Comments on: Three views of the negative binomial distribution http://www.johndcook.com/blog/2009/09/22/negative-binomial-distribution/ The blog of John D. Cook Thu, 18 Mar 2010 03:02:57 -0400 http://wordpress.org/?v=2.8.4 hourly 1 By: John Myles White http://www.johndcook.com/blog/2009/09/22/negative-binomial-distribution/comment-page-1/#comment-30648 John Myles White Tue, 12 Jan 2010 18:11:04 +0000 http://www.johndcook.com/blog/?p=3238#comment-30648 This was incredibly useful, John. I've known just a little about the negative binomial distribution for a while now after reading the start of "Inference and Disputed Authorship," but never fully appreciated the distribution's place. Thanks for giving such a useful buildup of motivating intuitions for it. This was incredibly useful, John. I’ve known just a little about the negative binomial distribution for a while now after reading the start of “Inference and Disputed Authorship,” but never fully appreciated the distribution’s place. Thanks for giving such a useful buildup of motivating intuitions for it.

]]> By: Jan Galkowski http://www.johndcook.com/blog/2009/09/22/negative-binomial-distribution/comment-page-1/#comment-30001 Jan Galkowski Sat, 02 Jan 2010 01:03:28 +0000 http://www.johndcook.com/blog/?p=3238#comment-30001 I'm wondering what the general expression for the variance of a mixture distribution might look like. To be specific, suppose X(1), X(2), ..., X(n) are r.v. each having means u(1), u(2), ..., u(n), respectively, and variances v(1), v(2), ..., v(n). The k-th distribution is chosen with probability p(k), and the choice is mutually exclusive. So, in each case, it's a multinomial gating which of the n distributions are used. In the case of n = 2, VAR[...] = p(1) v(1) + p(2) v(2) + p(1) p(2) [u(1) - u(2)]^2. The question is, what does this look like in general? It had been suggested to me that the general expression, given the intermediate definition, Y(i) = P(S == i) X(i) for the variance looks like SUM-over-i VAR[Y(i)] + 2 SUM-over-i-not-equal-j COV[Y(i),Y(j)] where, of course, COV[U,V] = E[U V] - E[U] E[V] Here, since Y(i) and Y(j) are mutually exclusive, E[Y(i) Y(j)] is presumably zero. But I cannot reconcile this in the case of two r.v. with n = 2 expression. Any suggestions? I’m wondering what the general expression for the variance of a mixture distribution might look like. To be specific, suppose X(1), X(2), …, X(n) are r.v. each having means u(1), u(2), …, u(n), respectively, and variances v(1), v(2), …, v(n). The k-th distribution is chosen with probability p(k), and the choice is mutually exclusive. So, in each case, it’s a multinomial gating which of the n distributions are used. In the case of n = 2,

VAR[...] = p(1) v(1) + p(2) v(2) + p(1) p(2) [u(1) - u(2)]^2.

The question is, what does this look like in general? It had been suggested to me that the general expression, given the intermediate definition,

Y(i) = P(S == i) X(i)

for the variance looks like

SUM-over-i VAR[Y(i)] + 2 SUM-over-i-not-equal-j COV[Y(i),Y(j)]

where, of course, COV[U,V] = E[U V] – E[U] E[V]

Here, since Y(i) and Y(j) are mutually exclusive, E[Y(i) Y(j)] is presumably zero.

But I cannot reconcile this in the case of two r.v. with n = 2 expression.

Any suggestions?

]]> By: EastwoodDC http://www.johndcook.com/blog/2009/09/22/negative-binomial-distribution/comment-page-1/#comment-26479 EastwoodDC Mon, 26 Oct 2009 02:09:05 +0000 http://www.johndcook.com/blog/?p=3238#comment-26479 I had not ever though about the NB being a generalization of the Poisson, but that is a useful was to think of it. A colleague of vast experience tells me he has never fit a Poisson model where overdispersion was not a problem. I had not ever though about the NB being a generalization of the Poisson, but that is a useful was to think of it. A colleague of vast experience tells me he has never fit a Poisson model where overdispersion was not a problem.

]]> By: John Cook y sus tres acercamientos a la distribución binomial negativa « Apuntes de Estadística http://www.johndcook.com/blog/2009/09/22/negative-binomial-distribution/comment-page-1/#comment-25047 John Cook y sus tres acercamientos a la distribución binomial negativa « Apuntes de Estadística Thu, 24 Sep 2009 17:39:29 +0000 http://www.johndcook.com/blog/?p=3238#comment-25047 [...] Cook plantea acá una interesante discusión acerca de la interpretación de la distribución binomial negativa. [...] [...] Cook plantea acá una interesante discusión acerca de la interpretación de la distribución binomial negativa. [...]

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