Chi-Squared Tests For and Against an Order Restriction on Multinomial Parameters
研究了多项分布参数在顺序约束下的卡方检验，给出了最小卡方估计量和Neyman修正卡方估计量的闭式表达式，并证明了这些检验与似然比检验的渐近等价性，通过赛马实例和模拟研究展示了其应用和性能。
Abstract Chi-squared tests are considered for testing homogeneity against an order restriction and for testing an order restriction against all alternatives on multinomial parameters. Closed-form expressions are obtained for the minimum chi-squared estimator and the Neyman modified chi-squared estimator. The asymptotic equivalence between these chi-squared tests and the likelihood ratio test is established. The latter has been studied by Robertson (1978). Their asymptotic null distribution is the chi-bar-squared distribution. These chi-squared tests are applied to a numerical example in horse racing. Also included are the results of some simulation studies conducted to investigate the rate of convergence to the limiting null distributions and to compare the powers of these tests. Chi-squared tests are widely used statistical tools for making inferences about a collection p 1, p 2, …, pk of multinomial parameters. Many such collections encountered in practice exhibit a trend. For instance, pi may represent Pr(ai-1 < X ≤ ai), where X is a continuous random variable with a support (ao, ak) on the real line and a1 < a2 < … < ak-1 are equally spaced points in (a 0, ak ). If the probability density function is unimodal but otherwise unknown, then there is a positive integer m such that p2 ≤ … ≤ Pm-1 ≤ pm ≥ … ≥ pk-1. For the horse racing example in Section 4, one might suppose that p1 ≥ p2 ≥ … p8. In the Mendelian theory of inheritance (Robertson 1978), one might assume that p1 ≥ p2 = p3 ≥ p4 or p1 ≥ p2, p1 ≥ p3, p2 ≥ p4, and p3 ≥ p4. There are many other interesting order restrictions on the multinomial parameters. Suppose that the observations comprise a multinomial random vector with parameters n and p = (p1, …, pk). Let O be an order restriction on the coordinates of p. Let p° be such that p°1 = p°2 = … = p°k = k-1. Then p° satisfies O. Define the hypothesis Hi: i = 0,1,2 by H0: p = p°, H1: p satisfies O, and H2 places no restriction on p. It is well known that one-sided procedures are more powerful than their two-sided counterparts provided that the ordering is satisfied (Bartholomew 1961, sec. 4). If one is interested in testing the simple null hypothesis H0 and if the restriction, O, is one's firm belief, it is advantageous to restrict the alternatives to H 1 – H 0. On the other hand, if one is interested in testing one's intuition about the shape of the underlying parameter one may test H 1 against H 2 – H 1. The hypothesis H 1 is not linear. The traditional approach to categorical data cannot be used. In this article, the theory given by Barlow, Bartholomew, Bremner, and Brunk (1972) will be used to treat the order-restricted statistical inference.