Finite Variance Research Articles

Optimal design unbiased strategy for estimating dichotomous finite population variance under Warner’s randomized response plan

ABSTRACTLet P be the proportion of individuals in a finite population possessing a sensitive attribute. We consider the problem of estimation of the population variance P(1 – P) under Warner’s randomized response plan and prove the optimality of a sampling strategy in a class of comparable design unbiased strategies under a super-population model.

A family of association measures for [formula omitted] contingency tables based on the [formula omitted]-divergence

The odds ratio is the predominant measure of association in 2×2 contingency tables, which, for inferential purposes, is usually considered on the log-scale. Under an information theoretic set-up, it is connected to the Kullback–Leibler divergence. Considering a generalized family of divergences, the ϕ divergence, alternative association measures are derived for 2×2 contingency tables. Their properties are studied and asymptotic inference is developed. For some members of this family, the estimated association measures remain finite in the presence of a sampling zero while for a subset of these members the estimators of these measures have finite variance as well. Special attention is given to the power divergence, which is a parametric family. The role of its parameter λ, in terms of the asymptotic confidence intervals’ coverage probability and average relative length, is further discussed. In special probability table structures, for which the performance of the asymptotic confidence intervals for the classical log odds ratio is poor, the measure corresponding to λ=1/3 is suggested as an alternative.

An efficient class of estimators of finite population variance using quartiles

ABSTRACTIn this paper, we have proposed a class of estimators of finite population variance using known values of parameters related to an auxiliary variable such as quartiles and its properties are studied in simple random sampling. The suggested class of ratio-type estimators has been compared with the usual unbiased, ratio estimators and the class of ratio-type estimators due to Singh et al. [Improved estimation of finite population variance using quartiles, Istatistik – J. Turkish Stat. Assoc. 6(3) (2013), pp. 166–121] and Solanki et al. [Improved ratio-type estimators of finite population variance using quartiles, Hacettepe J. Math. Stat. 44(3) (2015), pp. 747–754]. An empirical study is also carried out to judge the merits of the proposed estimator over other existing estimators of population variance using natural data set. It is found that the proposed class of ratio-type estimators ‘’ is superior to the usual unbiased estimator and the estimators recently proposed by Singh et al. [Improved estimation of finite population variance using quartiles, Istatistik – J. Turkish Stat. Assoc. 6(3) (2013), pp. 166–121] and Solanki et al. [Improved ratio-type estimators of finite population variance using quartiles, Hacettepe J. Math. Stat. 44(3) (2015), pp. 747–754].

On the Mathematical Foundations of Computational Photography

Flutter shutter (coded exposure) cameras allow to extend indefinitely the exposure time for uniform motion blurs. Recently, Tendero et al. (SIAM J Imaging Sci 6(2):813---847, 2013) proved that for a fixed known velocity v the gain of a flutter shutter in terms of root means square error (RMSE) cannot exceeds a 1.1717 factor compared to an optimal snapshot. The aforementioned bound is optimal in the sense that this 1.1717 factor can be attained. However, this disheartening bound is in direct contradiction with the recent results by Cossairt et al. (IEEE Trans Image Process 22(2), 447---458, 2013). Our first goal in this paper is to resolve mathematically this discrepancy. An interesting question was raised by the authors of reference (IEEE Trans Image Process 22(2), 447---458, 2013). They state that the "gain for computational imaging is significant only when the average signal level J is considerably smaller than the read noise variance $$\sigma _r^2$$?r2" (Cossairt et al., IEEE Trans Image Process 22(2), 447---458, 2013, p. 5). In other words, according to Cossairt et al. (IEEE Trans Image Process 22(2), 447---458, 2013) a flutter shutter would be more efficient when used in low light conditions e.g., indoor scenes or at night. Our second goal is to prove that this statement is based on an incomplete camera model and that a complete mathematical model disproves it. To do so we propose a general flutter shutter camera model that includes photonic, thermal (The amplifier noise may also be mentioned as another source of background noise, which can be included w.l.o.g. in the thermal noise) and additive [The additive (sensor readout) noise may contain other components such as reset noise and quantization noise. We include them w.l.o.g. in the readout.] (sensor readout, quantification) noises of finite variances. Our analysis provides exact formulae for the mean square error of the final deconvolved image. It also allows us to confirm that the gain in terms of RMSE of anyflutter shutter camera is bounded from above by 1.1776 when compared to an optimal snapshot. The bound is uniform with respect to the observation conditions and applies for any fixed known velocity. Incidentally, the proposed formalism and its consequences also apply to e.g., the Levin et al. motion-invariant photography (ACM Trans Graphics (TOG), 27(3):71:1---71:9, 2008; Method and apparatus for motion invariant imag- ing, October 1 2009. US Patent 20,090,244,300, 2009) and variant (Cho et al. Motion blur removal with orthogonal parabolic exposures, 2010). In short, we bring mathematical proofs to the effect of contradicting the claims of Cossairt et al. (IEEE Trans Image Process 22(2), 447---458, 2013). Lastly, this paper permits to point out the kind of optimization needed if one wants to turn the flutter shutter into a useful imaging tool.

