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There are 3 types of flowers that can grow from planting a seed. $$P(\text{Daisy}) = \theta_1$$ $$P(\text{Rose}) = (1-\theta_1)\theta_2$$ $$P(\text{Sunflower}) = (1-\theta_1)(1-\theta_2)$$

The total number of flowers at the end is $n.$ If $X=(X_1, X_2, X_3)$ is the number of daisies, roses and sunflowers respectively, find the MLEs for $\theta_1$ & $\theta_2$ and $E[X_1]$ and $E[X_2]$ in term of the parameters $\theta_1$ & $\theta_2$.

I am aware that this is a multinomial distribution but I don't know how to go about finding the estimators for the parameters since there are 2 of them.

Dale C
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1 Answers1

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If we're given that $X=\left(X_1,X_2,X_3\right)$ follows a multinomial distribution with parameters $\theta=\left(\theta_1,\left(1-\theta_1\right)\theta_2,\left(1-\theta_1\right)\left(1-\theta_2\right)\right)$, then the likelihood is

\begin{align*} L\left(X; \theta\right) &\propto \theta_1^{X_1}\left(1-\theta_1\right)^{X_2}\theta_2^{X_2}\left(1-\theta_1\right)^{X_3}\left(1-\theta_2\right)^{X_3}\\ \end{align*}

The log-likelihood is then

\begin{align*} \ell\left(X; \theta\right) &\approx X_1\log\left(\theta_1\right)+\left(X_2+X_3\right)\log\left(1-\theta_1\right)+X_2\log\left(\theta_2\right)+X_3\log\left(1-\theta_2\right)\\ \end{align*}

Taking derivatives gives

\begin{align*} \frac{\partial \ell}{\partial \theta_1} &= \frac{X_1}{\theta_1}-\frac{X_2+X_3}{1-\theta_1}\\ \frac{\partial \ell}{\partial \theta_2} &= \frac{X_2}{\theta_2}-\frac{X_3}{1-\theta_2}\\ \end{align*}

Setting the derivatives to 0 and solving gives

\begin{align*} \hat{\theta}_1 &= \frac{X_1}{X_1+X_2+X_3}\\ \hat{\theta}_2 &= \frac{X_2}{X_2+X_3}\\ \end{align*}

assumednormal
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