Integrated likelihood inference in multinomial distributions

Consider a random vector $$(N_1, N_2, \ldots , N_m)$$ ( N 1 , N 2 , … , N m ) with a multinomial distribution such that $${{\,\textrm{E}\,}}\left( N_j ; \theta \right) = n p_j(\theta )$$ E N j ; θ = n p j ( θ ) , $$ j=1, \ldots , m$$ j = 1 , … , m , where $$p_1, \cdots , p_m$$ p 1 , ⋯ , p m are known functions of an unknown d-dimensional parameter, satisfying $$p_1(\theta ) + \cdots + p_m(\theta ) = 1$$ p 1 ( θ ) + ⋯ + p m ( θ ) = 1 . This paper considers non-Bayesian likelihood inference for a real-valued parameter of interest $$\psi = g(\theta )$$ ψ = g ( θ ) , for a known function g, using an integrated likelihood function. The integrated likelihood function is constructed using the zero-score expectation (ZSE) parameter, proposed by Severini (Biometrika 94:529–524, 2007); thus, the integrated likelihood function has a number of important properties, such as approximate score- and information-unbiasedness. The methodology is illustrated on the problem of inference for the entropy of the distribution.