= Solution
For unrestricted category probabilities, the <log-likelihood> is $\sum_i n_i\log\pi_i$ subject to $\sum_i\pi_i=1$. A multiplier gives $n_i/\pi_i$ constant, hence the unique maximum, because all counts are positive and the <log-likelihood> is strictly concave:
$$
\boxed{\widehat\pi_i=n_i/n}.
$$
The <likelihood-ratio test>[likelihood ratio test] compares this saturated maximum with the maximum under the parametric model. Twice the log ratio is
$$
2\sum_i n_i\log\frac{n_i/n}{\pi_i(\widehat\theta)}=\Lambda(\widehat\theta).
$$
For $k$ bins and one regular fitted parameter, its usual large-sample reference distribution is $\chi^2_{k-2}$; reject for a sufficiently large value. Under a fully specified parameter $\theta_0$, use $\Lambda(\theta_0)$ and $\chi^2_{k-1}$. This relies on an identifiable interior fit and adequate expected bin counts.
For individual ungrouped abundances, let $S$ be the number of species and $T$ the total number of individuals. The logarithmic-series model gives
$$
\ell(\theta)=-S\log[-\log(1-\theta)]+T\log\theta+C,\qquad0<\theta<1.
$$
Its score vanishes when
$$
\boxed{\frac{-\widehat\theta}{(1-\widehat\theta)\log(1-\widehat\theta)}=\frac TS}.
$$
The left side increases continuously from one to infinity. One way to see strict increase is to set $t=-\log(1-\theta)>0$, making it $(e^t-1)/t$, whose derivative is positive because $e^t(t-1)+1$ increases from zero. The score has the sign of $T/S$ minus this increasing <function>, so this is the unique maximum. Bisection on $(0,1)$ is a robust numerical method; Newton's method is another option.
There are two fits implicit in the supplied presentation. Using the ungrouped sufficient statistics gives $\widehat\theta_{\rm full}\approx0.944228$. By contrast the displayed ten-bin deviance is minimized near \b[$0.93$], so that is the table-based estimate $\widehat\theta_{\rm grouped}$. Grouping the upper tail discards its detailed abundance total; minimizing the grouped deviance maximizes the grouped likelihood, not the ungrouped likelihood used in the preceding score equation. The estimates therefore need not agree. The following table-based decisions use its grouped-fit convention.
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