Lognormal Distributions and Geometric Averages of Symmetric Positive Definite Matrices

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (SPD) matrices, seen as a matrix‐variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. SPD matrices: the log‐Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of SPD matrices. The limiting distributions of these averages are used to provide large‐sample confidence regions and two‐sample tests for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability.     

Sample Design and Allocation for Random Digit Dialling

Sample design and sample allocation methods are developed for random digit dialling in household telephone surveys. The proposed method is based on a two-way stratification of telephone numbers. A weighted probability proportional to size sample allocation technique is used, with auxiliary variables about the telephone coverage rates, within local telephone exchanges of each substrata. This makes the sampling design nearly “self-weighting” in residential numbers when the prior information is well assigned. A computer program generates random numbers for the local areas within the existing phone capacities. A simulation study has shown greater sample allocation gain by the weighted probabilities proportional to size measures over other sample allocation methods. The amount of dialling required to obtain the sample is less than for proportional allocation. A decrease is also observed on the gain in sample allocation for some methods through the increasing sample sizes.     

Random Sets: Models and Statistics

This paper surveys aspects of the theory of random closed sets, focussing on issues of practical and current interest. First, some historical remarks on this part of probability theory are made, where the important role of Georges Matheron is emphasized. Then, fundamental characteristics of the distribution of random closed sets are introduced. The very important Boolean model serves as an example for discussing mathematical and statistical problems. A number of further models is then considered, namely excursion sets of random fields, the system of edges of the Poisson Voronoi tessellation and various random systems of non-overlapping spheres. Finally, some ideas of particle statistics are presented, including some models of random compact sets.     

Bridging the gap between a stationary point process and its Palm distribution

In the context of stationary point processes measurements are usually made from a time point chosen at random or from an occurrence chosen at random. That is, either the stationary distribution P or its Palm distribution P° is the ruling probability measure. In this paper an approach is presented to bridge the gap between these distributions. We consider probability measures which give exactly the same events zero probability as P°, having simple relations with P . Relations between P and P° are derived with these intermediate measures as bridges. With the resulting Radon-Nikodym densities several well-known results can be proved easily. New results are derived. As a corollary of cross ergodic theorems a conditional version of the well-known inversion formula is proved. Several approximations of P° are considered, for instance the local characterization of P^o as a limit of conditional probability measures P° N The total variation distance between P° and P₁ can be expressed in terms of the P-distribution function of the forward recurrence time.     

Matching with externalities: The role of prudence and social connectedness in stability

In matching problems with externalities, prudence measures the importance an agent gives to others’ potential reactions when she considers deviating and social connectedness measures others’ capacity to react to the agent’s deviation. Assuming that externalities and preferences are random, the roles of prudence and social connectedness are studied. It is shown that asymptotic stability – a property that never holds in the absence of externalities – is achieved when the product of social connectedness and prudence grows at least exponentially with the population. Since social connectedness increases quickly in both the marriage and roommates markets, stability obtains even when prudence vanishes (sufficiently slowly).     

Random scheduling with deadlines under dichotomous preferences

A group of individuals share a deterministic server which is capable of serving one job per unit of time. Every individual has a job and a cut off time slot (deadline) where service beyond this slot is as worthless as not getting any service at all. Individuals are indifferent between slots which are not beyond their deadlines (compatible slots). A schedule (possibly random) assigns the set of slots to individuals by respecting their deadlines. We only consider the class of problems where for every set of relevant slots (compatible with at least one individual) there are at least as many individuals who have a compatible slot in that set: we ignore the case of underdemand. For this class, we characterize the random scheduling rule which attaches uniform probability to every efficient deterministic schedule (efficient uniform rule) by Pareto efficiency, equal treatment of equals, and probabilistic consistency (Chambers, 2004). We also show that a weaker version of the probabilistic consistency axiom is enough to achieve our result. Finally we show that efficient uniform rule is strategyproof.     

Statistical analysis of change in networks

A survey is given of random graphs and random graph processes which can be used to describe and analyze networks that are changing with time. Marko-vian change over time, log-linear models for change, and conditionally uniform models for change are described. It is noted that estimation is usually complex if the random graph involves dependent dyads. Models with deterministic change over time may be a way to avoid the difficulties implied by dependent dyads. Logit regression methods are described that can be used to estimate such models.     

