Higher Order Asymptotic Sufficiency, Asymptotic Ancillarity in Time Series Analysis

Abstract

Let X_n = (X₁,⋯,X_n)' be an n-dimensional random vector which has an unknown parameter θ ∈ Θ ⊂ R¹. For the case when each X_j is independently and identically distributed, several works discussed the asymptotic sufficiency along the following directions. For an appropriate estimator ${\hat \theta _n}$ we define ${t_n}\, = \,{\hat \theta _n}\, + \,{\left\{ {nF\left( {{{\hat \theta }_n}} \right)} \right\}^{ - 1}}\, \cdot \,L_n^{\left( 1 \right)}\left( {{X_n},{{\hat \theta }_n}} \right)$, where F (θ) is the Fisher information and L⁽¹⁾_n(X_n,θ) ) is the derivative of the log-likelihood with respect to θ. Then LeCam(1956) showed that t_n is asymptotically sufficient in the sense that t_n is sufficient for a family {Q_n,θ; θ ∈ Θ} of probability measures and that ‖ P_n,θ - Q_n,θ‖ = o (1), uniformly on any compact set of Θ, where ‖ ⋅ ‖ is the variation norm. Under slight different conditions from those of LeCam, Pfanzagl(1972) showed that $\left\| {{P_{n,\theta }}} \right.\, - \,\,\left. {{Q_{n,\theta }}} \right\|\, = \,O\,\left( {{n^{ - \frac{1}{2}}}} \right)$, uniformly on every compact set of Θ, which improves the above result. Let ${s_n}\, = \,\left\{ {{{\hat \theta }_n},L_n^{\left( 1 \right)}\left( {{X_n},{{\hat \theta }_n}} \right),...,L_n^{\left( k \right)}\left( {{X_n},{{\hat \theta }_n}} \right)} \right\}$, where $L_n^{\left( j \right)}\left( {{X_n},\theta } \right)$ is the jth derivative of L_n(X_n, θ) with respect to θ. Suzuki(1978) proved that s_n is asymptotically sufficient up to order o(n-(k-1)/2) in the sense that s_n is sufficient for a family {Q_n,θ; θ ∈ Θ} of probability measures and that

$$\left\| {{P_{n,\theta }}} \right.\, - \,\left. {{Q_{n,\theta }}} \right\|\, = \,o\left( {{n^{ - \frac{{k - 1}}{2}}}} \right)\,$$

, uniformly on any compact set of Θ. Using the maximum likelihood estimator $\hat \theta _{ML}^{\left( n \right)}$ instead of ${\hat \theta _n}$, Ghosh and Subramanyam(1974) mentioned that $\left( {\hat \theta _{ML}^{\left( n \right)},L_n^{\left( 2 \right)}\left( {{X_n},\hat \theta _{ML}^{\left( n \right)}} \right),L_n^{\left( 3 \right)}\left( {{X_n},\hat \theta _{ML}^{\left( n \right)}} \right)L_n^{\left( 4 \right)}\left( {{X_n},\hat \theta _{ML}^{\left( n \right)}} \right)} \right)$ is asymptotically sufficient up to order o(n^-1).