An Entropy-Based Approach for Nonparametrically Testing Simple Probability Distribution Hypotheses

In this paper, we introduce a flexible and widely applicable nonparametric entropy-based testing procedure that can be used to assess the validity of simple hypotheses about a specific parametric population distribution. The testing methodology relies on the characteristic function of the population probability distribution being tested and is attractive in that, regardless of the null hypothesis being tested, it provides a unified framework for conducting such tests. The testing procedure is also computationally tractable and relatively straightforward to implement. In contrast to some alternative test statistics, the proposed entropy test is free from user-specified kernel and bandwidth choices, idiosyncratic and complex regularity conditions, and/or choices of evaluation grids. Several simulation exercises were performed to document the empirical performance of our proposed test, including a regression example that is illustrative of how, in some contexts, the approach can be applied to composite hypothesis-testing situations via data transformations. Overall, the testing procedure exhibits notable promise, exhibiting appreciable increasing power as sample size increases for a number of alternative distributions when contrasted with hypothesized null distributions. Possible general extensions of the approach to composite hypothesis-testing contexts, and directions for future work are also discussed.

Decision fusion for composite hypothesis testing in wireless sensor networks over a shared and noisy collision channel

We consider the composite hypothesis testing problem of time-bandwidth-constrained distributed detection. In this scenario, the probability distribution of the observed signal when the event of interest is happening is unknown. In addition, local decisions are censored and only those uncensored local decisions will be sent to the fusion center over a shared and noisy collision channel. The fusion center also has a limited time duration to collect transmitted decisions and make a final decision. Two types of medium access control that the sensor nodes apply to send their decisions are investigated: time division multiple access and slotted-Aloha. Unlike using the time division multiple access protocol, the slotted-Aloha-based distributed detection will experience packet collisions. However, in this article, since only uncensored decisions are sent, packet collisions are informative. We derive fusion rules according to generalized likelihood ratio test, Rao test, and Wald test for both the time division multiple access–based distributed detection and the slotted-Aloha-based distributed detection. We see that the fusion rules for the slotted-Aloha-based distributed detection here also exploit packet collisions in the final decision-making. In addition, the asymptotic performances and energy consumption of both schemes are analyzed. Extensive simulation and numerical results are provided to compare the performances of these two schemes. We show that, for a given time delay, the slotted-Aloha-based distributed detection can outperform the time division multiple access–based distributed detection by increasing the number of sensor nodes which results in higher energy consumption.