什麼是無信息統計量及如何消去多餘參數
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(3) What is Noninformative Statistics and How to Eliminate the Nuisiance Parameter NSC 87-2118-M-009-003 86 8 1
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(6) . Abstract. ! "#$% & ' ()*+,-./01*2345 67*89 :;"< =>?@A BC*89 D E FGHIJK=LM *NO PQ= Jeffrey's R?S T/;U"<V WXC*YZ :/"<[6\ ]%E F^_` *abc/ Kde/"<* fg4*eC*R?STh i j`/E F
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(11) * *^NO . is a difficult problem. For the single parameter case, we have many good results in our early work. But for the many nuisance parameter cases, this problem is not clear yet. The purpose of this project is trying to solve this problem in several steps. It is well known that the Jeffrey's prior is a good prior and have many good properties when the parameter is single. However, if the parameter is multi dimensional, then the Jeffrey's prior is no longer good. So the first step in this project is to find a good prior on the high dimensional parameter space. In the second step, we will study the orthogonality of parameters. Up to now, people define the orthogonality of parameters only through mathematics formula, but we think that we should look at the orthogonality form the statistical point of view such that the orthogonality can be applied to the irregular or discrete situation. The last thing we want to do in this project is to discuss the. As we know, how to eliminate the effect of nuisance parameters in a statistical model. klYh . situations in which the data does not contain any information about one parameter. We 1.
(12) sample size is unity. In this research, we will discuss the relationship between the average likelihood and the noninformative statistic in the group transformation models.. believe that if we can solve the above problems, then we know how to eliminate the effect of the nuisance parameter. Keywords: nuisance parameters, likelihood function, group transformation model, orthogonality.
(13) . (1) In a group transformation model, if the group G can be embedded into the real line such that the composition and inverse operators are both continuous functions, then there exists h() such that h() =g-1 and , i.e., is orthogonal to g h().. Let a statistical model be parameterized as ( , ) where is the parameter of interest and is the nuisance parameter. It is quite possible that a single observation does not contain any information about. For. (2) Let X1 and X2 have densities p(x), q(x) respectively. Suppose h is a one-to-one transformation of X, and let p'(x), q'(x) denote the densities of h(X1) and h(X2). example, if X is a random variable with normal distribution with mean and variance then it is reasonable to say that X does not contain any information about. If the data contain no information about,. respectively, then (3) Let (dg) denote (g|)dg, then (dg). then we will hope that the likelihood function for is a constant function.. is a left invariant measure on G. Since all the left invariant measures on G are up to some constant and the choice of weighting function in average likelihood is also up to some constant, we can choose (dg) to be independent of , say (dg).. To give a definition of "no information" is a difficult problem. Here, we give a sufficient condition for a random variable to contain no information about as follows (This idea can also be found in Barndorff-Nielsen (1976) and Dawid (1975)): If X satisfies the condition (N) Hung and Wong (1996), then we can say that X contains no information about.. (4) Let G be a unimodular group and Y f(y; ,g) satisfies condition (N), and conditions in Lemma 2.1, then the average likelihood function ofis a constant function.. This definition can be justified by the idea of invariant test (Cox and Hinkley 1974), see Hung and Wong (1996). From experience we know that if we can get the "right" likelihood function when we have only a single observation, then we will get the right likelihood function for all sample sizes. Therefore, we believe that one should pay more attention on the case when the. (5) Let X1, X2, ... , Xn are i.i.d. f(x|, g), such that for eachthe family f(x|, g) satisfies the condition (N). Then the marginal distribution of (X1-1 X2, X1-1 X_3, ... , X1-1 Xn ) depends only on. And the conditional distribution of X1 given (X1-1 . 2.
(14) X2, X1-1 X3,..., X1-1 Xn). satisfies the. condition (N). (6) Assume X1 X2, ..., Xn satisfies the conditions in 5. Then the average likelihood function of is proportion to the marginal density of (X1-1 X2, X1-1 X3,..., X1-1 Xn). (7) In the group transformation model, we require that the group is unimodular. This is a reasonable assumption. In fact, the compact groups (e.g. rotation group), finite groups (e.g. finite permutation group), denumerable discrete groups (e.g. integer) and abelian groups (e.g. scale and location groups) are all unimodular. Barndorff-Nielsen, O.(1973), "On Mancillary", Journal of Biometrika(1973), 60, 3, p.447-455. Barndorff-Nielsen, O.(1976), "Nonformation", Journal of Biometrika(1976), 63, 3, p.567-571. Fisher, R. A.(1935) "The logic of inductive inference", Journal of the Royal Statistic Society(1935), 98, p.39-54. Hung, H.N. and Wong W.H. (1996) "Average Likelihood". Technical report, University of Chicago. Sprott, D. A.(1975), "Marginal conditional sufficiency", Journal Biometrika, 62, 3, p.599-605.. and of. 3.
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