I'm working on a project to forecast the distribution of a baseball player's "At Bats per game" (a baseball statistic w/ integer domain) using a player's position in his team's batting order as a predictor (1st up to bat, 2nd up to bat etc).
I went with my initial thought, which was to train a logistic regression on historical data w/ a poisson assumption, plug in the batting order for the game I want to predict and just nab the predicted lambda from the model without actually attempting to predict a value.
I then decided to plot the density of my "fitted" poisson model over the empirical density from my data. Every time I do this the poisson distribution has much more variance than the empirical density. I have several thousand data points so I suspect my empirical density should be pretty good.
I've attached a picture of the empirical density(proportion of all at bats for players with a batting order = 2) plotted against a poisson with lambda predicted for batting order = 2 from my glm. Is there another technique I can use to better predict a distribution of count data? I'm assuming a negative binomial glm will give roughly the same results. I know it's somewhat taboo but it seems like I could get a much better fit for the distribution using a normal distribution from an ols. Thanks for any advice!