Error and Reliability in Stochastic Processes and Psychological Measurement

1972 ◽  
Vol 31 (1) ◽  
pp. 131-140 ◽  
Author(s):  
Donald W. Zimmerman
Keyword(s):  
Stochastic Processes ◽  
Random Error ◽  
Random Sampling ◽  
Probability Model ◽  
Random Variables ◽  
Formal Structure ◽  
Intermediate Structure ◽  
Psychological Measurement ◽  
Joint Distributions ◽  
True Values

The concepts of random error and reliability of measurements that are familiar in traditional theories based on the notions of “true values” and “errors” can be represented by a probability model having a simpler formal structure and fewer special assumptions about random sampling and independence of measurements. In this model formulas that relate observable events are derived from probability axioms and from primitive terms that refer to observable events, without an intermediate structure containing variances and correlations of “true” and “error” components of scores. While more economical in language and formalism, the model at the same time is more general than classical theories and applies to stochastic processes in which joint distributions of many dependent random variables are of interest. In addition, it clarifies some long-standing problems concerning “experimental independence” of measurements and the relation of sampling of individuals to sampling of measurements.

Probability Spaces and the Theory of Error of Measurement

1971 ◽  
Vol 28 (1) ◽  
pp. 291-301 ◽  
Author(s):  
Donald W. Zimmerman
Keyword(s):  
Probability Model ◽  
Random Variables ◽  
Measurement Procedure ◽  
Psychological Tests ◽  
Random Variability ◽  
Probability Spaces ◽  
True Values ◽  
Suitable Product ◽  
Error Of Measurement

A model of variability in measurement, which is sufficiently general for a variety of applications and which includes the main content of traditional theories of error of measurement and psychological tests, can be derived from the axioms of probability, without introducing “true values” and “errors.” Beginning with probability spaces (Ω, P1) and (φ, P2), the set Ω representing the outcomes of a measurement procedure and the set * representing individuals or experimental objects, it is possible to construct suitable product probability spaces and collections of random variables which can yield all results needed to describe random variability and reliability. This paper attempts to fill gaps in the mathematical derivations in many classical theories and at the same time to overcome limitations in the language of “true values” and “errors” by presenting explicitly the essential constructions required for a general probability model.

A normal form theorem for Lω1p, with applications

1982 ◽  
Vol 47 (3) ◽  
pp. 605-624 ◽  
Author(s):  
Douglas N. Hoover
Keyword(s):  
Normal Form ◽  
Probability Model ◽  
Random Variables ◽  
Interpolation Theorem ◽  
Complete Theory ◽  
De Finetti’S Theorem ◽  
Form Theorem ◽  
Normal Form Theorem ◽  
De Finetti's Theorem

AbstractWe show that every formula of Lω1P is equivalent to one which is a propositional combination of formulas with only one quantifier. It follows that the complete theory of a probability model is determined by the distribution of a family of random variables induced by the model. We characterize the class of distribution which can arise in such a way. We use these results together with a form of de Finetti’s theorem to prove an almost sure interpolation theorem for Lω1P.

Laws of large numbers for q-dependent random variables and nonextensive statistical mechanics

2017 ◽  
Vol 31 (15) ◽  
pp. 1750117
Author(s):  
Marco A. S. Trindade
Keyword(s):  
Statistical Mechanics ◽  
Stochastic Processes ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Nonextensive Statistical Mechanics ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Large Numbers

In this work, we prove a weak law and a strong law of large numbers through the concept of [Formula: see text]-product for dependent random variables, in the context of nonextensive statistical mechanics. Applications for the consistency of estimators are presented and connections with stochastic processes are discussed.

Fourier Transforms in Probability, Random Variables and Stochastic Processes

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Fourier Analysis ◽  
Stochastic Processes ◽  
Probability Density ◽  
Density Function ◽  
Fourier Transforms ◽  
Random Variables ◽  
Cumulative Distribution ◽  
The Face

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to the outcome of a random experiment. We can, for example, flip a coin and assign an outcome of a heads as X = 1 and a tails X = 0. Often the number is equated to the numerical outcome of the experiment, such as the number of dots on the face of a rolled die or the measurement of a voltage in a noisy circuit. The cumulative distribution function is defined by FX(x) = Pr[X ≤ x]. (4.1) The probability density function is the derivative fX(x) = d /dxFX(x). Our treatment of random variables focuses on use of Fourier analysis. Due to this viewpoint, the development we use is unconventional and begins immediately in the next section with discussion of properties of the probability density function.

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (02) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

scholarly journals Random Variables and Stochastic Processes

2000 ◽  
Vol 33 (16) ◽  
pp. 403-414
Author(s):  
Vladimir I. Zubov
Keyword(s):  
Stochastic Processes ◽  
Random Variables
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