一类半参数回归模型的经验似然

Empirical likelihood for a semiparametric regression model

考虑一类半参数回归模型yi=xi′β+g(xi)+ei,1≤i≤n,其中xi=(xi1,xi2,…,xik)′是固定设计点列,β=(β1,β2,…,βk)′是未知待估参数,g(.)是未知函数,ei是随机误差,且E(ei)=0,E(ei2)=σ2<∞.首先假设β已知,用非参数权函数方法估计非参数部分g,然后应用经验似然方法构造似然函数,证明了经验对数似然比统计量渐近服从卡方分布.结果可用于构造半参数回归模型中参数向量的置信区域.

The application of semiparametric regression model yi=xi′ β＋g（xi）＋ei,1≤i≤n is considered,where xi=（xi1,xi2,…,xik）′ is non-random design variables,β=（β1,β2,…,βk）′ is unknown parameter,g（·） is unknown function.ei is random error and E（ei）=0,E（ei^）=σ^2〈∞.At the first,suppose β is known,g is estimated by nonparameter weight functions,then likelihood function is structured by empirical likelihood method.It is proved that the empirical log-likelihood ratio statistic is asymptotically chisquared distribute.The empirical likelihood confidence regions can be constructed accordingly.