Finite Means Research Articles

Continued Logarithms and Associated Continued Fractions

ABSTRACTWe investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the continued logarithm to base b. We show convergence for them using equivalent forms of their corresponding continued fractions. Through numerical experimentation, we discover that, for one such formulation, the exponent terms have finite arithmetic means for almost all real numbers. This set of means, which we call the logarithmic Khintchine numbers, has a pleasing relationship with the geometric means of the corresponding continued fraction terms. While the classical Khintchine's constant is believed not to be related to any naturally occurring number, we find surprisingly that the logarithmic Khintchine numbers are elementary.

Optimal Policy for a Stochastic Scheduling Problem with Applications to Surgical Scheduling

We consider the stochastic, single‐machine earliness/tardiness problem (SET), with the sequence of processing of the jobs and their due‐dates as decisions and the objective of minimizing the sum of the expected earliness and tardiness costs over all the jobs. In a recent paper, Baker ( 2014 ) shows the optimality of the Shortest‐Variance‐First (SVF) rule under the following two assumptions: (a) The processing duration of each job follows a normal distribution. (b) The earliness and tardiness cost parameters are the same for all the jobs. In this study, we consider problem SET under assumption (b). We generalize Baker's result by establishing the optimality of the SVF rule for more general distributions of the processing durations and a more general objective function. Specifically, we show that the SVF rule is optimal under the assumption of dilation ordering of the processing durations. Since convex ordering implies dilation ordering (under finite means), the SVF sequence is also optimal under convex ordering of the processing durations. We also study the effect of variability of the processing durations of the jobs on the optimal cost. An application of problem SET in surgical scheduling is discussed.

Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes

Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = EXk* ≠ EXk*+1 = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations Δk = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that $${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n $$ if 0 ≤ t 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before.

A strong law of large numbers for nonnegative random variables and applications

For a sequence of nonnegative random variables {Xn,n≥1} with finite means and partial sums Sn=∑i=1nXi,n≥1, and a sequence of positive numbers {bn,n≥1} with bn↑∞, sufficient conditions are given under which (Sn−ESn)/bn→0 almost surely. Our result generalizes the strong law of large numbers obtained by Korchevsky (2015). Some applications for dependent random variables are also provided.

Towards an evolutionary or a transformational energy transition? Transition concepts and roadmaps in European Union policy discourse

To develop a new typology of energy transitions, the article firstly clarifies the ambiguities in the transition debate and establishes fault lines that divide rival conceptualizations of energy transitions. The main fault lines relate to a product versus a process emphasis, a deterministic versus an open-ended perspective, and near-term versus long-term orientations. Two “ideal type” positionings towards energy transition(s) are developed. An evolutionary variety is based on incremental energy transitions over the long term which combine “old” and “new” energy sources and providers, whereas a transformational variety recommends a “grand transition” based on a near-term switch to 100% renewables. In its second section, the article uses this twofold typology to investigate transition thinking in the energy policy discourse of the European Commission. A key finding is that the Commission’s policy documents indicate a leaning towards evolutionary transitions, with reliance on conventional energy sources set to continue for some time. Several explanations for this leaning are identified: a shortage of political will, finite available means, limits to knowledge, as well as institutional incapacity. An additional finding is that tensions in energy policy formulation emerging from the EU’s multilevel system of governance are leading to a clash between pioneer member states aiming for an ambitious energy transition and the Commission’s more modest orientations. In consequence, an analytical typology exploring conflict between rival conceptualizations of energy transitions will be relevant for the future.

Mixed Poisson process with Pareto mixing variable and its risk applications∗

This paper considers a mixed Poisson process with Pareto mixing variable, Exp-Pareto, and Erlang–Pareto distributions. New important properties of these distributions are derived.Approximations of the random time transformed Cramer–Lundberg collective risk model with Pareto mixing variable and different claim sizes, with finite or infinite means are given. The resulting risk model has dependent, Exp-Pareto interarrival times with Archimedean copulas.

Longitudinal Survey, Nonmonotone, Nonresponse, Imputation, Nonparametric Regression

The study focuses on the imputation for the longitudinal survey data which often has nonignorable nonrespondents. Local linear regression is used to impute the missing values and then the estimation of the time-dependent finite populations means. The asymptotic properties (unbiasedness and consistency) of the proposed estimator are investigated. Comparisons between different parametric and nonparametric estimators are performed based on the bootstrap standard deviation, mean square error and percentage relative bias. A simulation study is carried out to determine the best performing estimator of the time-dependent finite population means. The simulation results show that local linear regression estimator yields good properties.

