Concept Of Order Statistics Research Articles

Inferring maximum magnitudes from the ordered sequence of large earthquakes.

The largest magnitude earthquake in a sequence is often used as a proxy for hazard estimates, as consequences are often predominately from this single event (in small seismic zones). In this article, the concept of order statistics is adapted to infer the maximum magnitude ([Formula: see text]) of an earthquake catalogue. A suite tools developed here can discern [Formula: see text] influences through hypothesis testing, quantify [Formula: see text] through maximum likelihood estimation (MLE) or select the best [Formula: see text] prediction amongst several models. The efficacy of these tools is benchmarked against synthetic and real-data tests, demonstrating their utility. Ultimately, 13 cases of induced seismicity spanning wastewater disposal, hydraulic fracturing and enhanced geothermal systems are tested for volume-based [Formula: see text]. I find that there is no evidence of volume-based processes influencing any of these cases. On the contrary, all these cases are adequately explained by an unbounded magnitude distribution. This is significant because it suggests that induced earthquake hazards should also be treated as unbounded. On the other hand, if bounded cases exist, then the tools developed here will be able to discern them, potentially changing how an operator mitigates these hazards. Overall, this suite of tools will be important for better-understanding earthquakes and managing their risks. This article is part of the theme issue 'Induced seismicity in coupled subsurface systems'.

Spectral, Energy, and Economic Efficiency of 5G Multicell Massive MIMO Systems With Generalized Spatial Modulation

This paper studies generalized spatial modulation (Gen-SM) schemes in multicell multiuser massive multiple-input multiple-output (MIMO) systems as a promising high-throughput and energy-efficient technique for fifth-generation (5G) wireless networks. A detection algorithm for such systems is proposed based on linear processing techniques. By applying the concept of order statistics, a general framework for approximating the achievable sum rates with linear detection is also given. The probability of detecting antenna combinations is analyzed and is used to approximate the sum-rate performance with practical channel conditions, such as antenna correlation, imperfect channel information, and pilot contamination. The fundamental trade-off between spectral efficiency (SE) and energy efficiency (EE) is also investigated. Despite offering less SE, spatial modulation (SM) with a single active antenna per user is shown to be the most energy-efficient transmission mode among the Gen-SM class. Within the operating range of SM (i.e., in the low-to-moderate SE regime), SM is demonstrated to achieve better EE compared with conventional massive MIMO schemes. The performance in terms of economic efficiency, indicating economic profitability (in monetary unit per second), is also analyzed and is shown to serve as a complementary performance metric, enabling an implicit trade-off between SE and EE.

Order statistics of uncertain random variables with application to k-out-of-n system

Uncertain random variables are tools to deal with a mixture of uncertainty and randomness. A new concept of order statistics associated with uncertain random variables is proposed, and is applied to analyze k-out-of-n systems with uncertain random lifetimes. The chance distributions of order statistics of uncertain random variables are derived from the operational law of uncertain random variables. Finally, the reliability of k-out-of-n systems with uncertain random lifetimes is discussed.

Multivariate L-estimation

In one dimension, order statistics and ranks are widely used because they form a basis for distribution free tests and some robust estimation procedures. In more than one dimension, the concept of order statistics and ranks is not clear and several definitions have been proposed in the last years. The proposed definitions are based on different concepts of depth. In this paper, we define a new notion of order statistics and ranks for multivariate data based on density estimation. The resulting ranks are invariant under affinc transformations and asymptotically distribution free. We use the corresponding order statistics to define a class of multivariate estimators of location that can be regarded as multivariate L-estimators. Under mild assumptions on the underlying distribution, we show the asymptotic normality of the estimators. A modification of the proposed estimates results in a high breakdown point procedure that can deal with patches of outliers. The main idea is to order the observations according to their likelihoodf(X 1),...,f(X n ). If the densityf happens to be cllipsoidal, the above ranking is similar to the rankings that are derived from the various notions of depth. We propose to define a ranking based on a kernel estimate of the densityf. One advantage of estimating the likelihoods is that the underlying distribution does not need to have a density. In addition, because the approximate likelihoods are only used to rank the observations, they can be derived from a density estimate using a fixed bandwidth. This fixed bandwidth overcomes the curse of dimensionality that typically plagues density estimation in high dimension.

Engineering Analysis of Extreme Value Data: Selection of Models

This paper discusses some of the existing statistical models for the analysis of extreme value data in the case of independence, pointing out their excellence and possible sources of error. Initially, the concept of order statistics is introduced, and the joint distribution of any set of order statistics is given. As simple examples, the distribution of the maximum, the minimum, any single order statistic, or any pair of order statistics are derived. Then, the problem of limit distribution is raised and carefully analyzed making a clear distinction between maxima and minima. It is shown that all models can be grouped in the Von Mises‐Jenkinson families, which include the three classical families. Several methods for selecting an adequate limit distribution based on data, including probability papers, least‐squares methods, and the curvature method are described. To clarify concepts, several illustrative examples of applications are included. Finally, a practical method for determining the limit distribution is described in detail.