## GENERAL THEORY OF INFERENTIAL MODELS II. MARGINAL INFERENCE

Citations: | 1 - 0 self |

### BibTeX

@MISC{Martin_generaltheory,

author = {Ryan Martin and Jing-shiang Hwang and Chuanhai Liu and Academia Sinica},

title = {GENERAL THEORY OF INFERENTIAL MODELS II. MARGINAL INFERENCE},

year = {}

}

### OpenURL

### Abstract

This paper is a continuation of the authors ’ theoretical investigation of inferential model (IMs); see Martin, Hwang and Liu (2010). The fundamental idea is that prior-free posterior probability-like inference with desirable long-run frequency properties can be achieved through a system based on predicting unobserved auxiliary variables. In Part I, an intermediate conditioning step was proposed to reduce the dimension of the auxiliary variable to be predicted, making the construction of efficient IMs more manageable. Here we consider the problem of inference in the presence of nuisance parameters, and we show that such problems admit a further auxiliary variable reduction via marginalization. Unlike classical procedures that use optimization or integration, the proposed framework eliminates nuisance parameters via a set union operation. Sufficient conditions are given for when this marginalization operation can be performed without loss of information, and in such cases we prove that an appropriately constructed IM is calibrated, in a frequentist sense, for marginal inference. In problems where these sufficient conditions are not met, we propose a marginalization technique based on parameter expansion that leads to conservative marginal inference. The marginal IM approach is illustrated on a number of examples, including Stein’s problem and the Behrens-Fisher problem.

### Citations

2456 |
A Mathematical Theory of Evidence
- Shafer
- 1976
(Show Context)
Citation Context ...n detail in Zhang and Liu (2010), Martin, Zhang and Liu (2010), and Martin, Hwang and Liu (2010). Here we give a quick review for completeness. 3.1. Belief functions. Belief functions (Dempster 1967; =-=Shafer 1976-=-) are similar to, but more general than, probability measures. The DempsterShafer (DS) theory for statistical inference constructs a belief function on Θ as follows. For observed data X = x, define th... |

419 |
Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univerisity
- Bickel, Klaassen, et al.
- 1993
(Show Context)
Citation Context ...s) of the parameter vector θ are of interest. Linear regression, with θ = (β, σ 2 ), is one such example where primary interest is in the vector β of regression coefficients. Semiparametric problems (=-=Bickel et al. 1998-=-), such as the Cox proportional hazards model, form another important class of examples. More formally, suppose θ can be decomposed as θ = (ψ, ξ), where ψ is the parameter of interest and ξ is the nui... |

294 | Nonparametrics: Statistical Methods Based on Ranks. Upper Saddle River - Lehmann - 1998 |

273 | On the mathematical foundations of theoretical statistics - Fisher - 1922 |

169 |
Theory of Statistical Estimation
- Fisher
- 1925
(Show Context)
Citation Context ...also prove a “conditional version” of the fundamental Theorem 3.1 of Zhang and Liu (2010), and draw parallels between this conditional IM framework and Fisher’s notion of sufficiency and ancillarity (=-=Fisher 1925-=-, 1934, 1935) in group transformation models. This conditioning step, reviewed in Section 3, will often reduce the dimension of the auxiliary variable to that of the parameter. But in marginal inferen... |

87 | The generalization of student’s problem when several different population variances are involved - Welch - 1947 |

86 |
An empirical Bayes approach to statistics
- Robbins
- 1956
(Show Context)
Citation Context ...d to as Stein’s paradox (Stein 1956). Attempts to understand this phenomenon have led, at least in part, to the development of empirical Bayes methods and shrinkage estimation (Efron and Morris 1977; =-=Robbins 1956-=-, 1964). The basic a-equation for this problem is (4.1) X = ψξ + U, U ∼ Nn(0, I). file: MHL-marg.tex date: August 17, 2010MARGINAL IMS 9 The parameter ξ is not of interest. In fact, there is no less ... |

79 | The logic of inductive inference - Fisher - 1935 |

37 | Two New Properties of Mathematical Likelihood - FISHER - 1934 |

34 | Principles of Statistical Inference - Cox - 2006 |

17 | Practical Solutions of the Behrens-Fisher Problem - Scheffé - 1970 |

11 | The functional-model basis of fiducial inference - Dawid, Stone - 1982 |

10 | Dempster-Shafer theory and statistical inference with weak beliefs - Martin, Zhang, et al. - 2010 |

6 | Pivotal models and the fiducial argument - Barnard - 1995 |

6 | On the Behrens-Fisher Problem: A Review - Kim, Cohen - 1998 |

6 | Fudicial Distribution of Several Parameters with Application to a Normal System - Segal - 1938 |

3 | Contributions to the theory of Student’s t-Test as applied to the problem of two samples - Hsu - 1938 |

2 | Exact solutions to the BehrensFisher problem: asymptotically optimal and finite sample efficient choice among - Dudewicz, J, et al. - 2007 |

1 | 10 1054–1074. With discussion. MR673643 - Statist - 1967 |

1 | Inference 137 1584–1605. MR2339261 - Plann - 1977 |

1 | A weak belief approach to inference on constrained parameters: elastic beliefs. Working paper - Leaf, D, et al. - 2010 |

1 | file: MHL-marg.tex date: August 17, 2010 IMS 25 - Fraser - 1968 |

1 | The empirical Bayes approach to statistical decision problems - unknown authors - 1964 |