Numerical tools for obtaining power-law representations of heavy-tailed datasets

Many empirical datasets have highly skewed, non-Gaussian, heavy-tailed distributions, dominated by a relatively small number of data points at the high end of the distribution. Consistent with their role as stable distributions, power laws have frequently been proposed to model such datasets. However there are physical situations that require distributions with finite means. Such situations may call for power laws with high-end cutoffs. Here, I present a maximum-likelihood technique for determining an optimal cut-off power law to represent a given dataset. I also develop a new statistical test of the quality of fit. Results are demonstrated for a number of benchmark datasets. Non-power-law datasets can frequently be represented by power laws, but this is a trivial result unless the dataset spans a broad domain. Nevertheless, I demonstrate that there are non-power-law distributions, including broad log-normal distributions, whose tails can be fit to power laws over many orders of magnitude. Therefore, caution is called for whenever power laws are invoked to represent empirical data.

From Ideality to Historicity, What Happens?

1. Historicity and Origin of Geometry1.1 as Prototype of Pure HistoricityIn his lecture about Philosophy and Crisis of European Humanity (1935), also known as Vienna Lecture Edmund Husserl describes development of philosophy and science as an irreversible transformation of human civilization. This transformation was due to introduction of a sense of Along with which established a system of ideal objects and deductive reasoning, there arrived possibility of manipulating universal and infinite objects by finite means of human reason.2 Infinite horizons for universal validity in knowledge, ideas and ideal objects that are not subjected to transformation and becoming, that stand eternally and omnitemporally both outside historical becoming and as regulative horizons for historical progression, were made possible for first time. But introduction of infinity not only entailed a shift in science and philosophy. Human values and human art, ethical and political life, also begin to be shaped and guided by infinite horizons. Thus Greek miracle involved a transformation of way in which human beings produce their own civilization. Human culture became a culture of trans-historical or trans-generational objects, and began producing humanity itself according to such infinite objects, indeed as infinite objects.3The introduction of infinity made possible concept of historicity itself. As Derrida puts it in his commentary on Husserl's posthumous essay concerning of published as an appendix to Crisis, the openness of infinite is only . . . openness of history itself, in utmost depths and purity of its essence (Derrida 1962: 143; 1989: 131). Historicity is not senseless mortality or senseless finitude; it is not opposed to infinite realization of human values; it is not realm of delusion and mere appearance against background of absolute being and truth. Historicity is realm in which infinity becomes actual, concrete and true for first time. Historicity is infinite horizon for infinite realization of humanity's true knowledge and absolute value.Husserl's phenomenological investigations into concept of history take as a prototype of science and philosophy in general, and therefore, ultimately, as a prototype of transformation giving rise to new idea of historicity defining European humanity. For that reason, question about origin of geometry, as surprising as it may seem at first sight, must be understood as a question concerning concept of historicity itself. The origin of geometry is origin of historicity, hence origin of any possible understanding of historical origins in general. Geometry will therefore be treated as being much more than, or something totally different from, a simple mathematical discipline or theory. It will not be taken, for instance, as general concept of normal way in which science is practiced, that is, as a 'paradigm' in Thomas Kuhn's sense (Kuhn 1996). It is not even taken as one horizon or one infinite task, among others. Geometry is analysed as infinite horizonality, as ideality as such, as possibility itself of an idea of infinite progression and infinite task, in sum as methodological basis upon which any historicity of truth is thinkable and therefore any concept of science and of history of science is possible, including idea of science as a cultural practice historically articulated through shift of paradigms that are incommensurable to each other.4 is considered a prototype of ultimate horizon for any trans-generational practice of meaning and for any formation of world-views or world-conceptions capable of providing representations of historical progression (Heidegger 2002; Derrida 2013: 196-202).The problem of origin of geometry is crucial for understanding formation and development of Derrida's early conception of historicity (1989). …

A QUANTILE-BASED PROBABILISTIC MEAN VALUE THEOREM

For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.

Reversibility and asymmetric conflict in event structures

Reversible computation has attracted increasing interest in recent years, with applications in hardware, software and biochemistry. We introduce reversible forms of prime event structures and asymmetric event structures. In order to control the manner in which events are reversed, we use asymmetric conflict on events. We prove a number of results about reachable configurations; for instance, we show under what conditions reachable configurations which are finite are reachable by purely finite means. We discuss, with examples, reversing in causal order, where an event is only reversed once all events it caused have been reversed, as well as forms of non-causal reversing.

Five fundamental constraints on theories of the origins of music.

The diverse forms and functions of human music place obstacles in the way of an evolutionary reconstruction of its origins. In the absence of any obvious homologues of human music among our closest primate relatives, theorizing about its origins, in order to make progress, needs constraints from the nature of music, the capacities it engages, and the contexts in which it occurs. Here we propose and examine five fundamental constraints that bear on theories of how music and some of its features may have originated. First, cultural transmission, bringing the formal powers of cultural as contrasted with Darwinian evolution to bear on its contents. Second, generativity, i.e. the fact that music generates infinite pattern diversity by finite means. Third, vocal production learning, without which there can be no human singing. Fourth, entrainment with perfect synchrony, without which there is neither rhythmic ensemble music nor rhythmic dancing to music. And fifth, the universal propensity of humans to gather occasionally to sing and dance together in a group, which suggests a motivational basis endemic to our biology. We end by considering the evolutionary context within which these constraints had to be met in the genesis of human musicality.

