Como definimos um estimador para dados provenientes de uma distribuição binomial? Para bernoulli, posso pensar em um estimador estimando um parâmetro p, mas no binômio não consigo ver quais parâmetros estimar quando n caracterizamos a distribuição?
Atualizar:
Por estimador, entendo uma função dos dados observados. Um estimador é usado para estimar os parâmetros da distribuição que gera os dados.
estimation
binomial
Rohit Banga
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Respostas:
I guess what you are looking for is the probability generating function. A derivation of the probability generating function of the binomial distribution can be found under
http://economictheoryblog.com/2012/10/21/binomial-distribution/
However, having a look at Wikipedia is nowadays always a good idea, although I have to say that the specification of the binomial could be improved.
https://en.wikipedia.org/wiki/Binomial_distribution#Specification
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Every distribution have some unknown parameter(s). For example in the Bernoulli distribution has one unknown parameter probability of success (p). Likewise in the Binomial distribution has two unknown parameters n and p. It depends on your objective which unknown parameter you want to estimate. you can fix one parameter and estimation other one. For more information see this
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Say you have datak1,…,km∼iid binomial(n,p) .
You could easily derive method-of-moment estimators by settingk¯=n^p^ and s2k=n^p^(1−p^) and solving for n^ and p^ .
Or you could calculate MLEs (perhaps just numerically), eg using
optim
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I think we could use method of moments estimation to estimate the parameters of the Binomial distribution by the mean and the variance.
Using the method of moments estimation to estimate The parametersp and m .
[{\hat{p}}_n=\frac{\overline{X}-S^2}{\overline{X}}][\hat{m}_n=\frac{\overline{X}^2}{\overline{X}-S^2}]
Proof
The estimators of the parameters m and p by the Method of Moments are the solutions of the system of equations
mp=X¯,mp(1−p)=S2.
Hence our equations for the method of moments are:
[\overline{X}=mp]
[S^2=mp (1-p).]
Simple arithmetic shows: [S^2 = mp\left(1 - p\right) = \bar{X}\left(1 - p\right)] [S^2=\bar{X}-\bar{X} p] [\bar{X}p=\bar{X}-S^2, \mbox{ therefore } \hat{p}=\frac{\bar{X}-S^2}{\bar{X}}.] Then, [\bar{X} = mp, \mbox{ that is, } m \left(\frac{\bar{X}-S^2}{\bar{X}}\right)] [\bar{X}=m\left(\frac{\bar{X}-S^2}{\bar{X}}\right), \mbox{ or } \hat{m}=\frac{\bar{X}^2}{\bar{X}-S^2}. ]
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