The identifiability of mixtures of distributions

1969 ◽  
Vol 6 (2) ◽  
pp. 389-398 ◽  
Author(s):  
G. M. Tallis
Keyword(s):  
Distribution Function ◽  
Measurable Function ◽  
Borel Measurable Function ◽  
Mixtures Of Distributions ◽  
Image Position ◽  
Borel Measurable

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).

The identifiability of mixtures of distributions

1969 ◽  
Vol 6 (02) ◽  
pp. 389-398 ◽  
Author(s):  
G. M. Tallis
Keyword(s):  
Distribution Function ◽  
Measurable Function ◽  
Borel Measurable Function ◽  
Mixtures Of Distributions ◽  
Image Position ◽  
Borel Measurable

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).

scholarly journals Estimates for the green function and existence of positive solutions for higher-order elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-22
Author(s):  
Imed Bachar
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Measurable Function ◽  
Elliptic Equations ◽  
Nonlinear Problems ◽  
Borel Measurable Function ◽  
Continuous Solutions ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Borel Measurable

We establish a3G-theorem for the iterated Green function of(−∆)pm, (p≥1,m≥1), on the unit ballBofℝn(n≥1)with boundary conditions(∂/∂ν)j(−∆)kmu=0on∂B, for0≤k≤p−1and0≤j≤m−1. We exploit this result to study a class of potentials𝒥m,n(p). Next, we aim at proving the existence of positive continuous solutions for the following polyharmonic nonlinear problems(−∆)pmu=h(‧,u), inD(in the sense of distributions),lim|x|→1((−∆)kmu(x)/(1−|x|)m−1)=0, for0≤k≤p−1, whereD=BorB\{0}andhis a Borel measurable function onD×(0,∞)satisfying some appropriate conditions related to𝒥m,n(p).

On planar sets with prescribed packing dimensions of line sections

2001 ◽  
Vol 130 (3) ◽  
pp. 523-539 ◽  
Author(s):  
MARIANNA CSÖRNYEI
Keyword(s):  
Measurable Function ◽  
Packing Dimension ◽  
Borel Measurable Function ◽  
Packing Dimensions ◽  
Borel Measurable

We prove that for an arbitrary Borel measurable function f on the space of all planar lines there exists a set which intersects almost every line [lscr ] in a set of packing dimension f([lscr ]).

Uniform approximation by universal series on arbitrary sets

2008 ◽  
Vol 144 (1) ◽  
pp. 207-216 ◽  
Author(s):  
VANGELIS STEFANOPOULOS
Keyword(s):  
Uniform Approximation ◽  
Measurable Function ◽  
Continuous Functions ◽  
Partial Sums ◽  
Borel Measurable Function ◽  
Universal Series ◽  
Space Of Continuous Functions ◽  
Almost Everywhere ◽  
Bounded Functions ◽  
Borel Measurable

AbstractBy considering a tree-like decomposition of an arbitrary set we prove the existence of an associated series with the property that its partial sums approximate uniformly all elements in a relevant space of bounded functions. In a topological setting we show that these sums are dense in the space of continuous functions, hence in turn any Borel measurable function is the almost everywhere limit of an appropriate sequence of partial sums of the same series. The coefficients of the series may be chosen in c0, or in a weighted ℓp with 1 < p < ∞, but not in the corresponding weighted ℓ1.

scholarly journals On rearranging maximal functions in Rn

1975 ◽  
Vol 19 (4) ◽  
pp. 363-369 ◽  
Author(s):  
P. L. Walker
Keyword(s):  
Distribution Function ◽  
Lebesgue Measure ◽  
Measurable Function ◽  
Maximal Functions ◽  
Decreasing Rearrangement ◽  
Image Position

Denote by f a positive measurable function on Rn, and by λ the distribution function of denotes the Lebesgue measure of the set specified. We shall suppose that λ(y)<∞ for each y>0, and that λ(y)→0 as y→∞. The decreasing rearrangement f* of f is defined on (0, ∞) by