On degenerate sums of m-dependent variables

It is well known that the central limit theorem holds for partial sums of a stationary sequence (Xi) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if var(Xi) ≠ 0. We show that this happens only in the case when Xi – EXi = Yi – Yi–1 for an (m − 1)-dependent stationary sequence (Yi) with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to be degenerate. Two applications to subtree counts in random trees are given.

On degenerate sums of m-dependent variables

It is well known that the central limit theorem holds for partial sums of a stationary sequence (Xi ) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if var(Xi ) ≠ 0. We show that this happens only in the case when Xi – EXi = Yi – Yi –1 for an (m − 1)-dependent stationary sequence (Yi ) with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to be degenerate. Two applications to subtree counts in random trees are given.

Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of φ-Mixing Sequences

Abstract Let {Xn | n ≥ 1} be a strictly stationary sequence of φ-mixing random variables with zero mean and finite variance. Set we show the exact rates in the law of iterated logarithm of a kind of weighted infinite series of as ε ↘ 0.

Point process-based Monte Carlo estimation

This paper addresses the issue of estimating the expectation of a real-valued random variable of the form $$X = g(\mathbf {U})$$X=g(U) where g is a deterministic function and $$\mathbf {U}$$U can be a random finite- or infinite-dimensional vector. Using recent results on rare event simulation, we propose a unified framework for dealing with both probability and mean estimation for such random variables, i.e. linking algorithms such as Tootsie Pop Algorithm or Last Particle Algorithm with nested sampling. Especially, it extends nested sampling as follows: first the random variable X does not need to be bounded any more: it gives the principle of an ideal estimator with an infinite number of terms that is unbiased and always better than a classical Monte Carlo estimator--in particular it has a finite variance as soon as there exists $$k \in \mathbb {R}> 1$$k?R>1 such that $${\text {E}}\left[ X^k \right] < \infty $$EXk<?. Moreover we address the issue of nested sampling termination and show that a random truncation of the sum can preserve unbiasedness while increasing the variance only by a factor up to 2 compared to the ideal case. We also build an unbiased estimator with fixed computational budget which supports a Central Limit Theorem and discuss parallel implementation of nested sampling, which can dramatically reduce its running time. Finally we extensively study the case where X is heavy-tailed.

Exact Sampling of Stationary and Time-Reversed Queues

We provide the first algorithm that, under minimal assumptions, allows simulation of the stationary waiting-time sequence of a single-server queue backward in time, jointly with the input processes of the queue (interarrival and service times). The single-server queue is useful in applications of Dominated Coupling from the Past (DCFTP), which is a well-known protocol for simulation without bias from steady-state distributions. Our algorithm terminates in finite time, assuming only finite mean of the interarrival and service times. To simulate the single-server queue in stationarity until the first idle period in finite expected termination time, we require the existence of finite variance. This requirement is also necessary for such idle time (which is a natural coalescence time in DCFTP applications) to have finite mean. Thus, in this sense, our algorithm is applicable under minimal assumptions.

An exact asymptotic for the square variation of partial sum processes

Nous établissons une formule asymptotique exacte pour la variation quadratique de certains processus de sommes partielles. Soit $\{X_{i}\}$ une suite de variables indépendantes et identiquement distribuées de moyenne nulle et de variance finie $\sigma^{2}$ satisfaisant une condition de moments $\mathbb{E}[|X_{i}|^{2+\delta}]<\infty$ pour un $\delta>0$. Soit $\mathcal{P}_{N}$ l’ensemble de toutes les partitions possibles de l’intervalle $[N]$ en sous-intervalles, alors nous montrons que presque sûrement $\max_{\pi\in\mathcal{P}_{N}}\sum_{I\in\pi}|\sum_{i\in I}X_{i}|^{2}\sim2\sigma^{2}N\ln\ln(N)$. Ceci peut être interprété comme une amélioration de la loi du logarithme itéré et précise les résultats de J. Qian sur les sommes partielles et les processus empiriques. Quand $\delta=0$, nous obtenons une version plus faible, en probabilité, de ce résultat.

