Is there any statistical method testing the paired poisson data? Is there a statistical method which was used for testing the paired Poisson data? Just like the paired $t$-test for numerical data and paired(McNemar)-$\chi^2$ for binomial distributed data, I'm looking for a test for paired Poisson distributed data.
 A: Because you have not responded with information about your data, I will briefly
discuss a couple of possible tests using some fake data. (These are not the only possible tests, but should be appropriate for counts as large as those in the data shown.)
Consider the following data for AM, PM, and ten paired differences $d_i:$
       am pm   d
 [1,]  86 99 -13
 [2,]  96 90   6
 [3,] 108 77  31
 [4,]  95 97  -2
 [5,] 103 87  16
 [6,]  98 90   8
 [7,] 113 89  24
 [8,] 115 91  24
 [9,]  96 84  12
[10,]  96 90   6

Suppose we expect AM measurements to be larger than PM measurements, if there
is any difference. Thus we test $H_0: \lambda_a = \lambda_p$ against
the one-sided alternative $H_a: \lambda_a > \lambda_b.$
Because data are Poisson with fairly large means, it is reasonable to
assume that they are nearly normal. If the null hypothesis is true, it is
reasonable to assume that the usual statistic $T = \frac{\bar d - 0}{s_d/\sqrt{10}} = 2.6836$ for a paired t test is approximately distributed according to Student's t
distribution with $n - 1 = 10 - 1 = 9$ degrees of freedom. The critical
value for a test at the 5% significance level is $c = 2.262,$ so we
reject at the 5% level (the P-value is about 0.013).
Of course, the data are Poisson, not normal. So the critical value and the P-value may not be exactly correct. However, according to a Shapiro-Wilk test,
the data are consistent with normality. Also, the nonparametric (rank-based)
Wilcoxon signed-rank test rejects the null hypothesis that the median of the
$d_i$'s is zero (against the right-sided alternative) with P-value approximately 0.02; an exact P-value is not available because there happen to be ties among the $d_i.$
(The Wilcoxon test does not assume normality, but does assume that the $d_i$ are roughly symetical.)
Note: My fake data do not show significant correlation between AM and PM, and correlation would be typical with real paired data. But if anything, correlated data
with counts about the same size would tend to have even lower P-values in both the t and Wilcoxon tests. If you have questions about whether these tests
are appropriate for your data, please include some details along the lines
suggested in my earlier Comment, and I will take a look. 
