Finding confidence interval for a Poisson parameter that varies over time

I have data as $$t=[0, 1 ,2 ,3,4,5,6]$$ and $$X(t)=[395000,500000,400000,300000,300000,250000,300000]$$ and I am fitting to these data assuming $$X(t)$$~Poisson$$X_0e^{-\theta t}$$ to estimate the parameter value of $$\theta$$. Using the maximum likelihood method the estimate of $$\theta$$ was found.
Now I want to compute a confidence interval (CI) around $$\hat \theta$$.

1) I tried using the methods in here, but according to that post $$\lambda=X_0e^{-\theta t}$$ how can I estimate CI as the $$\lambda$$ value varies with time.
Since I don't have a large sample, I thought of using the chi square approach linked in that post in here but it is only for a single sample.
I don't understand how that could be used with my data where the Poisson parameter varies with time.

2) I also saw that there is a method called profile-likelihood based confidence intervals. I do not know about them so read what was given here. What I thought it meant was, first find the log-likelihood value $$(L(\theta))$$ at the maximum likelihood estimate $$\hat \theta$$ and then find $$(\beta_1,\beta_2)$$ such that it would be $$L(\beta_1)=L(\hat \theta)-1.92$$ and $$L(\beta_2)=L(\hat \theta)+1.92$$ where 1.92 comes from $$χ^2 (0.05,1)/2 = 1.92$$.
However, I don't think what I have understood is correct, as there cannot be $$L(\hat \theta)+1.92$$ as $$L(\hat \theta)$$ is the maximum.
Also, based on what is this $$χ^2 (0.05,1)$$ 1 degree of freedom used?

Can someone please help me understanding these and to find a way to estimate the confidence interval for $$\hat \theta$$

1 Answer

Assuming I have understood correctly, this is technically a Poisson regression, so why not just use the asymoptotic wald intervals in GLM theory? Your model assumes that the parameter of a poisson density takes form $$X_0\exp(-\theta t)$$, which can be rewritten as $$\exp(-\theta t + \theta_0)$$, where $$X_0 = \exp(\theta_0)$$.

Note that Poisson regression estimates a model in the form

$$\lambda = \exp(x^T\beta)$$

where $$y\sim \mbox{Poisson}(\lambda)$$

I'm sure the data will estimate a negative $$\theta$$, so no need to force it to be negative. Using R,

t = 0:6
X = c(395000,500000,400000,300000,300000,250000,300000)/1e3

plot(t,X)

model = glm(X~t, family ='poisson')
summary(model)
confint(model)

>>>Waiting for profiling to be done...
2.5 %      97.5 %
(Intercept)  6.0460950  6.17818844
t           -0.1111713 -0.07111918


Note I've scaled your data, and so the parameters should be interpreted accordingly.

• Thank you. However, I have other data which are censored. For example $X(t)= [200000,10,10,10,10,10,10]$. So I have written the log likelihood in matlab for censored regression and use fminsearch to find the MLE. Therefore, it is not possible to calculate the CI in this way. According to Wald interval CI is $\hat \lambda +/- Z_{\alpha/2} \sqrt({\hat \lambda \over n})$. So, here $\hat \lambda=\theta t+\theta_0$ right? If I am to manually calculate what should be done with this time variable $t$?. Also, when $\theta$<0 isn't it possible for square root to have a negative value? – clarkson May 6 '19 at 1:48
• @clarkson Censoring changes things. Can you say more about the structure of the problem? Also, as for calculating the wald interval manually, your estimate of $\lambda$ is $\exp(\theta t + \theta_0)$ which is always positive. – Demetri Pananos May 6 '19 at 2:49
• So,in $\exp(\theta t + \theta_0)$ what should the value of $t$ be because it is a vector of values from 0-6? If I understand what to do with this $t$ I think I can come up with the confidence interval. So, the structure of the problem is data is censored at 10. When calculating the log-likelihood, observations that are censored (values with 10) are accounted through the cumulative distribution while the other values are through the probability distribution of Poisson $(X_0e^{-\theta t})$ – clarkson May 6 '19 at 3:32
• Please edit your original question to describe the censoring. It might also help to include a description of exactly what it is you are modelling (i.e. the physical process) – Demetri Pananos May 6 '19 at 12:13
• I was able to find a way to compute confidence intervals using the profile-likelihood method. As you suggested I divided $X(t)$ by 10. In this case how would I interpret the MLE that I found using $X(t)$/10 data. What is the relationship between the MLE and CIs that I would have obtained had I not scaled the data by dividing by 10. – clarkson May 8 '19 at 0:47