Linked Questions

56
votes
7answers
8k views

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 ...
11
votes
2answers
1k views

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 ...
3
votes
3answers
141 views

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 ...
7
votes
1answer
436 views

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 $p$ (...
0
votes
1answer
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}\...
2
votes
1answer
460 views

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 ...
1
vote
1answer
579 views

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'...
4
votes
1answer
379 views

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 ...
4
votes
0answers
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 ...
2
votes
0answers
162 views

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 ...
1
vote
1answer
59 views

Clarification on a paper regarding estimating N from a Binomial Distribution

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 ...
2
votes
1answer
52 views

Estimating total events from buckets hit

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 ...