110,906 research outputs found
An Information-Theoretic Test for Dependence with an Application to the Temporal Structure of Stock Returns
Information theory provides ideas for conceptualising information and
measuring relationships between objects. It has found wide application in the
sciences, but economics and finance have made surprisingly little use of it. We
show that time series data can usefully be studied as information -- by noting
the relationship between statistical redundancy and dependence, we are able to
use the results of information theory to construct a test for joint dependence
of random variables. The test is in the same spirit of those developed by
Ryabko and Astola (2005, 2006b,a), but differs from these in that we add extra
randomness to the original stochatic process. It uses data compression to
estimate the entropy rate of a stochastic process, which allows it to measure
dependence among sets of random variables, as opposed to the existing
econometric literature that uses entropy and finds itself restricted to
pairwise tests of dependence. We show how serial dependence may be detected in
S&P500 and PSI20 stock returns over different sample periods and frequencies.
We apply the test to synthetic data to judge its ability to recover known
temporal dependence structures.Comment: 22 pages, 7 figure
Distributed Hypothesis Testing with Privacy Constraints
We revisit the distributed hypothesis testing (or hypothesis testing with
communication constraints) problem from the viewpoint of privacy. Instead of
observing the raw data directly, the transmitter observes a sanitized or
randomized version of it. We impose an upper bound on the mutual information
between the raw and randomized data. Under this scenario, the receiver, which
is also provided with side information, is required to make a decision on
whether the null or alternative hypothesis is in effect. We first provide a
general lower bound on the type-II exponent for an arbitrary pair of
hypotheses. Next, we show that if the distribution under the alternative
hypothesis is the product of the marginals of the distribution under the null
(i.e., testing against independence), then the exponent is known exactly.
Moreover, we show that the strong converse property holds. Using ideas from
Euclidean information theory, we also provide an approximate expression for the
exponent when the communication rate is low and the privacy level is high.
Finally, we illustrate our results with a binary and a Gaussian example
Distributed Binary Detection with Lossy Data Compression
Consider the problem where a statistician in a two-node system receives
rate-limited information from a transmitter about marginal observations of a
memoryless process generated from two possible distributions. Using its own
observations, this receiver is required to first identify the legitimacy of its
sender by declaring the joint distribution of the process, and then depending
on such authentication it generates the adequate reconstruction of the
observations satisfying an average per-letter distortion. The performance of
this setup is investigated through the corresponding rate-error-distortion
region describing the trade-off between: the communication rate, the error
exponent induced by the detection and the distortion incurred by the source
reconstruction. In the special case of testing against independence, where the
alternative hypothesis implies that the sources are independent, the optimal
rate-error-distortion region is characterized. An application example to binary
symmetric sources is given subsequently and the explicit expression for the
rate-error-distortion region is provided as well. The case of "general
hypotheses" is also investigated. A new achievable rate-error-distortion region
is derived based on the use of non-asymptotic binning, improving the quality of
communicated descriptions. Further improvement of performance in the general
case is shown to be possible when the requirement of source reconstruction is
relaxed, which stands in contrast to the case of general hypotheses.Comment: to appear on IEEE Trans. Information Theor
On the Reliability Function of Distributed Hypothesis Testing Under Optimal Detection
The distributed hypothesis testing problem with full side-information is
studied. The trade-off (reliability function) between the two types of error
exponents under limited rate is studied in the following way. First, the
problem is reduced to the problem of determining the reliability function of
channel codes designed for detection (in analogy to a similar result which
connects the reliability function of distributed lossless compression and
ordinary channel codes). Second, a single-letter random-coding bound based on a
hierarchical ensemble, as well as a single-letter expurgated bound, are derived
for the reliability of channel-detection codes. Both bounds are derived for a
system which employs the optimal detection rule. We conjecture that the
resulting random-coding bound is ensemble-tight, and consequently optimal
within the class of quantization-and-binning schemes
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