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Register a new userNonparametric inference for ergodic, stationary time series
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Added by
Gusztav Morvai
Publication date
2007
fields
Informatics Engineering
and research's language is
English
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The setting is a stationary, ergodic time series. The challenge is to construct a sequence of functions, each based on only finite segments of the past, which together provide a strongly consistent estimator for the conditional probability of the next observation, given the infinite past. Ornstein gave such a construction for the case that the values are from a finite set, and recently Algoet extended the scheme to time series with coordinates in a Polish space. The present study relates a different solution to the challenge. The algorithm is simple and its verification is fairly transparent. Some extensions to regression, pattern recognition, and on-line forecasting are mentioned.
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Let ${(X_i,Y_i)}$ be a stationary ergodic time series with $(X,Y)$ values in the product space $R^dbigotimes R .$ This study offers what is believed to be the first strongly consistent (with respect to pointwise, least-squares, and uniform distance) algorithm for inferring $m(x)=E[Y_0|X_0=x]$ under the presumption that $m(x)$ is uniformly Lipschitz continuous. Auto-regression, or forecasting, is an important special case, and as such our work extends the literature of nonparametric, nonlinear forecasting by circumventing customary mixing assumptions. The work is motivated by a time series model in stochastic finance and by perspectives of its contribution to the issues of universal time series estimation.
The forward estimation problem for stationary and ergodic time series ${X_n}_{n=0}^{infty}$ taking values from a finite alphabet ${cal X}$ is to estimate the probability that $X_{n+1}=x$ based on the observations $X_i$, $0le ile n$ without prior knowledge of the distribution of the process ${X_n}$. We present a simple procedure $g_n$ which is evaluated on the data segment $(X_0,...,X_n)$ and for which, ${rm error}(n) = |g_{n}(x)-P(X_{n+1}=x |X_0,...,X_n)|to 0$ almost surely for a subclass of all stationary and ergodic time series, while for the full class the Cesaro average of the error tends to zero almost surely and moreover, the error tends to zero in probability.
The forecasting problem for a stationary and ergodic binary time series ${X_n}_{n=0}^{infty}$ is to estimate the probability that $X_{n+1}=1$ based on the observations $X_i$, $0le ile n$ without prior knowledge of the distribution of the process ${X_n}$. It is known that this is not possible if one estimates at all values of $n$. We present a simple procedure which will attempt to make such a prediction infinitely often at carefully selected stopping times chosen by the algorithm. We show that the proposed procedure is consistent under certain conditions, and we estimate the growth rate of the stopping times.
The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but they all demand prohibitive amounts of input data. In this paper we consider real-valued time series and construct conditional distribution estimates that make much more efficient use of the input data. The estimates are consistent in a weak sense, and the question whether they are pointwise consistent is still open. For finite-alphabet processes one may rely on a universal data compression scheme like the Lempel-Ziv algorithm to construct conditional probability mass function estimates that are consistent in expected information divergence. Consistency in this strong sense cannot be attained in a universal sense for all stationary processes with values in an infinite alphabet, but weak consistency can. Some applications of the estimates to on-line forecasting, regression and classification are discussed.
Let ${X_n}_{n=0}^{infty}$ be a stationary real-valued time series with unknown distribution. Our goal is to estimate the conditional expectation of $X_{n+1}$ based on the observations $X_i$, $0le ile n$ in a strongly consistent way. Bailey and Ryabko proved that this is not possible even for ergodic binary time series if one estimates at all values of $n$. We propose a very simple algorithm which will make prediction infinitely often at carefully selected stopping times chosen by our rule. We show that under certain conditions our procedure is strongly (pointwise) consistent, and $L_2$ consistent without any condition. An upper bound on the growth of the stopping times is also presented in this paper.
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