Estimation in Two-Stage Models with Heteroscedasticity

A surprising number of important problems can be cast in the framework of estimating a mean and variance using data arising from a two-stage structure. The first stage is a random sampling of "units" with some quantity of interest associated with the unit. The second stage produces an estimate of that quantity and usually, but not always, an estimated standard error, which may change considerably across units. Heteroscedasticity in the estimates over different units can arise for a number of reasons, including variation associated with the unit and changing sampling effort over units. This paper presents a broad discussion of the problem of making inferences for the population mean and variance associated with the unobserved true values at the first stage of sampling. A careful discussion of the causes of heteroscedasticity is given, followed by an examination of ways in which inferences can be carried out in a manner that is robust to the nature of the within unit heteroscedasticity. Among the conclusions are that under any type of heteroscedasticity, an unbiased estimate of the mean and the variance of the estimated mean can be obtained by using the estimates as if they were true unobserved values from the first stage. The issue of using the mean versus a weighted average which tries to account for the heteroscedasticity is also discussed. An unbiased estimate of the population variance is given and the variance of this estimate and its covariance with the estimated mean is provided under various types of heteroscedasticity. The two-stage setting arises in many contexts including the one-way random effects models with replication, meta-analysis, multi-stage sampling from finite populations and random coefficients models. We will motivate and illustrate the problem with data arising from these various contexts with the goal of providing a unified framework for addressing such problems.     

Stochastic equilibria of an asset pricing model with heterogeneous beliefs and random dividends

We investigate dynamical properties of a heterogeneous agent model with random dividends and further study the relationship between dynamical properties of the random model and those of the corresponding deterministic skeleton, which is obtained by setting the random dividends as their constant mean value. Based on our recent mathematical results, we prove the existence and stability of random fixed points as the perturbation intensity of random dividends is sufficiently small. Furthermore, we prove that the random fixed points converge almost surely to the corresponding fixed points of the deterministic skeleton as the perturbation intensity tends to zero. Moreover, simulations suggest similar behaviors in the case of more complicated attractors. Therefore, the corresponding deterministic skeleton is a good approximation of the random model with sufficiently small random perturbations of dividends. Given that dividends in real markets are generally very low, it is reasonable and significant to some extent to study the effects of heterogeneous agents’ behaviors on price fluctuations by the corresponding deterministic skeleton of the random model.     

Asymptotic confidence intervals for the length of the shortt under random censoring

A short t of a one dimensional probability distribution is defined to be an interval which has at least probability t and minimal length. The length of a show and its obvious estimator are significant measures of scale of a distribution and the corresponding random sample, respectively. In this note a non-parametric asymptotic confidence interval for the length of the (uniqueness is assumed) short t is established in the random censorship from the right model. The estimator of the length of the short t is based on the product-limit (PL) estimator of the unknown distribution function. The proof of the result mainly follows from an appropriate combination of the Glivenko-Cantelli theorem and the functional central limit theorem for the PL estimator.     

VARIANCE-RATIO TESTS OF RANDOM WALK: AN OVERVIEW

Abstract.  This paper reviews the recent developments in the field of the variance-ratio (VR) tests of the random walk and martingale hypothesis. In particular, we present the conventional individual and multiple VR tests as well as their improved modifications based on power-transformed statistics, rank and sign tests, subsampling and bootstrap methods, among others. We also re-examine the weak-form efficiency for five emerging equity markets in Latin America.     

RANDOM COEFFICIENT MODELS: THEORY AND APPLICATIONS

Abstract. This paper provides an overview of the rationale behind, and the implementation, and uses of, the random coefficient approach to econometric modelling. A simple random coefficient model is presented, and methods for estimating, testing, and validating such a model are described. A more general model is then presented. The general model is shown to include several fixed-coefficient models as special cases and can be estimated incorporating a variety of judgements concerning simplification. Finally, the paper reviews recent applications of random coefficient estimation.     

Models of exchange rates : A comparison of forecasting results

In this note we compare the results of several published papers on exchange rate forecasting. With regard to univariate time series models, we confirm the result that such models, on average, do not outperform the simple random walk forecasting rule. This conclusion corrects results reported in this Journal by Alexander and Thomas (1987).     

Density estimation for randomly distributed circular objects

Summary Nearest neighbour methods traditionally used to estimate density of a sessile biological population treat individuals as points. The present paper suggests distance-based density estimators which treat individuals as circles with variable areas. Distribution of distance between a sample point and thek-th (k = 1, 2, 3, …) nearest circle is derived. Maximum likelihood estimator of density is obtained from a random sample of point tok-th order distances. Assuming a skewed distribution for the circle radius, moment estimators of density and mean circle area are derived.     

