1,213 research outputs found
Some new results on convolutions of heterogeneous gamma random variables
AbstractConvolutions of independent random variables often arise in a natural way in many applied areas. In this paper, we study various stochastic orderings of convolutions of heterogeneous gamma random variables in terms of the majorization order [p-larger order, reciprocal majorization order] of parameter vectors and the likelihood ratio order [dispersive order, hazard rate order, star order, right spread order, mean residual life order] between convolutions of two heterogeneous gamma sets of variables wherein they have both differing scale parameters and differing shape parameters. The results established in this paper strengthen and generalize those known in the literature
Schur properties of convolutions of gamma random variables
Sufficient conditions for comparing the convolutions of heterogeneous gamma
random variables in terms of the usual stochastic order are established. Such
comparisons are characterized by the Schur convexity properties of the
cumulative distribution function of the convolutions. Some examples of the
practical applications of our results are given
Bounds for mixtures of order statistics from exponentials and applications
AbstractThis paper deals with the stochastic comparison of order statistics and their mixtures. For a random sample of size n from an exponential distribution with hazard rate λ, and for 1≤k≤n, let us denote by Fk:n(λ) the distribution function of the corresponding kth order statistic. Let us consider m random samples of same size n from exponential distributions having respective hazard rates λ1,…,λm. Assume that p1,…,pm>0, such that ∑i=1mpi=1, and let U and V be two random variables with the distribution functions Fk:n(λ) and ∑i=1mpiFk:n(λi), respectively. Then, V is greater in the hazard rate order (or the usual stochastic order) than U if and only if λ≥∑i=1mpiλikk, and V is smaller in the hazard rate order (or the usual stochastic order) than U if and only if λ≤min1≤i≤mλi, for all k=1,…,n.These properties are used to find the best bounds for the survival functions of order statistics from independent heterogeneous exponential random variables. For the proof, we will use a mixture type representation for the distribution functions of order statistics
Enabling Explainable Fusion in Deep Learning with Fuzzy Integral Neural Networks
Information fusion is an essential part of numerous engineering systems and
biological functions, e.g., human cognition. Fusion occurs at many levels,
ranging from the low-level combination of signals to the high-level aggregation
of heterogeneous decision-making processes. While the last decade has witnessed
an explosion of research in deep learning, fusion in neural networks has not
observed the same revolution. Specifically, most neural fusion approaches are
ad hoc, are not understood, are distributed versus localized, and/or
explainability is low (if present at all). Herein, we prove that the fuzzy
Choquet integral (ChI), a powerful nonlinear aggregation function, can be
represented as a multi-layer network, referred to hereafter as ChIMP. We also
put forth an improved ChIMP (iChIMP) that leads to a stochastic gradient
descent-based optimization in light of the exponential number of ChI inequality
constraints. An additional benefit of ChIMP/iChIMP is that it enables
eXplainable AI (XAI). Synthetic validation experiments are provided and iChIMP
is applied to the fusion of a set of heterogeneous architecture deep models in
remote sensing. We show an improvement in model accuracy and our previously
established XAI indices shed light on the quality of our data, model, and its
decisions.Comment: IEEE Transactions on Fuzzy System
Some Unified Results on Comparing Linear Combinations of Independent Gamma Random Variables
In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9]
On the Temporal Effects of Mobile Blockers in Urban Millimeter-Wave Cellular Scenarios
Millimeter-wave (mmWave) propagation is known to be severely affected by the
blockage of the line-of-sight (LoS) path. In contrast to microwave systems, at
shorter mmWave wavelengths such blockage can be caused by human bodies, where
their mobility within environment makes wireless channel alternate between the
blocked and non-blocked LoS states. Following the recent 3GPP requirements on
modeling the dynamic blockage as well as the temporal consistency of the
channel at mmWave frequencies, in this paper a new model for predicting the
state of a user in the presence of mobile blockers for representative 3GPP
scenarios is developed: urban micro cell (UMi) street canyon and
park/stadium/square. It is demonstrated that the blockage effects produce an
alternating renewal process with exponentially distributed non-blocked
intervals, and blocked durations that follow the general distribution. The
following metrics are derived (i) the mean and the fraction of time spent in
blocked/non-blocked state, (ii) the residual blocked/non-blocked time, and
(iii) the time-dependent conditional probability of having blockage/no blockage
at time t1 given that there was blockage/no blockage at time t0. The latter is
a function of the arrival rate (intensity), width, and height of moving
blockers, distance to the mmWave access point (AP), as well as the heights of
the AP and the user device. The proposed model can be used for system-level
characterization of mmWave cellular communication systems. For example, the
optimal height and the maximum coverage radius of the mmWave APs are derived,
while satisfying the required mean data rate constraint. The system-level
simulations corroborate that the use of the proposed method considerably
reduces the modeling complexity.Comment: Accepted, IEEE Transactions on Vehicular Technolog
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