Convergence of bi-measure $\mathbb{R}$-trees and the pruning process

In [Aldous,Pitman,1998] a tree-valued Markov chain is derived by pruning off more and more subtrees along the edges of a Galton-Watson tree. More recently, in [Abraham,Delmas,2012], a continuous analogue of the tree-valued pruning dynamics is constructed along L\'evy trees. In the present paper, we provide a new topology which allows to link the discrete and the continuous dynamics by considering them as instances of the same strong Markov process with different initial conditions. We construct this pruning process on the space of so-called bi-measure trees, which are metric measure spaces with an additional pruning measure. The pruning measure is assumed to be finite on finite trees, but not necessarily locally finite. We also characterize the pruning process analytically via its Markovian generator and show that it is continuous in the initial bi-measure tree. A series of examples is given, which include the finite variance offspring case where the pruning measure is the length measure on the underlying tree.

Coalescing Brownian flows: A new approach

The coalescing Brownian flow on $\mathbb{R}$ is a process which was introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ. Wisconsin, Madison] and T\'{o}th and Werner [Probab. Theory Related Fields 111 (1998) 375-452], and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. This result holds under a finite variance assumption and is thus optimal. In previous works by Fontes et al. [Ann. Probab. 32 (2004) 2857-2883], Newman et al. [Electron. J. Probab. 10 (2005) 21-60], the topology and state-space required a moment of order $3-\varepsilon$ for this convergence to hold. The proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work - in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the noncrossing property.

Surrogate accelerated sampling of reservoir models with complex structures using sparse polynomial chaos expansion

Markov Chain Monte Carlo (MCMC) methods are often used to probe the posterior probability distribution in inverse problems. This allows for computation of estimates of uncertain system responses conditioned on given observational data by means of approximate integration. However, MCMC methods suffer from the computational complexities in the case of expensive models as in the case of subsurface flow models. Hence, it is of great interest to develop alterative efficient methods utilizing emulators, that are cheap to evaluate, in order to replace the full physics simulator. In the current work, we develop a technique based on sparse response surfaces to represent the flow response within a subsurface reservoir and thus enable efficient exploration of the posterior probability density function and the conditional expectations given the data.Polynomial Chaos Expansion (PCE) is a powerful tool to quantify uncertainty in dynamical systems when there is probabilistic uncertainty in the system parameters. In the context of subsurface flow model, it has been shown to be more accurate and efficient compared with traditional experimental design (ED). PCEs have a significant advantage over other response surfaces as the convergence to the true probability distribution when the order of the PCE is increased can be proved for the random variables with finite variances. However, the major drawback of PCE is related to the curse of dimensionality as the number of terms to be estimated grows drastically with the number of the input random variables. This renders the computational cost of classical PCE schemes unaffordable for reservoir simulation purposes when the deterministic finite element model is expensive to evaluate. To address this issue, we propose the reduced-terms polynomial chaos representation which uses an impact factor to only retain the most relevant terms of the PCE decomposition. Accordingly, the reduced-terms polynomial chaos proxy can be used as the pseudo-simulator for efficient sampling of the probability density function of the uncertain variables.The reduced-terms PCE is evaluated on a two dimensional subsurface flow model with fluvial channels to demonstrate that with a few hundred trial runs of the actual reservoir simulator, it is feasible to construct a polynomial chaos proxy which accurately approximates the posterior distribution of the high permeability zones, in an analytical form. We show that the proxy precision improves with increasing the order of PCE and corresponding increase of the number of initial runs used to estimate the PCE coefficient.

The Divisible Sandpile at Critical Density

The divisible sandpile starts with i.i.d. random variables ("masses") at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses at most 1. The process stabilizes almost surely if m<1 and it almost surely does not stabilize if m>1, where $m$ is the mean mass per vertex. The main result of this paper is that in the critical case m=1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete biLaplacian Gaussian field.

