;\alpha,\beta)$ and joint probability function $F(x_1,...,x_N,y_1,...,y_N;\alpha, \beta)$

The first order derivatives are $\frac{\partial F}{\partial \alpha}$ and $\frac{\partial F}{\partial \beta}$

Is the first order condition

$\frac{\partial F}{\partial \alpha}=0$ and $\frac{\partial F}{\partial \beta}=0$

OR

$\frac{\partial F}{\partial \alpha}=0$ or $\frac{\partial F}{\partial \beta}=0$

Do they need to satisfy at the same time or just one of them?

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The first variant. Maximum likelihood estimation maximises the likelihood, which is a multivariable function. For point to be considered local minimum or maximum point of a function it must be stationary point, i.e. the point at which all first order partial derivatives are zero.

The intuition transfers from one variable functions. The stationary point is the point where derivative is zero. For multivariable functions the derivative is vector. Vector is equal to zero, when its all elements are zero.

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  • $\begingroup$ Bonus point for finding this information on wikipedia. Bizarelly enough only one variable functions are considered. $\endgroup$ – mpiktas Dec 13 '11 at 11:57

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