## Models with both observation and process error

Statistical models with only observation error, or only process error, make the assumption that stochasticity predominantly influences one aspect of the population dynamics. However, there are situations where both process and observation errors are sufficiently large to impact inference about population dynamics in a measurable way. Owing to the presence of process error, such statistical models are easier to study for discrete biological models. A popular statistical model of discrete population growth with both process and observation error is

N — N, - iexp(« + ¿log(N, - i))exp(£,) Y, — N exp(F,) where E ~ g(a2) and F ~ f (r2)

where g(a2) and fY2) are normal distributions with mean zero and variances a2 and y2, respectively. Statistical models ofthis form are referred to as state-space models. The model shown in eqn [10] is the Gompertz state-space model, which can be thought of as a logarithmic form of the logistic growth model. The likelihood function for models with both observation and process error is more challenging to derive. For the Gompertz state-space model, the likelihood function is n 1 n /

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