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Integrated likelihood inference in multinomial distributions

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Abstract

Consider a random vector \((N_1, N_2, \ldots , N_m)\) with a multinomial distribution such that \({{\,\textrm{E}\,}}\left( N_j ; \theta \right) = n p_j(\theta )\), \( j=1, \ldots , m\), where \(p_1, \cdots , p_m\) are known functions of an unknown d-dimensional parameter, satisfying \(p_1(\theta ) + \cdots + p_m(\theta ) = 1\). This paper considers non-Bayesian likelihood inference for a real-valued parameter of interest \(\psi = g(\theta )\), 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.

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Acknowledgements

The comments of the Associate Editor and a referee are gratefully acknowledged.

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  1. Department of Statistics, Northwestern University, 2006 Sheridan Road, Evanston, IL, 60208, USA

    Thomas A. Severini

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  1. Thomas A. Severini
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Correspondence to Thomas A. Severini.

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Severini, T.A. Integrated likelihood inference in multinomial distributions. METRON 81, 131–142 (2023). https://doi.org/10.1007/s40300-022-00236-x

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