12 questions linked to/from Estimating parameters for a binomial
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### Examples where method of moments can beat maximum likelihood in small samples?

Maximum likelihood estimators (MLE) are asymptotically efficient; we see the practical upshot in that they often do better than method of moments (MoM) estimates (when they differ), even at small ...
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### Is population size a parameter, or sample size a statistic?

The definitions of a parameter and statistic pretty much agree: parameters and statistics are numerical characteristics or numerical summaries of a population and sample, respectively, for a given ...
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### Maximum likelihood estimator of $n$ when $X \sim \mathrm{Bin}(n,p)$

Given a random variable $X\sim Bin(n,p)$, where $p$ is known $p\in (0,1)$ , $n$ is an unknown positive integer and $x\in\{0,1,2,....n\}$, what is the maximum likelihood estimator of $n$? I ...
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### Estimating $n$ and $p$ for Binomial distribution, repeated counting of partly hidden population

A brief motivation: $n$ critters live in an aquarium, where sadly they often hide in, under or behind things. When the aquarium is observed, each critter is only seen with probability $p$ (...
683 views

### Hypergeometric distribution when K is unknown

The probability to have $k$ white balls in a sample of size $n$ taken from an urn of $N$ balls with $K$ of them being white is equal to:  P(k|n,N,K) = \frac{{{n}\choose{k}}{{N-n}\choose{K-k}}}{{{N}\...
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### Profile likelihood of N in binomial model

Let $(n_1,...,n_k)$ be a sample from a Binomial$(N,p)$ where both parameters are unknown. In many cases, the profile likelihood of $N$ is asymptotic in the sense that it never decays to $0$. An ...
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### Likelihood of two dice. One fair the other unfair

first of all sorry about the title. I couldn't think anything better. Let me describe the problem first and ask later. Imagine I have 1500 observations of the results of two dice being played (I don'...
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### What can we say about hypergeometric distribution with unknown $N$?

Hypergeometric distribution describes outcome of $n$ draws without replacement from the urn containing $K$ white balls and $N-K$ black balls. Binomial distribution describes outcome of $n$ draws with ...
456 views

### Multiple imputation of glm binomial size parameter

Suppose we have a generalized linear model with a binomial response $y_i\sim \mathrm{bin}(n_i,p_i)$ where $p_i$ is determined by the linear predictor in the usual way via some link function. Is there ...
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### Binomial GLM with unknown totals?

Essentially, I have some covariate data X, and a dependent variable Y consisting of proportions of a sample that shown a certain ...
I was wondering if someone could clarify the following for me. In the paper "Inference for the binomial $N$ parameter" by Adrian Raftery, his first example outlines the posterior of $N$ given $x$ as ...
I'm working on a project that will run $n=10000$ experiments. In this experiment, $j$ events will occur (an unknown number). Each of the events has a value $E_j$ attached to it. We expect these values ...