Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRegression estimation from an individual stable sequence
88
0
0.0
(
0
)
Added by
Gusztav Morvai
Publication date
2007
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We consider univariate regression estimation from an individual (non-random) sequence $(x_1,y_1),(x_2,y_2), ... in real times real$, which is stable in the sense that for each interval $A subseteq real$, (i) the limiting relative frequency of $A$ under $x_1, x_2, ...$ is governed by an unknown probability distribution $mu$, and (ii) the limiting average of those $y_i$ with $x_i in A$ is governed by an unknown regression function $m(cdot)$. A computationally simple scheme for estimating $m(cdot)$ is exhibited, and is shown to be $L_2$ consistent for stable sequences ${(x_i,y_i)}$ such that ${y_i}$ is bounded and there is a known upper bound for the variation of $m(cdot)$ on intervals of the form $(-i,i]$, $i geq 1$. Complementing this positive result, it is shown that there is no consistent estimation scheme for the family of stable sequences whose regression functions have finite variation, even under the restriction that $x_i in [0,1]$ and $y_i$ is binary-valued.
rate research
Read More
This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that (i) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and (ii) there is a known upper bound for the variation of the density on an increasing sequence of intervals. A simple estimation scheme is proposed, and is shown to be $L_1$ consistent when (i) and (ii) apply. In addition it is shown that there is no consistent estimation scheme for the set of individual sequences satisfying only condition (i).
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.
We describe estimators $chi_n(X_0,X_1,...,X_n)$, which when applied to an unknown stationary process taking values from a countable alphabet ${cal X}$, converge almost surely to $k$ in case the process is a $k$-th order Markov chain and to infinity otherwise.
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.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
