0

I have trouble finding the likelihood function in an applied problem involving the binomial distribution.

Given are $N$ independent random variables having identical binomial distributions with the parameters $\theta$ and $n = 3$ where $n_0$ of them take on the value $0$, $n_1$ take on the value $1$, $n_2$ take on the value $2$, and $n_3$ take on the value $3$.

Now the Method of Maximum Likelihood should be used to find a formula for estimating $\theta$.

I started off from the probability distribution function of a general binomial random variable and the derivation of the maximum likelihood estimator in the general case. However, the case is now different and I got stuck already in the beginning. How should one proceed here?

Anna D.
  • 525

1 Answers1

0

$$L(\theta|n_i\vert_{i=0}^{n}, n) = \prod_{j=0}^{n} P(X=j)^{n_j} = \prod_{j=0}^{n}\left(\binom{n}{j}\theta^{j}(1-\theta)^{n-j}\right)^{n_j} = \left(\prod_{j=0}^{n}\binom{n}{j}\right)\theta^{N}(1-\theta)^{N_0-N}$$

where $N = \sum_{j=0}^{n}jn_j$ and $N_0 = n\sum_{j=0}^{n}n_j$

$$\log L(\theta|n_i\vert_{i=0}^{n}, n) = C + N\log(\theta) + (N_0-N) \log(1-\theta)$$

where $C$ is independent of $\theta$.

Take derivative wrt $\theta$ and equate to $0$,

$$N/\theta - (N_0-N)/(1-\theta) = 0 \implies \theta = \frac{N}{N_0}$$

Take double derivative wrt $\theta$ and verify that it is negative with above value of $\theta$.

Dhruv Kohli
  • 5,216
  • In the iid case the joint probability distribution is the product of the individual marginal distributions, that explains the product. But why is the power $P(X = j)^{n_j}$ there? – Anna D. Aug 12 '17 at 17:30
  • Actually it should be $\prod_{j=1}^{num_samples}P(X=x_j)$ where $x_j \in {0,1,\ldots,n}$.

    $n_k = \sum_{j=1}^{num_samples}1_{x_j=k}$.

     – Dhruv Kohli Aug 13 '17 at 15:40