Contiguity and the statistical invariance principle.

*(English)*Zbl 0659.62029
Stochastics Monographs, 1. New York: Gordon & Breach Science Publishers. VIII, 236 p.; $ 39.00 (1985).

This book is in two parts, both concerned with the asymptotic behaviour of sequences of likelihood ratios. In the first part necessary and sufficient conditions are derived for the contiguity of two sequences of measures; in the second part necessary and sufficient conditions are determined for the log-likelihood process to be asymptotically Gaussian.

The notion of the contiguity of two sequences of measures was introduced by L. Le Cam [Univ. California Publ. Statist. 3, 37-98 (1960; Zbl 0104.127)] and is useful in the study of asymptotic properties of tests. It can be thought of as a generalization of the absolute continuity of two measures. Let \((\Omega^ n,{\mathcal F}^ n)_{n\geq 1}\) be a sequence of measurable spaces with \(P^ n\), \(Q^ n\) probability measures on \((\Omega^ n,{\mathcal F}^ n)\). The sequence of measures \(\{Q^ n\}\) is said to be contiguous to the sequence \(\{P^ n\}\) if for any sequence of sets \((A^ n)_{n\geq 1}\), \(A^ n\in {\mathcal F}^ n\), \(\lim_{n}P^ n(A^ n)=0\) implies \(\lim_{n}Q^ n(A^ n)=0\). When \(\Omega^ n=\Omega\), \({\mathcal F}^ n={\mathcal F}\), \(P^ n=P\) and \(Q^ n=Q\), the contiguity of \(\{Q^ n\}\) with respect to \(\{P^ n\}\) is simply the requirement that Q be absolutely continuous with respect to P.

At the opposite extreme to the absolute continuity of two measures is the possibility of their mutual singularity. This can be generalized to the idea of entire separability. The sequences of measures \(\{P^ n\}\) and \(\{Q^ n\}\) are said to be asymptotically entirely separated if there exist a sequence \(k_ n\) and sets \(A^{k_ n}\in {\mathcal F}^{k_ n}\) such that \(\lim_{n}P^{k_ n}(A^{k_ n})=0\) whereas \(\lim_{n}Q^{k_ n}(A^{k_ n})=1\). In statistical terms, \(\{P^ n\}\) and \(\{Q^ n\}\) are entirely separable if and only if there exists a sequence of tests of \(\{P^ n\}\) versus \(\{Q^ n\}\) which is consistent and has asymptotic size zero.

The authors study the case where the measurable spaces \((\Omega^ n,{\mathcal F}^ n)\) have further structure, each having an associated filtration (\({\mathcal F}^ n_ k)_{k\geq 0}\). This occurs naturally when n is the size of a random sample, \(\Omega^ n={\mathbb{R}}^ n\), \({\mathcal F}^ n={\mathcal B}^ n\) and \({\mathcal F}^ n_ k\) is the sub-\(\sigma\)-field generated by the first k observations in the sample. Denote the restrictions of \(P^ n\) and \(Q^ n\) to \({\mathcal F}^ n_ k\) by \(P^ n_ k\) and \(Q^ n_ k\), respectively. Necessary and sufficient conditions for the contiguity (entire separability) of \(\{Q^ n\}\) with respect to \(\{P^ n\}\) are derived in terms of the conditional likelihood ratios \[ \alpha^ n_ k=(dQ^ n_ k/dP^ n_ k)/(dQ^ n_{k-1}/dP^ n_{k-1}). \] The \(\alpha^ n_ k\) are \({\mathcal F}^ n_ k\)-measurable whereas the conditions derived are “predictable”. For example, in order that \(\{Q^ n\}\) be continguous to \(\{P^ n\}\) it is necessary and sufficient that \[ \lim_{N}\lim_{n}Q^ n(\sup_{k}\alpha^ n_ k\geq N)=0\quad and\quad \lim_{N}\lim_{n}Q^ n(\sum^{\infty}_{k=1}E_{P^ n}[(1- \sqrt{\alpha^ n_ k})^ 2| {\mathcal F}^ n_{k-1}]\geq N)=0. \] In the second half of the book, the same structure as before is retained, namely, a sequence of measurable spaces together with a filtration \((\Omega^ n,{\mathcal F}^ n,({\mathcal F}^ n_ k)_{k\geq 0})\) of sub- \(\sigma\)-fields. Again two sequences of measures \(\{P^ n\}\) and \(\{Q^ n\}\) are given on \((\Omega^ n,{\mathcal F}^ n,({\mathcal F}^ n_ k)_{k\geq 0})\) and their restrictions to \({\mathcal F}^ n_ k\) are denoted by \(P^ n_ k\) and \(Q^ n_ k\), respectively. A continuously parametrized process of likelihood ratios is constructed by setting \(Z^ n_ t=dQ^ n_{[nt]}/dP^ n_{[nt]}\). Necessary and sufficient conditions (again in predictable form) are derived for log \(Z^ n_ t\) to converge in distribution to a Gaussian martingale. This is achieved by applying recent results on the martingale central limit theorem which have been derived by the authors.

