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Being inspired by the biological eye, event camera is a novel asynchronous technology that poses a paradigm shift in acquisition of visual information. This paradigm enables event cameras to capture pixel-size fast motions much more naturally compared to classical cameras.

In this paper, we present a new asynchronous event-driven algorithm for detection of high-frequency pixel-size periodic signals using an event camera. Development of such new algorithms to efficiently process the asynchronous information of event cameras is essential to utilize its special properties and potential, and being a main challenge in the research community.

It turns out that this algorithm, which was developed in order to satisfy the new paradigm, is related to an untreated theoretical problem in probability: Let $0\le {\tau }_1\le {\tau }_2\le \cdots \le {\tau }_m\le 1$ originated from an unknown distribution. Let also $𝜖,\delta \in ℝ$, and $d\in \mathbb{N}$. What can be said about the probability $\mathrm{\Phi }(m,d)$ of having more than $d$ adjacent ${\tau }_i$-s pairs that the distance between them is $\delta$, up to an error $𝜖$? This problem, which reminds the area of order statistic, shows how the new visualization paradigm is also an opportunity to develop new areas and problems in mathematics.

Uncertainty estimation methods using deep learning approaches strive against separating how uncertain the state of the world manifests to us via measurement (objective end) from the way this gets scrambled with the model specification and training procedure used to predict such state (subjective means) — e.g., number of neurons, depth, connections, priors (if the model is bayesian), weight initialization, etc. This poses the question of the extent to which one can eliminate the degrees of freedom associated with these specifications and still being able to capture the objective end. Here, a novel non-parametric quantile estimation method for continuous random variables is introduced, based on the simplest neural network architecture with one degree of freedom: a single neuron. Its advantage is first shown in synthetic experiments comparing with the quantile estimation achieved from ranking the order statistics (specifically for small sample size) and with quantile regression. In real-world applications, the method can be used to quantify predictive uncertainty under the split conformal prediction setting, whereby prediction intervals are estimated from the residuals of a pre-trained model on a held-out validation set and then used to quantify the uncertainty in future predictions — the single neuron used here as a structureless “thermometer” that measures how uncertain the pre-trained model is. Benchmarking regression and classification experiments demonstrate that the method is competitive in quality and coverage with state-of-the-art solutions, with the added benefit of being more computationally efficient.

Often, the duration of a reliability growth development test is specified in advance and the decision to terminate or continue testing is conducted at discrete time intervals. These features are normally not captured by reliability growth models. This paper adapts a standard reliability growth model to determine the optimal time for which to plan to terminate testing. The underlying stochastic process is developed from an Order Statistic argument with Bayesian inference used to estimate the number of faults within the design and classical inference procedures used to assess the rate of fault detection. Inference procedures within this framework are explored where it is shown the Maximum Likelihood Estimators possess a small bias and converges to the Minimum Variance Unbiased Estimator after few tests for designs with moderate number of faults. It is shown that the Likelihood function can be bimodal when there is conflict between the observed rate of fault detection and the prior distribution describing the number of faults in the design. An illustrative example is provided.

In the literature, median type control charts have been widely investigated as easy and efficient means to monitor the process mean when observations are from a normal distribution. In this work, a Variable Sampling Interval (VSI) Exponentially Weighted Moving Average (EWMA) median control chart is proposed and studied. The Markov chains are used to calculate the average run length to signal (ARL). A performance comparison with the original EWMA median control chart is made. The numerical results show that the proposed chart is considerably more effective as it is faster in detecting process shifts. Finally, the implementation of the proposed chart is illustrated with an example in food production process.

In this paper, we developed a method for constructing confidence intervals for the parameters of lifetime distributions based on progressively type II censored data. The method produces closed form expressions for the bounds of the confidence intervals for several special cases of parameters and lifetime distributions. Closed form approximations are derived for the intervals for the parameters of the location or scale families of distributions. The method is illustrated with several examples and analyses of real data sets are included to illustrate the application of the method.

This paper presents a new four-parameter lifetime model by generalizing the Inverse Weibull (IW) distribution using Transmuted Kumaraswamy family of distributions. The proposed model explains the high probability at the tail effectively. The induction of additional parameters enhance the potentiality of the IW distribution and make it a pliant model that provides a vast range of existing distributions as special cases. Several probabilistic properties of the proposed model including moments, probability weighted moments, moment generating function, quantile function, reliability measures, and order statistics are discussed. The maximum likelihood (ML) procedure is adopted for the estimation of model parameters. The efficiency of ML estimates is to testify through simulation study. Four datasets from the field of reliability science are used to expound the competence of the proposed model.

