What is the statistical method that I should use for the following problem? I have been given a problem and need to find the best statistical method that would help. The problem is as follows:
A business has approx. 100,000 active customers who spend between £500 per annum and £200,000 per annum.  The smaller ones transact irregularly, once per year, the larger ones maybe once per week.  They are distributed across 15 different market sectors.
The business has started to decline, and whilst we have implemented a process for making offers to customers once they have closed their account, it is thought this would be much more effective if we could predict which customers are likely to close, and make offers to keep them before they close.
How would you go about determining which customers to make offers to?
 A: I would consider a Hidden Markov Model for the underlying state of the customer. Learn the parameters of the model based on the history of your customers, and then try to see if there is any feature in the hidden variables that is a predictor for the customer closing the account. Then finding the probability that your customer is in each one of the hidden states, given his/her past history of purchases, is a simple application of the forward algorithm.
Easier said than done, but I find it feasible.
A: Maybe a logistic mixed model where you predict whether a customer will close an account or not (that's the response or y variable) using what you know about the customers, like how much they spend, how many transactions they do and anything else you may know. You could include market sector as a random effect because it's a categorical variable with many classes that may account for baseline variation in whether a customer will close an account.
Close account ~ spendingPerAnnum + TransactionRate + (1|marketSector)
If there are a lot of variables you could use to predict whether people will close accounts, you could also explore machine learning techniques too. Your goal seems to be prediction, so I would try a few techniques and see where you can get the best accuracy, without concerning yourself with ease of interpreting the model. 