Unbiased Estimation with Square Root Convergence for SDE Models

In many settings in which Monte Carlo methods are applied, there may be no known algorithm for exactly generating the random object for which an expectation is to be computed. Frequently, however, one can generate arbitrarily close approximations to the random object. We introduce a simple randomization idea for creating unbiased estimators in such a setting based on a sequence of approximations. Applying this idea to computing expectations of path functionals associated with stochastic differential equations (SDEs), we construct finite variance unbiased estimators with a “square root convergence rate” for a general class of multidimensional SDEs. We then identify the optimal randomization distribution. Numerical experiments with various path functionals of continuous-time processes that often arise in finance illustrate the effectiveness of our new approach.

Families of Estimators for Finite Population Variance Using Auxiliary Character Under Double Sampling the Non-respondents

In this paper, we have suggested the families of estimators for estimating the finite population variance using auxiliary character under incomplete information due to non-response. The asymptotic expressions of bias and mean square error of the proposed families of estimators are obtained and their properties are studied. Theoretical and empirical studies are performed to show the relative performance of the suggested families of estimators.

Lower and upper bounds on the variances of spacings

For classic i.i.d. samples with an arbitrary nondegenerate and finite variance distribution, Papadatos (1995, Annals of the Institute of Statistical Mathematics, 47, 185–193) presented sharp lower and upper bounds on the variances of order statistics, expressed in population variance units. We provide here analogous results for spacings. Also, we describe the parent distributions which attain the bounds.

Global Clustering Coefficient in Scale-Free Weighted and Unweighted Networks

AbstractIn this article, we present a detailed analysis of the global clustering coefficient in scale-free graphs. Many observed real-world networks of diverse nature have a power-law degree distribution. Moreover, the observed degree distribution usually has an infinite variance. Therefore, we are especially interested in such degree distributions. In addition, we analyze the clustering coefficient for both weighted and unweighted graphs. There are two well-known definitions of the clustering coefficient of a graph: the global and the average local clustering coefficients. There are several models proposed in the literature for which the average local clustering coefficient tends to a positive constant as a graph grows. However, there are no models of scale-free networks with an infinite variance of the degree distribution and with an asymptotically constant global clustering coefficient. Models with constant global clustering and finite variance were also proposed. Therefore, in this work we focus only ...

Real eigenvalues of non-Gaussian random matrices and their products

We study the properties of the eigenvalues of real random matrices and their products. It is known that when the matrix elements are Gaussian-distributed independent random variables, the fraction of real eigenvalues tends to unity as the number of matrices in the product increases. Here we present numerical evidence that this phenomenon is robust with respect to the probability distribution of matrix elements, and is therefore a general property that merits detailed investigation. Since the elements of the product matrix are no longer distributed as those of the single matrix nor they remain independent random variables, we study the role of these two factors in detail. We study numerically the properties of the Hadamard (or Schur) product of matrices and also the product of matrices whose entries are independent but have the same marginal distribution as that of normal products of matrices, and find that under repeated multiplication, the probability of all eigenvalues to be real increases in both cases, but saturates to a constant below unity showing that the correlations amongst the matrix elements are responsible for the approach to one. To investigate the role of the non-normal nature of the probability distributions, we present a thorough analytical treatment of the 2 × 2 single matrix for several standard distributions. Within the class of smooth distributions with zero mean and finite variance, our results indicate that the Gaussian distribution has the maximum probability of real eigenvalues, but the Cauchy distribution characterized by infinite variance is found to have a larger probability of real eigenvalues than the normal. We also find that for the two-dimensional single matrices, the probability of real eigenvalues lies in the range .

Cramér–Rao Bound for Sparse Signals Fitting the Low-Rank Model with Small Number of Parameters

In this letter, we consider signals with a low-rank covariance matrix which reside in a low-dimensional subspace and can be written in terms of a finite (small) number of parameters. Although such signals do not necessarily have a sparse representation in a finite basis, they possess a sparse structure which makes it possible to recover the signal from compressed measurements. We study the statistical performance bound for parameter estimation in the low-rank signal model from compressed measurements. Specifically, we derive the Cramer-Rao bound (CRB) for a generic low-rank model and we show that the number of compressed samples needs to be larger than the number of sources for the existence of an unbiased estimator with finite estimation variance. We further consider the applications to direction-of-arrival (DOA) and spectral estimation which fit into the low-rank signal model. We also investigate the effect of compression on the CRB by considering numerical examples of the DOA estimation scenario, and show how the CRB increases by increasing the compression or equivalently reducing the number of compressed samples.