Moving shift averages for ergodic transformations

The equality between the expectation value of a random variable and its shift average under an ergodic transformation is shown·to hold true if the end points of the interval along which the shifts are taken do not increase faster than a linear function of the interval length. If the increase is too fast, however, the moving shift average may, with probability 1, not converge to the expectation value.     

Forecasting Bitcoin with technical analysis: A not-so-random forest?

This paper uses data sampled at hourly and daily frequencies to predict Bitcoin returns. We consider various advanced non-linear models based on a multitude of popular technical indicators that represent market trend, momentum, volume, and sentiment. We run a robust empirical exercise to observe the impact of forecast horizon, model type, time period, and the choice of inputs (predictors) on the forecast performance of the competing models. We find that Bitcoin prices are weakly efficient at the hourly frequency. In contrast, technical analysis combined with non-linear forecasting models becomes statistically significantly dominant relative to the random walk model on a daily horizon. Our comparative analysis identifies the random forest model as the most accurate at predicting Bitcoin. The estimated measures of the relative importance of predictors reveal that the nature of investing in the Bitcoin market evolved from trend-following to excessive momentum and sentiment in the most recent time period.     

蒙特卡洛模拟法下已知服务水平的定量订货法研究

库存是企业经营不可缺少的部分,运用科学的库存管理方法可以为企业节约成本。利用Visual FoxPro V6.0软件,采用蒙特卡洛模拟法,通过计算机随机数取日销售量和订货前置期的长度,进而计算出前置期的销售量,这样重复N次,由此计算出各自的概率分布及累计概率,在给定的库存满足率的情况下就可以求出订货点,实现定量订货法两个指标的控制。     

Forecasting solar stock prices using tree-based machine learning classification: How important are silver prices?

Solar energy is one of the fastest growing sources of electricity generation. Forecasting solar stock prices is important for investors and venture capitalists interested in the renewable energy sector. This paper uses tree-based machine learning methods to forecast the direction of solar stock prices. The feature set used in prediction includes a selection of well-known technical indicators, silver prices, silver price volatility, and oil price volatility. The solar stock price direction prediction accuracy of random forests, bagging, support vector machines, and extremely randomized trees is much higher than that of logit. For a forecast horizon of between 8 and 20 days, random forests, bagging, support vector machines, and extremely randomized trees achieve a prediction accuracy greater than 85%. Although not as prominent as technical indicators like MA200, WAD, and MA20, oil price volatility and silver price volatility are also important predictors. An investment portfolio trading strategy based on trading signals generated from the extremely randomized trees stock price direction prediction outperforms a simple buy and hold strategy. These results demonstrate the accuracy of using tree-based machine learning methods to forecast the direction of solar stock prices and adds to the broader literature on using machine learning techniques to forecast stock prices.     

Much Ado About Nothing: the Mixed Models Controversy Revisited

We consider a well-known controversy that stems from the use of two mixed models for the analysis of balanced experimental data with a fixed and a random factor. It essentially originates in the different statistics developed from such models for testing that the variance parameter associated to the random factor is null. The corresponding hypotheses are interpreted as that of null random factor main effects in the presence of interaction. The controversy is further complicated by different opinions regarding the appropriateness of such hypothesis. Assuming that this is a sensible option, we show that the standard test statistics obtained under both models are really directed at different hypotheses and conclude that the problem lies in the definition of the main effects and interactions. We use expected values as in the fixed effects case to resolve the controversy showing that under the most commonly used model, the test usually associated to the inexistence of the random factor main effects addresses a different hypothesis. We discuss the choice of models, and some further problems that occur in the presence of unbalanced data.     

Pitfalls in simulation-based evaluation of forecast monitoring schemes

To evaluate the performance of a forecast monitoring scheme, forecasters have traditionally used a simulation-based estimator of some characteristic of the associated run length distribution. The most frequently cited performance measures are the average run length and the probability that the run length does not exceed a user-specified cutoff point. However, there is disagreement about the definition of run length that is appropriate in the context of forecasting. In this note we present the precise relationships between conflicting formulations both of the average run length and of the probability distribution function for the run length. The practical significance of each of these relationships is discussed. These results bear directly on the way in which simulation experiments should be designed and executed to compare alternative monitoring schemes.