Let $$b(y|\mu,\phi)=\frac{\Gamma(\phi)}{\Gamma(\mu\phi)\Gamma((1-\mu)\phi)}y^{\mu\phi-1}(1-y)^{(1-u)\phi-1}$$ where $0<\mu<1$ and $\phi>0$ a reparametrization of beta distribution. Then the density of beta rectangular is $$f(y|\mu,\phi,\theta)=\theta+(1-\theta)b(y|\mu,\phi)$$ where $0\leq\theta\leq 1$ is a mixture parameter.
I need to find the log-likelihood function of $f(.)$ then I did $$log(f(y|\mu,\phi,\theta)=log\theta+log(1-\theta)+log(\Gamma(\phi))-log(\Gamma(\mu\phi))-log(\Gamma((1-\mu)\phi))+(\mu\phi-1)log(y)+((1-\mu)\phi-1)log(1-y)$$
Is this log-likelihood correct? I am doing some calculations that are based on it and comparing with the results of the conventional beta, but the results are not as I expected.
The joint likelihood of the parameters, given a sample $\boldsymbol y$ from the distribution with density $f$ is $$\mathcal L(\mu, \phi, \theta \mid \boldsymbol y) = \prod_{i=1}^n \left( \theta + (1-\theta) b(y_i \mid \mu, \phi) \right).$$ Thus the log-likelihood is $$\ell(\mu, \phi, \theta \mid \boldsymbol y) = \sum_{i=1}^n \log \left(\theta + (1-\theta) b(y_i \mid \mu, \phi) \right).$$ The error you are making is that $$\log (x + y) \ne \log x + \log y.$$ This is one reason why mixture distributions are not easy to work with.