The exponential moment tail of inhomogeneous renewal process

Let θ1,θ2,… be a sequence of nonnegative not necessarily identically distributed and independent random variables having finite means and satisfying some additional conditions. We consider the asymptotic behavior of the quantity E(bΘ(t)1Θ(t)>at), where a and b are suitable positive constants, and Θ(t) is an inhomogeneous renewal process generated by the sequence θ1,θ2,…. We also present a few corollaries concerning elementary renewal theorems for the above process.

Doubly robust imputation procedures for finite population means in the presence of a large number of zeros

AbstractSingle imputation is often used in surveys to compensate for item nonresponse. In some cases, the variable requiring imputation contains a large amount of zeros. This is especially frequent in business surveys that collect economic variables. Motivated by a mixture regression model, we propose three imputation procedures and study their properties in terms of bias and variance. We show that these procedures are doubly robust, leading to consistent estimators of the finite population mean if either the imputation model or the nonresponse model is well specified. For the proposed procedures, we consider a jackknife variance estimator, which is consistent for the true variance, provided the overall sampling fraction is negligible. Finally, the results of a simulation study comparing the performance of point and variance estimators in terms of relative bias and mean square error are presented. The Canadian Journal of Statistics 42: 650–669; 2014 © 2014 Statistical Society of Canada

Equivalence of the Symbol Grounding and Quantum System Identification Problems

The symbol grounding problem is the problem of specifying a semantics for the representations employed by a physical symbol system in a way that is neither circular nor regressive. The quantum system identification problem is the problem of relating observational outcomes to specific collections of physical degrees of freedom, i.e., to specific Hilbert spaces. It is shown that with reasonable physical assumptions these problems are equivalent. As the quantum system identification problem is demonstrably unsolvable by finite means, the symbol grounding problem is similarly unsolvable.

Star-Type Systematic Sampling Schemes for Finite Populations

Two new sampling schemes namely, Star-Type Systematic (STS) sampling without replacement and Modified Star-Type Systematic (MSTS) sampling without replacement for estimation of finite population means are introduced. The relative performances of the proposed star-type systematic sample means along with those of the simple random and systematic sample means are assessed for a hypothetical population with a linear trend and also for certain natural populations. Furthermore, the usefulness of the proposed sampling schemes in quality control and for constructing partial diallel crosses in mating designs are briefly break discussed.

Stable Paretian Versus Student'stStock Market Hypothesis

This article investigates the types of probability distributions that can best represent equity returns using a large sample of daily S&P500 index returns. The competing models, Stable Paretian and Pearson families, are compared using Bayesian methods. The evidence against Stable Paretian as a model of S&P500 index returns is overwhelming. The distribution that best fits the data is Pearson Type IV, and Student's t fits almost as well. One implication is that a Bayesian decision maker should strongly shift beliefs in favor of a Pearson distribution with finite means and variances as a model of daily changes in the S&P500 stock index.

Evidence of recursion in tool use

We discuss the discovery of technologies involving knotted netting, such as textiles, basketry, and cordage, in the Upper Paleolithic. This evidence, in our view, suggests a new way of connecting toolmaking and syntactic structure in human evolution, because these technologies already exhibit an "infinite use of finite means," which we take to constitute the key transition to human cognition.

Objective Bayesian Analysis of a Measurement Error Small Area Model

We consider small area estimation under a nested error linear regression model with measurement errors in the covariates. We propose an objective Bayesian analysis of the model to estimate the finite population means of the small areas. In particular, we derive Jeffreys’ prior for model parameters. We also show that Jeffreys’ prior, which is improper, leads, under very general conditions, to a proper posterior distribution. We have also performed a simulation study where we have compared the Bayes estimates of the finite population means under the Jeffreys’ prior with other Bayesian estimates obtained via the use of the standard flat prior and with non-Bayesian estimates, i.e., the corresponding empirical Bayes estimates and the direct estimates.

New examples of heavy-tailed O-subexponential distributions and related closure properties

Let L and S denote the classes of distributions with long tails and subexponential tails respectively. Let OS denote the class of distributions with O-subexponential tails, which means the distributions with the tails having the same order as the tails of their 2-fold convolutions. In this paper, we first construct a family of distributions without finite means in L ∩ OS ∖ S . Next some distributions in L ∩ OS ∖ S , which possess finite means or even finite higher moments, are also constructed. In connection with this, we prove that the class OS is closed under minimization of random variables. However, it is not closed under maximization of random variables.