Estimating Density Dependence, Process Noise, and Observation Error

We describe a discrete-time, stochastic population model with density dependence, environmental-type process noise, and lognormal observation or sampling error. The model, a stochastic version of the Gompertz model, can be transformed into a linear Gaussian state-space model (Kalman filter) for convenient fitting to time series data. The model has a multivariate normal likelihood function and is simple enough for a variety of uses ranging from theoretical study of parameter estimation issues to routine data analyses in population monitoring. A special case of the model is the discrete-time, stochastic exponential growth model (density independence) with environmental-type process error and lognormal observation error. We describe two methods for estimating parameters in the Gompertz state-space model, and we compare the statistical qualities of the methods with computer simulations. The methods are maximum likelihood based on observations and restricted maximum likelihood based on first differences. Both offer adequate statistical properties. Because the likelihood function is identical to a repeated-measures analysis of variance model with a random time effect, parameter estimates can be calculated using PROC MIXED of SAS. We use the model to analyze a data set from the Breeding Bird Survey. The fitted model suggests that over 70% of the noise in the population's growth rate is due to observation error. The model describes the autocovariance properties of the data especially well. While observation error and process noise variance parameters can both be estimated from one time series, multimodal likelihood functions can and do occur. For data arising from the model, the statistically consistent parameter estimates do not necessarily correspond to the global maximum in the likelihood function. Maximization, simulation, and bootstrapping programs must accommodate the phenomenon of multimodal likelihood functions to produce statistically valid results.