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my question actually revolves around applying the secant method (optimization) to find the maximum likelihood estimator for $\theta$ where $$ \begin{align} p_1 &= (2+\theta)/4,\\p_2 &= (1-\theta)/4,\\p_3 &= (1-\theta)/4,\\ p_4 &= \theta/4 \end{align} $$

I don't want to ruin this exercise for myself, but I'm stumped on how to find the gradient of the pdf, $$ \begin{align} f(x) = \binom{m}{x_1,x_2,x_2,x_4}p_1^{x_1}p_2^{x_2}p_3^{x_3}p_4^{x_4} \end{align} $$ Does anyone know how to do this without using advanced methods only known to experienced statisticians (No gaussian methods or anything)?Maybe I'm forgetting something basic from calculus, but I'm not sure how to approach this.Thank you for any assistance.

FoiledIt24
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Jdev
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  • The likelihood given a sample of size $n$ writes as $L(\theta)= (p_1(\theta))^{n_1}(p_2(\theta))^{n_2}(p_3(\theta))^{n_3}(p_4(\theta))^{n_4}$ for $\theta\in (0,1)$ and $n_1+n_2+n_3+n_4=n$. Here $n_i$ is the number of observations in the sample corresponding to $i$th category. – StubbornAtom Feb 20 '21 at 06:13
  • Sure, but I thought the gradient doesn't use the likelihood function? – Jdev Feb 20 '21 at 19:47

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