This is a research monograph, giving a detailed account of recent and definitive work in two related areas. In the words of the introduction to the series in which it is published, it provides “timely and authoritative coverage of areas of current research in a more extended and expository form than is possible within the confines of a journal article”.

The notion of the contiguity of two sequences of measures was introduced by L. Le Cam [Univ. California Publ. Statist. 3, 37-98 (1960; Zbl 0104.127)] and is useful in the study of asymptotic properties of tests. It can be thought of as a generalization of the absolute continuity of two measures. Let \((\Omega^ n,{\mathcal F}^ n)_{n\geq 1}\) be a sequence of measurable spaces with \(P^ n\), \(Q^ n\) probability measures on \((\Omega^ n,{\mathcal F}^ n)\). The sequence of measures \(\{Q^ n\}\) is said to be contiguous to the sequence \(\{P^ n\}\) if for any sequence of sets \((A^ n)_{n\geq 1}\), \(A^ n\in {\mathcal F}^ n\), \(\lim_{n}P^ n(A^ n)=0\) implies \(\lim_{n}Q^ n(A^ n)=0\). When \(\Omega^ n=\Omega\), \({\mathcal F}^ n={\mathcal F}\), \(P^ n=P\) and \(Q^ n=Q\), the contiguity of \(\{Q^ n\}\) with respect to \(\{P^ n\}\) is simply the requirement that Q be absolutely continuous with respect to P.

At the opposite extreme to the absolute continuity of two measures is the possibility of their mutual singularity. This can be generalized to the idea of entire separability. The sequences of measures \(\{P^ n\}\) and \(\{Q^ n\}\) are said to be asymptotically entirely separated if there exist a sequence \(k_ n\) and sets \(A^{k_ n}\in {\mathcal F}^{k_ n}\) such that \(\lim_{n}P^{k_ n}(A^{k_ n})=0\) whereas \(\lim_{n}Q^{k_ n}(A^{k_ n})=1\). In statistical terms, \(\{P^ n\}\) and \(\{Q^ n\}\) are entirely separable if and only if there exists a sequence of tests of \(\{P^ n\}\) versus \(\{Q^ n\}\) which is consistent and has asymptotic size zero.

The authors study the case where the measurable spaces \((\Omega^ n,{\mathcal F}^ n)\) have further structure, each having an associated filtration (\({\mathcal F}^ n_ k)_{k\geq 0}\). This occurs naturally when n is the size of a random sample, \(\Omega^ n={\mathbb{R}}^ n\), \({\mathcal F}^ n={\mathcal B}^ n\) and \({\mathcal F}^ n_ k\) is the sub-\(\sigma\)-field generated by the first k observations in the sample. Denote the restrictions of \(P^ n\) and \(Q^ n\) to \({\mathcal F}^ n_ k\) by \(P^ n_ k\) and \(Q^ n_ k\), respectively. Necessary and sufficient conditions for the contiguity (entire separability) of \(\{Q^ n\}\) with respect to \(\{P^ n\}\) are derived in terms of the conditional likelihood ratios \[ \alpha^ n_ k=(dQ^ n_ k/dP^ n_ k)/(dQ^ n_{k-1}/dP^ n_{k-1}). \] The \(\alpha^ n_ k\) are \({\mathcal F}^ n_ k\)-measurable whereas the conditions derived are “predictable”. For example, in order that \(\{Q^ n\}\) be continguous to \(\{P^ n\}\) it is necessary and sufficient that \[ \lim_{N}\lim_{n}Q^ n(\sup_{k}\alpha^ n_ k\geq N)=0\quad and\quad \lim_{N}\lim_{n}Q^ n(\sum^{\infty}_{k=1}E_{P^ n}[(1- \sqrt{\alpha^ n_ k})^ 2| {\mathcal F}^ n_{k-1}]\geq N)=0. \] In the second half of the book, the same structure as before is retained, namely, a sequence of measurable spaces together with a filtration \((\Omega^ n,{\mathcal F}^ n,({\mathcal F}^ n_ k)_{k\geq 0})\) of sub- \(\sigma\)-fields. Again two sequences of measures \(\{P^ n\}\) and \(\{Q^ n\}\) are given on \((\Omega^ n,{\mathcal F}^ n,({\mathcal F}^ n_ k)_{k\geq 0})\) and their restrictions to \({\mathcal F}^ n_ k\) are denoted by \(P^ n_ k\) and \(Q^ n_ k\), respectively. A continuously parametrized process of likelihood ratios is constructed by setting \(Z^ n_ t=dQ^ n_{[nt]}/dP^ n_{[nt]}\). Necessary and sufficient conditions (again in predictable form) are derived for log \(Z^ n_ t\) to converge in distribution to a Gaussian martingale. This is achieved by applying recent results on the martingale central limit theorem which have been derived by the authors.

This is a research monograph, giving a detailed account of recent and definitive work in two related areas. In the words of the introduction to the series in which it is published, it provides “timely and authoritative coverage of areas of current research in a more extended and expository form than is possible within the confines of a journal article”.

##### MSC:

62F05 | Asymptotic properties of parametric tests |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62B99 | Sufficiency and information |

62E20 | Asymptotic distribution theory in statistics |

60F17 | Functional limit theorems; invariance principles |

60G42 | Martingales with discrete parameter |