Suppose 100 seeds were planted and it is recorded that $x < 11$ seeds germinated. Assuming a binomial model we obtain the following likelihood: binomial image
I believe the curve can be calculated with the following r-code:
curve(pbinom(11, 100, y), xname = "y")
We have that:
$$L(\theta) = \sum_{x=0}^{10} \binom{n}{x}\theta^x(1-\theta)^{n-x}$$
The MLE is $\hat{\theta}=0$ However, how do I calculate this MLE?
I have tried:
$$L(\theta;x) = \prod_{i=1}^{10}f(x_i;\theta) = \prod_{i=1}^{10}\sum_{x=0}^{10}\binom{n}{x}\theta^x(1-\theta)^{n-x} \\ \implies \sum_{x=0}^{10}\binom{n}{\prod_{i=1}^{10}x}\theta^{\sum_{i=1}^{10}x_i}(1-\theta)^{10n-\sum_{i=1}^nx_i}$$
Then taking the logarithm: $$\log L(\theta;x) = \log \left(\sum_{x=0}^{10}\binom{n}{\prod_{i=1}^{10}x}\right)+\sum_{i=1}^{10}x_i\log \left(\sum_{x=0}^{10}\theta\right)+(10n-\sum_{i=1}^{10}x_i) \log \left(\sum_{x=0}^{10}(1-\theta)\right)$$
We know that the MLE of a binomial distribution is given as $\hat{\theta} = \frac{x}{n}$ so I cannot understand how the MLE for the likelihood is 0. I suppose my way of indexing summation and products is wrong here?
Taken from: Pawitan, Y. 2013. In All Likelihood: Statistical Modelling and Inference using Likelihood, pg 40