An Exact Test for Multiple Inequality and Equality Constraints in the Linear Regression Model
线性回归中多重不等式与等式约束的精确检验AI 解读 ↓
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线性回归中多重不等式与等式约束的精确检验
- 研究问题
- 在正态线性回归中，如何对回归系数的多重不等式约束及等式与不等式混合约束构造精确检验并计算临界值？
- 方法与思路
- 作者借助二次规划对偶理论推导检验统计量的等价形式，讨论由加权 F 分布构成的精确分布及临界值、概率值的计算。
- 材料陈述的贡献
- 摘要称，研究建立多类约束检验的精确分布，并通过拉格朗日乘子把相关检验与多元单侧假设检验联系起来。
- 阅读关注点
- 应核对原 PDF 中各组原假设与备择、未知方差的处理、混合分布权重的确定，以及对偶表示和计算程序的条件。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；假设表达式、参数上下标及符号存在歧义，需查原 PDF，不能重构公式或推断片段外结果及刊本变化。
原文摘要
In this article we consider the linear regression model y = Xβ + ε, where ε is N(0, σ2I). In this context we derive exact tests of the form H: Rβ ≥ r versus K: β ∈ RK for the case in which θ2 is unknown. We extend these results to consider hypothesis tests of the form H: R1β ≥ r1 and R2β = r2 versus K: (β ∈ RK . For each of these hypotheses tests we derive several equivalent forms of the test statistics using the duality theory of the quadratic programming. For both tests we derive their exact distribution as a weighted sum of Snedecor's F distributions normalized by the numerator degrees of freedom of each F distribution of the sum. A methodology for computing critical values as well as probability values for the tests is discussed. The relationship between this testing framework and the multivariate one-sided hypothesis testing literature is also discussed. In this context we show that for any size of the hypothesis test H: λ = 0 versus K: β ∈ RK the test statistic and critical value obtained are the same as those from the hypothesis test H: λ = 0 versus K: λ ≥ 0, where λ is the expectation of the Lagrange multiplier arising from the estimation of β subject to the equality constraints Rβ = r. In this way we link the multivariate inequality constraints test to the much studied multivariate one-sided hypothesis test, H: μ = 0 versus K: μ ≥ 0, where μ is the mean of a multivariate normal random vector. We also show that the test H: R 1β ≥ r 1 and R 2β = r 2 versus K: β ∈ RK has the following equivalent test in terms of λ, H: λ = 0 versus K: λ1 ≥ 0, and λ2 ≠ 0, where λ1 is the subvector of λ corresponding to R1β ≥ r1 and λ2 corresponds to R 2β = r 2. Extensions of recent work in one-sided hypothesis testing for the coefficients of the linear regression model are also derived. For the normal linear regression model we derive exact tests for the hypothesis testing problems H: Rβ = r versus K: Rβ ≥ r and H: Rβ = r versus K: R 1β ≥ r, and R 2β ≠ r 2.