Using an arrangement monotonicity property of the parametrized family of hazard rate ordered random variables, and a bivariate characterization of the hazard rate ordering, we obtain some new stochastic arrangement inequalities for the random variables that are hazard rate ordered. Some applications of optimal allocation and assembly are also discussed.

Classical regression approaches are not robust when errors are heavy-tailed or asymmetric. That may be due to the non-existence of the mean or variance of the error distribution. Estimation based on trimmed data, which ignored outlier or leverage points, has an old history and frequently used. This procedure chooses fixed cut-off points. In this work, we use this idea recently applied for initial estimates of regression coefficients with heavy-tailed stable errors. We propose an effective procedure to calculate the cut-off points based on the tail index and skewness parameters of errors. We use the property of the existence of some moments of stable distribution order statistics. Data are trimmed based on ordered residuals of a least square regression. However, the trimmed data’s optimal number is determined based on the number of error order statistics whose variance exists. Then, we use the rest of the ordered data to estimate the regression coefficients. Based on these order statistics’ joint distribution, we analytically compute the bias and variance of the introduced estimator of regression parameters that was impossible for regression with stable errors.

In this paper, we investigate a multivariate regression model with multivariate heavy-tailed stable errors. Since in heavy-tailed data, especially in multivariate stable distributions, some moments do not exist; classical multivariate regression methods do not perform well. We suggest using an effective property of the existence of some moments of order statistics stable distribution. We propose a method for trimming the data set using this property. Then, we estimate the regression coefficients based on the rest of the ordered data. We calculate the trimmed data set based on the error’s tail index and skewness parameters. Also, we analytically compute the bias and variance of the introduced estimators of the regression parameters. Finally, we study the performance of the proposed methods with ordinary least squares via a simulation study and a real data set.

Least Trimmed Squares (LTS) is a robust regression method with respect to outliers. It is based on performing Ordinary Least Squares (OLS) estimates on sub-datasets and determining the optimal solution corresponding to the minimum sum of squared residuals. Since the method of LTS is based on OLS, errors in regression models have finite variance. This work aims to generalize LTS for heavy tail data with infinite variance. When errors have infinite variance, it is impossible to benefit from OLS estimates. We use the property of variance existence of most ordered errors to find an initial robust OLS estimate. We polish the LTS method with a maximum likelihood estimator based on the density function of order statistics and determine the optimal solution for stable regression models. The proposed algorithm is implemented for linear regression models.

In this paper we consider a generalize dynamic entropy measure and prove that this measure characterizes the distribution function uniquely. Also we propose cumulative residual Rényi entropy of order statistics and prove that it also determines the distribution function uniquely. Applications of entropy concepts to DNA sequence analysis, the ultimate support for the biological systems, have been widely explored by researchers. The entropy measures discussed here can be applied for analysis of ordered DNA sequences.

Assume that $n$ mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors $i$ from $1$ to $n,$ where the contribution of the $i^{\mathrm{\text{th}}}$ sensor is its displacement from the current location to the anchor equidistant point $t_i=\frac{i}{n}-\frac{1}{2n},$ raised to the $a^{\mathrm{\text{th}}}$ power, when $a$ is an odd natural number.

As a consequence, we derive the following asymptotic identity. Fix $a$ positive integer. Let $X_{i:n}$ denote the $i^{\mathrm{\text{th}}}$ order statistic from a random sample of size $n$ from the Uniform$(0,1)$ population. Then

In this paper, a convenient and parsimonious family of the quantile functions is proposed for modeling time-to-event data. Detailed mathematical arguments are presented for deriving the explicit expressions of the quantile density, quantile hazard and quantile mean remaining life functions. Their functional properties are also explored. Some important members of the family are provided. A special case, the so-called additive power logarithmic quantile family, is discussed with detailed properties, parameter estimation and application to time-to-event data. Measures of skewness and kurtosis for the proposed quantile function are derived. Important statistical measures useful for reliability analysis are also provided with their explicit expressions. Stochastic ordering, tail weight and order statistics are also discussed. L-moments are explicitly derived for the proposed quantile function. Estimation is approached using the likelihood function and percentiles. A real data set consists of times between occurrence of the coal-mining accidents is analyzed for illustration purposes.

