# What distributions are possible for an arrival rate?

I'm really struggling to find a good statistical distribution. I've tried Poisson and Gamma so far, but without success (best I've got was a p-value of 0,00005 with a Pearson Chi-Square test). So I really hope you can send me in the right direction.

The case is as follows: I'm studying the arrival rate for the application for mortgages. I'm trying to determine the arrival rate per hour (thus the number of applications that arrive in a certain hour). This data are the total number of arrivals in a specific hour, in this example between 13:00 AM and 14:00 AM. This is the data: Example data

I'm trying to determine the arrival rate per hour. These data are the total number of arrivals in a specific hour, in this example between 13:00 AM and 14:00 AM. This is the dataset: Example data

As an example I've taken a set with a relative high N.

I got the following metadata of the distribution:

Mean 15,60
St Error 0,32
Median 14,50
Mode 16,00
Standard Deviation 9,27
Variance 85,92
Kurtosis 5,49
Skewness 1,54
Range 68,00
Minimum 1,00
Maximum 69,00
Sum 12853,00
Count 824,00

I also have a histogram: I've rejected Poisson, since the variance is not the same as the mean. Furthermore I've tried two-parameter gamma with alpha = mean^2/Variance and beta = Variance/mean, but without success.

• Welcome to Cross Validated! For a start would you explain what you're in fact measuring? The Poisson distribution is for a random variable that takes non-negative integer values (e.g. counts per a fixed time interval); the gamma for a continuous non-negative r.v. (e.g. time in minutes between successive events): they can't both be appropriate. Your summary data suggest the former case. Note also that the horizontal-axis labels are missing from your histogram. Jan 11 '16 at 9:28
• Can you provide data sample as an example?
– Tim
Jan 11 '16 at 9:30
• Please be more precise: You wrtie "best I've got was a p-value of 0,00005", probability of what? You write "I've rejected, ..." What did you reject? Jan 11 '16 at 9:44
• I'm trying to determine the arrival rate per hour. This data are the total number of arrivals in a specific hour, in this example between 13:00 AM and 14:00 AM. This is the data: Example data Jan 11 '16 at 9:46
• Are you trying to predict the number of arrivals for the next day,week or month ? Are you trying to detect an usual value when it arrives ? Similarly we have seen the question "what is the probability that the most recent value comes from (is generated by) the historical/observed hisorical distribution ? Are you trying to find out if the distribution for specific hours are statistically different from each other ? Are you trying to find out if the distribution has changed over time ? Jan 11 '16 at 13:26

Gamma is continuous, so I wouldn't (at least not to begin with) consider it for count data.

When variance tends to be larger than mean, one common choice is the negative binomial; it can be regarded as a mixture of Poissons (where the Poisson rates come from a gamma distribution). As a result it can often be suitable for situations where you have a populations which may be heterogeneous.

A negative binomial with the same mean and variance as your sample looks like this: This seems more or less reasonable.

[However, in your case it may be that a different mixture of Poissons could work better, perhaps a finite mixture with only two or three components could work.]

• Thank you very much for your clear and extensive. I could really work with this! Jan 11 '16 at 15:37
• How did you determine the parameters for this function? I've used Statsmodels in Python, but got some clearly different distribution Jan 12 '16 at 14:55
• The answer to that question is already stated in my answer ("with the same mean and variance as your sample") -- that is, I matched the first two moments (since that information was readily available in your question). If you use MLE you will get different parameter estimates, but with such a large count and a not so far from reasonably negative-binomialish shape it shouldn't be very different. Which negative binomial is statsmodels in Python fitting? What parameter values do you have? Jan 12 '16 at 14:58
• I've used Negative Binomial Regressions, with an array of ones as the independent variable. The output I've got is: const 2.747162 alpha 0.283321 If I interpret it right, the p-value of the distribution is the alpha and the constant is the r-value is the constant? Or do I misinterpret this Jan 12 '16 at 15:53
• I can't tell what those parameters are representing (surely the documentation tells you?), but it looks unlikely that your interpretation could be right -- that would imply a mean of less than 1.09 Jan 12 '16 at 16:12