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Higher order density approximations for solutions to estimating equations

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  • Almudevar, Anthony

Abstract

General formulae for the intensity function of a point process defined by the solution set of a system of smooth random equations are widely available in the literature, offering a precise characterization of a type of random process arising naturally in many fields. Almost all are modifications or generalizations of an original formula derived by Rice (1945) and share the same relatively simple structure. Related methods have been applied to the evaluation of the density of the solution to multidimensional estimating equations arising in statistical inference (Skovgaard, 1990; Jensen and Wood, 1998; Almudevar et al., 2000). This approach has been able to verify or extend a variety of known approximation methods, but has otherwise not been commonly used in the area of small sample asymptotic theory, despite its potential for the development of approximation methods of considerable generality. This article develops a general order O(1/n) density approximation method for solutions to multidimensional estimating equations which are sums of continuous independent, non-identically distributed random vectors. Two issues in particular which arise in the application of the Rice formula are addressed. Validation of this formula is often technically challenging, so a set of general conditions motivated specifically by the application to estimating equations is developed. In addition, the Rice formula includes a conditional expectation which would be difficult to evaluate for non-Gaussian processes. To address this issue, a general order O(1/n) approximation for expectations conditioned on random sums is derived, which may be directly used in the Rice formula under the hypothesis considered here. The method is demonstrated using the negative exponential regression model, a type of non-canonical generalized linear model.

Suggested Citation

  • Almudevar, Anthony, 2016. "Higher order density approximations for solutions to estimating equations," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 424-439.
    • Handle: RePEc:eee:jmvana:v:143:y:2016:i:c:p:424-439
    DOI: 10.1016/j.jmva.2015.09.014
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    References listed on IDEAS

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    1. M.R. Leadbetter & G.V. Spaniolo, 2004. "Reflections on Rice's Formulae for Level Crossings - History, Extensions and Use," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 46(1), pages 173-180, March.
    2. Ib M. Skovgaard, 2001. "Likelihood Asymptotics," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(1), pages 3-32, March.
    3. Chris Field & John Robinson & Elvezio Ronchetti, 2008. "Saddlepoint approximations for multivariate M-estimates with applications to bootstrap accuracy," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(1), pages 225-227, March.
    4. Chris Field & John Robinson & Elvezio Ronchetti, 2008. "Saddlepoint approximations for multivariate M-estimates with applications to bootstrap accuracy," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(1), pages 205-224, March.
    5. Jens Jensen & Andrew Wood, 1998. "Large Deviation and Other Results for Minimum Contrast Estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(4), pages 673-695, December.
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