The call auction is a widely used trading mechanism, especially during the opening and closing periods of financial markets. In this paper, we study a standard call auction problem where orders are submitted according to Poisson processes, with random prices distributed according to a general distribution F, and may be cancelled at any time. We compute the analytical expressions of the distributions of the traded volume, of the lower and upper bounds of the clearing prices, and of the price range of these possible clearing prices of the call auction. Using results from the theory of order statistics and a theorem on the limit of sequences of random variables with independent random indices, we derive the weak limits of all these distributions. In this setting, traded volume and bounds of the clearing prices are found to be asymptotically normal, while the clearing price range is asymptotically exponential. All the parameters of these distributions are explicitly derived as functions of the parameters of the incoming orders' flows.

The high spectral resolution of hyperspectral data allows us to solve important applied tasks with improved accuracy, but the costs of transmission, storage, processing, and recognition of such data are high. However, hyperspectral data contain substantial redundancy, so the important problem is to reduce the dimensionality of such data while preserving the improved quality of the solution of applied tasks.

In this chapter, to solve the above problem, we focus on a well-known nonlinear mapping technique, which is based on the principle of preserving pairwise Euclidean distances between vectors. We compare this technique to other dimensionality reduction techniques. To address the problem of the high computational complexity, we use a variety of methods, including stochastic gradient descent, interpolation, and space partitioning.

Next, we extend the considered technique in a natural way to use it with different spectral dissimilarity measures, such as spectral angle, spectral correlation, and spectral information divergence. In particular, several dimensionality reduction techniques based on the principle of preserving the selected pairwise dissimilarity measures are described in this chapter.

Finally, for hyperspectral images, we consider how the spatial information contained in images can be used in nonlinear mapping. In particular, we modify the dissimilarity measures to take into account the spatial context of image pixels using the order statistics.

All the experiments in this chapter are conducted using well-known open hyperspectral images.

In this paper, we consider the estimation of the three-parameter Weibull distribution. We construct numerical algorithms to estimate the location, scale and shape parameters by maximal likelihood and simplified analytical methods. Convergence of these algorithms is studied theoretically and by computer modeling. Computer modeling results confirm practical applicability of estimates proposed. Recommendations for implementation of the estimates are discussed, too. Results from simulation studies assessing the performance of our proposed method are included.

The gamma distribution, having location (threshold), scale and shape parameters, is used as a model for distributions of life spans, reaction time, and for other types of non-symmetrical data. It has been said that the inference for the three-parameter gamma distribution is difficult because of nonregularity in maximum likelihood estimation although numerous papers have appeared over the years. On the other hand, the methodology for inference for the two-parameter gamma distribution have been established over the years. It is usual to avoid fitting the three-parameter gamma distribution and to fit the two-parameter gamma distribution to data in practice. In this article, we propose a new method of estimation of the shape parameter of the gamma distribution based on the data transformation free from location and scale parameters. The method is easily implemented with the aid of table or graph. A simulation study shows that the proposed estimator performs better than the maximum likelihood estimator of the shape parameter of the two-parameter gamma distribution when the threshold is existent even though that is close to zero.

By deriving two recurrence relations which express the single and double moments of order statistics from a symmetric distribution in terms of the corresponding quantities from its folded distribution, Govindarajulu (1963, 1966) determined means, variances and covariances of Laplace order statistics (using the results on exponential order statistics) for sample sizes up to 20. He also tabulated the BLUE's (Best Linear Unbiased Estimators) of the location and scale parameter of the Laplace distribution based on complete and symmetrically Type-II censored samples. In this paper, we first establish similar relations for the computation of triple and quadruple moments. We then use these results to develop Edgeworth approximations for some pivotal quantities which will enable one to develop inference for the location and scale parameters. Next, we show that this method provides close approximations to percentage points of the pivotal quantities determined by Monte Carlo simulations. Finally, we present an example to illustrate the method of inference developed in this paper.

Some properties of models for the outcomes of races are described, these properties being consequences of a stochastic ordering of the permutations which define the outcomes of a race. Order statistics models which lead to stochastic ordering are also discussed — particular cases of these are the first-order model of Plackett (1975) and the normal model of Upton and Brook (1974). An approximation for the normal model is suggested.

In this paper, we propose a method of estimation for the parameters of a bivariate Birnbaum-Saunders distribution based on Type-II censored samples. The distributional relationship between the bivariate normal and bivariate Birnbaum-Saunders distribution is used for the development of these estimators. The performance of the estimators are then assessed by means of Monte Carlo simulations. Finally, an example is used to illustrate the method of estimation developed here.