CANONICAL BEHAVIOR OF BOREL FUNCTIONS ON SUPERPERFECT RECTANGLES

2001 ◽  
Vol 01 (02) ◽  
pp. 173-220 ◽  
Author(s):  
OTMAR SPINAS
Keyword(s):  
Equivalence Relation ◽  
Measurable Function ◽  
Borel Measurable Function ◽  
Canonical Function ◽  
Borel Functions ◽  
Borel Measurable

We describe a list of canonical functions from (ωω)2 to ℝ such that every Borel measurable function from (ωω)2 to ℝ, on some superperfect rectangle, induces the same equivalence relation as some canonical function.

scholarly journals Convergence of ergodic averages for many group rotations

2015 ◽  
Vol 36 (7) ◽  
pp. 2107-2120
Author(s):  
ZOLTÁN BUCZOLICH ◽  
GABRIELLA KESZTHELYI
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Measurable Function ◽  
Dual Group ◽  
Ergodic Averages ◽  
Abelian Topological Group ◽  
Compact Abelian Topological Group ◽  
Image Position ◽  
Rotation Set ◽  
Group Rotations

Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f:G\rightarrow \mathbb{R}$ is a measurable function. Given $(n_{k})$, a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages $$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$ The $f$-rotation set is $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$We prove that if $G$ is a compact locally connected Abelian group and $f:G\rightarrow \mathbb{R}$ is a measurable function then from $m(\unicode[STIX]{x1D6E4}_{f})>0$ it follows that $f\in L^{1}(G)$. A similar result is established for ordinary Birkhoff averages if $G=Z_{p}$, the group of $p$-adic integers. However, if the dual group, $\widehat{G}$, contains ‘infinitely many multiple torsion’ then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, $f(x+n_{k}\unicode[STIX]{x1D6FC})/k$, $k=1,\ldots ,$ for almost every $x$ for many $\unicode[STIX]{x1D6FC}$; hence, some of our theorems are stated by using instead of $\unicode[STIX]{x1D6E4}_{f}$ slightly larger sets, denoted by $\unicode[STIX]{x1D6E4}_{f,b}$.

scholarly journals Some Remarks on a Recent Paper by Borch

1961 ◽  
Vol 1 (5) ◽  
pp. 265-272 ◽  
Author(s):  
Paul Markham Kahn
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Fixed Number ◽  
Point Of View ◽  
Optimum Amount ◽  
Measurable Transformation ◽  
The Difference ◽  
Image Position ◽  
Stop Loss ◽  
Condition B

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such thatwhere c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

scholarly journals A Note on the Most “Dangerous” and Skewest Class of Distributions

1963 ◽  
Vol 2 (3) ◽  
pp. 387-390 ◽  
Author(s):  
Gunnar Benktander
Keyword(s):  
Distribution Function ◽  
Theoretical Models ◽  
Stochastic Variable ◽  
Classical Definition ◽  
Image Position ◽  
Third Moment

In the classical definition skewness is departure from symmetry. It was therefore natural to measure skewness by using a normalized third moment μ3/σ3. This condensed measure, however, is not refined enough to be used as an operational instrument for studying various functions which might be used to describe actual claim distributions. This is true especially when the interest is concentrated towards the higher values of the variate.In their paper (1) Benktander-Segerdahl have suggested that the average excess claim m(x) as a function of the priority x should be used to reveal the characteristics of the tail of the distribution where P(x) = 1 — H(x) denotes the distribution function.This statistic is very apt when comparing actual claim distributions with possible theoretical models. It is also useful when classifying these models.If, however, emphasis mainly is laid on classifying distributions according to their skewness, another statistic might be preferable. Let μ(x)dx denote the probability that a stochastic variable which is known to be at least equal to x, does not exceed x+dx. In other words, μ(x)dx represents the probability that a claim or the corresponding stochastic variable, which, when observed from the bottom, is “alive” at x, “dies” in the interval (x, x + dx). The lower this claims rate of mortality, the skewer and more dangerous is the claim distribution.

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