# Estimating parameters for shifted Poisson distribution

Suppose I have a data vector v, in which all values are greater than zero.

Now, I want to see if it follows a shifted Poisson distribution.

Can I do it by shifting everything to the left first, and fitting a Poisson in the usual fashion? For example, in R I can fit a Poisson by using the "fitdistr" function. Can I do vec <- v - 1 to shift everything to the left first (so that we can treat it like normal Poisson distribution)? And then run fitdistr(vec, "poisson").

Second Question: Now suppose I want to carry out a chisq.test(). The frequency table of the actual data vector vecis as follow:

table(vec)
0    1    2    3    4    5    6    7    8    9   10   11   12   13   14   16   17   18   19   20   21   22
611 1297 1893 2730 2759 2198 1525  927  456  243   75   39   18    7    2    1    2    4   12   20   34   34
23   24   25   26   27   30   34   35   36   37   38   39   40   41   42   43   44   47   48   52   54   55
19   13    8    2    1    3    1    4    7    7   10    8    5    2    8    3    3    3    1    1    1    2
56   57   58   67   70   71   75  188
2    1    3    1    1    1    1    1


And to construct chi-squared goodness of fit test, I did:

p1 <- dpois(0:188, lambda = 4.33657562)
p1 <- c(p1, 1 - sum(p1))
names(p1) <- c(0:188, "more")
chisq.test(table(factor(vec, levels = c(0:188, "more") )), p = p1)


I do not know why I got p value = NA at the end. Please explain to me what is going wrong here ?

X-squared = NaN, df = 189, p-value = NA

The distribution of discrete data is as follows, with the purple line - the shifted poisson distribution ($\lambda = 4.33$)

• Suddenly, this looks very familiar. Is this for some class? – Glen_b -Reinstate Monica Sep 8 '13 at 5:56
• No, but I am preparing myself for class in future – user1769197 Sep 8 '13 at 6:00
• Then it would appear to count as self-study, but perhaps it's not absolutely critical here. – Glen_b -Reinstate Monica Sep 8 '13 at 6:01
• I have been stuck in this p value = NA problem for almost the whole day. I just wanted to know what went wrong. It would be helpful, if you can give me some help in this problem, because I think I am still not grasping the concept of chisq test well enough. – user1769197 Sep 8 '13 at 6:04
• Try working out the contribution to chi-square of the last cell by hand. That would be my guess as to why you get NA. Incidentally, I think the occurrence of a value of 188 with that sample size rules out a Poisson(4.33) pretty comprehensively. – Glen_b -Reinstate Monica Sep 8 '13 at 6:06

Yes, that strategy - shifting back, then estimating in the usual way - should work fine (with the proviso that you then need to shift the resulting fitted distribution back again if you want to write it as a shifted Poisson on the original data).

An example in R, using fitdistr:

 v <- rpois(110,3.2)+1  # make up some data
library(MASS)
vfit <- fitdistr(v-1,"poisson")
vfit
lambda
3.1909091
(0.1703181)
obf <- as.vector(table(v)/sum(table(v)))
plot(1:8,obf,type="h",lwd=2,xlim=c(0,10),ylim=c(0,0.25))
lines((1:10)+.05,dpois(0:9,3.191),col=2,type="h",lwd=2)
abline(h=0,col=8)


• Thanks. Actually, there is one more question I want to ask. Please see edited question in 30 minutes. – user1769197 Sep 8 '13 at 5:29
• I hope the discussion in comments under the question is now sufficient to cover that issue for you. – Glen_b -Reinstate Monica Sep 8 '13 at 16:54