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How is the Fisher information of a binomial distribution defined?
Fisher information of a Binomial distribution. The Fisher information is defined as $\\mathbb{E}\\Bigg( \\frac{d \\log f(p,x)}{dp} \\Bigg)^2$, where $f(p,x)={{n}\\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the log-likelihood function is $L'(p,x) = \\frac{x}{p} – \\frac{n-x}{1-p}$.
Which is the inverse of the Fisher information?
So that when you apply the classic result about the asymptotic distribution of the MLE, you have that the variance is simply the inverse of the Fisher information: p ( 1 − p) n . Therefore, the asymptotic variances coincide from both perspectives.
When does the Fisher information take the form of an n × 1 vector?
When there are N parameters, so that θ is an N × 1 vector = […], then the Fisher information takes the form of an N × N matrix. This matrix is called the Fisher information matrix (FIM) and has typical element
What is the Fisher information for one Bernoulli trial?
The Fisher information for a single Bernoulli trial is 1 p (1 − p). When you have n trial, the asymptotic variance indeed becomes p (1 − p) n. When you consider the Binomial resulting from the sum of the n Bernoulli trials, you have the Fisher information that (as the OP shows) is n p (1 − p).
If we are tossing a coin, then the negative binomial distribution can give the number of heads (“success”) we are likely to encounter before we encounter a certain number of tails (“failure”). In the probability mass function below, p is the probability of success, and (1- p) is the probability of failure.
When is the negative binomial distribution called the Pascal distribution?
For the special case where r is an integer, the negative binomial distribution is known as the Pascal distribution. It is the probability distribution of a certain number of failures and successes in a series of independent and identically distributed Bernoulli trials.