Optimal Stopping of a Random Sequence with Unknown Distribution

The subject of this paper is the problem of optimal stopping of a sequence of independent and identically distributed random variables with unknown distribution. We propose a stopping rule that is based on relative ranks and study its performance as measured by the maximal relative regret over suitable nonparametric classes of distributions. It is shown that the proposed rule is first-order asymptotically optimal and nearly rate optimal in terms of the rate at which the relative regret converges to zero. We also develop a general method for numerical solution of sequential stopping problems with no distributional information and use it in order to implement the proposed stopping rule. Some numerical experiments illustrating performance of the rule are presented as well.

Improving Log-Likelihood Ratio Estimation with Bi-Gaussian Approximation under Multiuser Interference Scenarios

Accurate estimation of channel log-likelihood ratio (LLR) is crucial to the decoding of modern channel codes like turbo, low-density parity-check (LDPC), and polar codes. Under an additive white Gaussian noise (AWGN) channel, the calculation of LLR is relatively straightforward since the closed-form expression for the channel likelihood function can be perfectly known to the receiver. However, it would be much more complicated for heterogeneous networks where the global noise (i.e., noise plus interference) may be dominated by non-Gaussian interference with an unknown distribution. Although the LLR can still be calculated by approximating the distribution of global noise as Gaussian, it will cause performance loss due to the non-Gaussian nature of global noise. To address this problem, we propose to use bi-Gaussian (BG) distribution to approximate the unknown distribution of global noise, for which the two parameters of BG distribution can easily be estimated from the second and fourth moments of the overall received signals without any knowledge of interfering channel state information (CSI) or signaling format information. Simulation results indicate that the proposed BG approximation can effectively improve the word error rate (WER) performance. The gain of BG approximation over Gaussian approximation depends heavily on the interference structure. For the scenario of a single BSPK interferer with a 5 dB interference-to-noise ratio (INR), we observed a gain of about 0.6 dB. The improved LLR estimation can also accelerate the convergence of iterative decoding, thus involving a lower overall decoding complexity. In general, the overall decoding complexity can be reduced by 25 to 50%.

Exponential and Hypoexponential Distributions: Some Characterizations

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n≥2, X1,X2,…,Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables.

Informational Cycles in Search Markets

I show that market participants’ equilibrium beliefs can create fluctuations in the volume of trading, even in a stationary environment. I study a sequential search model where buyers face an unknown distribution of offers. Each buyer learns about the distribution by observing whether a randomly chosen buyer traded yesterday. A cyclical equilibrium exists where the informational content of observing a trade fluctuates, which leads to fluctuations in the volume of trading. The cyclical equilibrium is more efficient than steady-state equilibria. The efficiency result holds also if buyers get a signal about past transaction prices or past trading volumes. (JEL D82, D83)

Asymptotical problems of sequential interval and point estimation

The accuracy of interval estimation systems is usually measured using interval lengths for given covering probabilities. The confidence intervals are the intervals of a fixed width if the length of the interval is determined, i.e., not random, and tends to zero for a given covering probability. We consider two important directions of statistical analysis -sequential interval estimation with confidence intervals of fixed width and sequential point estimation with asymptotically minimum risk. Two statistical models are used to describe the basis problems of sequential interval estimation by confidence intervals of a fixed width and point estimation. A review of data on nonparametric sequential estimation is carried out and new original results obtained by the authors are presented. Sequential analysis is characterized by the fact that the moment of termination of observations (stopping time) is random and is determined depending on the values of the observed data and on the adopted measure of optimality of the constructed statistical estimate. Therefore, to solve the asymptotic problems of sequential estimation, the methods of summation of random variables are used. To prove the asymptotic consistency of the confidence intervals of a fixed width, we used a method based on application of limit theorems for randomly stopped random processes. General conditions of the consistency and efficiency of sequential interval estimation of a wide class of functionals of an unknown distribution function are obtained and verified by sequential interval estimation of an unknown probability density of asymptotically uncorrelated and linear processes. Conditions of the regularity are specified that provide the property of being an estimate with an asymptotically minimum risk for a wide class of estimates and loss functions. Those conditions are verified by sequential point estimation of an unknown distribution function.

Multi-scale vector quantization with reconstruction trees

We propose and study a multi-scale approach to vector quantization (VQ). We develop an algorithm, dubbed reconstruction trees, inspired by decision trees. Here the objective is parsimonious reconstruction of unsupervised data, rather than classification. Contrasted to more standard VQ methods, such as $k$-means, the proposed approach leverages a family of given partitions, to quickly explore the data in a coarse-to-fine multi-scale fashion. Our main technical contribution is an analysis of the expected distortion achieved by the proposed algorithm, when the data are assumed to be sampled from a fixed unknown distribution. In this context, we derive both asymptotic and finite sample results under suitable regularity assumptions on the distribution. As a special case, we consider the setting where the data generating distribution is supported on a compact Riemannian submanifold. Tools from differential geometry and concentration of measure are useful in our analysis.

Time-of-arrival–based localization algorithm in mixed line-of-sight/non-line-of-sight environments

A novel time-of-arrival–based localization algorithm in mixed line-of-sight/non-line-of-sight environments is proposed. First, an optimization problem of target localization in the known distribution of line-of-sight and non-line-of-sight is established, and mixed semi-definite and second-order cone programming techniques are used to transform the original problem into a convex optimization problem which can be solved efficiently. Second, a worst-case robust least squares criterion is used to form an optimization problem of target localization in unknown distribution of line-of-sight and non-line-of-sight, where all links are treated as non-line-of-sight links. This problem is also solved using the similar techniques used in the known distribution of line-of-sight and non-line-of-sight case. Finally, computer simulation results show that the proposed algorithms have better performance in both the known distribution and the unknown distribution of line-of-sight and non-line-of-sight environments.