While reading the text from Keith H. Thompson on the Estimation of the Proportion of Vectors in a Natural Population of Insects, I came across the following part where I don't understand everything.
Let $p$ be the proportion of viruliferous insects and $h(p) = (1-p)^k$ where $k$ is the number of insects on each plant, a Bernoulli random variable $X$ may be defined with probability function $$b[x;p,k] = [h(p)]^x[1-h(p)]^{1-x}$$ such that $X = 1$ if a plant is not infected and $X = 0$ if a plant is infected. In $n$ trials ($n$ plants are tested), the maximum likelihood estimators of $h(p)$ and $p$ are obtained from the likelihood function $L = \prod_{i=1}^n b[x_i; p, k]$ as $$\hat{h}(p) = \frac{\sum X_i}{n}$$ and $$\hat{p} = 1 - \left[ \frac{\sum X_i}{n} \right]^{1/k}$$ respectively. [...] and $\hat{p}$ is a biased estimator of $p$ with mean $$E(\hat{p}) = 1 - \sum_{i=0}^n \left(\frac{i}{n}\right)^{1/k} \binom{n}{i}[(1-p)^k]^i[1-(1-p)^k]^{n-i}.$$ [...] As $n$ approaches infinity, the maximum likelihood estimator $\hat{p}$ is distributed asymptotically normally and converges in probability to $p$. $\hat{p}$ is also asymptotically efficient, and its asymptotic variance is determined as $$\lim_{n \to \infty} V(\hat{p}) = \frac{1}{nE\left[ \frac{ \partial \log b(x; p, k)}{\partial p}\right]^2}.$$
I never had a class on statistics, so I am unfamiliar with the notions of estimators, but I am a math student so I am familiar with the notions of expectancy, variance, and laws.
I could find some references on how to derive $\hat{h}(p)$ and we derive $\hat{p}$ from $\hat{h}(p)$ so my questions are the following:
- I could not derive $E(\hat{p})$, so how is this done?
- Is $\hat{p}$ is distributed asymptotically normally because of the central limit theorem?
- Is the asymptotic variance derived from a specific formula?
Thanks for any answer