Related papers: Taylor's Theorem and Mean Value Theorem for Random…
The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor's classical theorem is often needed, but only…
We generalize Taylor's theorem by introducing a stochastic formulation based on an underlying Poisson point process model. We utilize this approach to propose a novel non-linear regression framework and perform statistical inference of the…
Using the recently defined concept of Taylor measures, we propose a generalization of Taylor's theorem to measurable, non-analytic functions, that do not require differentiation. We study consequences of the generalization, including the…
As a rigorous statistical approach, statistical Taylor expansion extends the conventional Taylor expansion by replacing precise input variables with random variables of known distributions and sample counts to compute the mean, the…
For nonnegative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows to construct new distributions with support (0,1), and to obtain a…
In this note a general a Cauchy-type mean value theorem for the ratio of functional determinants is offered. It generalizes Cauchy's and Taylor's mean value theorems as well as other classical mean value theorems.
This paper examines the foundational concept of random variables in probability theory and statistical inference, demonstrating that their mathematical definition requires no reference to randomization or hypothetical repeated sampling. We…
Ever since the proof of asymptotic normality of maximum likelihood estimator by Cramer (1946), it has been understood that a basic technique of the Taylor series expansion suffices for asymptotics of $M$-estimators with…
A mid-point theorem is proved in an elementary way for the U type shape of functions that arise out of exponential quadratic functions. These results are inspired from epidemic patterns and growth over a time period. Key words: natural…
The Taylor hypothesis which allows surrogating spatial measurements requiring many experimental probes by time series from one or two probes is examined on the basis of a simple analytic model of turbulent statistics. The main points are as…
In this paper, we explore two fundamental theorems of differential calculus: Rolle's Theorem and the Mean Value Theorem (MVT). These theorems play a crucial role in the development of theoretical and practical results in mathematics,…
The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy Mean Value Theorem is taken at a point which has a well-determined position in the interval. As an application of this…
If a real-valued function is continuous on a real interval and it takes on two different values, then it will also take any value in between those two, by the Intermediate Value Theorem. It is not immediately clear what would be a natural…
Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in…
Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most…
Asymptotic statistical theory for estimating functions is reviewed in a generality suitable for stochastic processes. Conditions concerning existence of a consistent estimator, uniqueness, rate of convergence, and the asymptotic…
Taylor's law (TL) states that the variance $V$ of a non-negative random variable is a power function of its mean $M$, i.e. $V=a M^b$. The ubiquitous empirical verification of TL, typically displaying sample exponents $b \simeq 2$, suggests…
Taylor's law, also known as fluctuation scaling in physics and the power-law variance function in statistics, is an empirical pattern widely observed across fields including ecology, physics, finance, and epidemiology. It states that the…
Signal processing makes extensive use of point estimators and accompanying error bounds. These work well up until the likelihood function has two or more high peaks. When it is important for an estimator to remain reliable, it becomes…
There is a growing need for the ability to analyse interval-valued data. However, existing descriptive frameworks to achieve this ignore the process by which interval-valued data are typically constructed; namely by the aggregation of…