I am trying to estimate the probability $q$ that a household with certain known covariates will move to a new home in the following year, by estimating an event rate $\lambda$ dependending on some covariates in an exponential distributon model. The data I have available comes from a large survey which includes the covariates and how long a household has lived in their current home up until the survey date (I'll call it “lvd”).
The trouble is, all reported “lvd” values are right censored survival times. Most (but likely not all) households will move some time after the survey date. If I look at a histogram of “lvd” data, I see something resembling an exponential distribution. Thinking about this distribution, if the actual event rate was very large (i.e. everybody moves almost all the time), you would expect the distribution of “lvd” to have most of its density close to 0. If the actual event rate was close to zero, you would expect a very large spread in “lvd” times. My guess was that there should be a way to estimate the event rate based purely on these right censored data, but I don’t know how. I tried to use a maximum-likelihood estimator much like in the answer to this question: [ ML estimate of exponential distribution (with censored data) ] but, I ran into the same problem as mentioned there: the MLE gives $\lambda = 0$ when all data is right censored.
Can anyone give a suggestion how one could estimate the event rate $\lambda$ (or $q$ directly) when only right censored data is available, or if this is at all possible?
(*I’m using an exponential model because I wish to apply the model to situations where I don’t know how long a household has lived in their current home, so I want some general/average event rate $\lambda$ for young vs older famillies, or married vs. single etc. Also, I’m pretty new to survival analysis and non-parametric models, so I just stuck with what I knew. If some other approach is preferred, I wouldn’t mind using that.)