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Title:

Passage times of random walks and Levy processes across power law boundaries

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Probability Theory and Related Fields

Year of Publication:

2005-05-03

Source:

Probability Theory and Related Fields. 2005; 133(1).

Probability Theory and Related Fields. 2005; 133(1). Minimize

Document Type:

Journal article

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http://www.manchester.ac.uk/escholar/uk-ac-man-scw:1a9910

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Title:

Passage of Levy processes across power law boundaries .at small times.

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Source:

Annals of Probabability. (1):160-197.

Annals of Probabability. (1):160-197. Minimize

Document Type:

Original research ; Academic journal article ; Journal article

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http://www.manchester.ac.uk/escholar/uk-ac-man-scw:148235

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Title:

Renewal theorems and stability for the reflected process

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Year of Publication:

2009

Source:

Stochastic processes and their applications. 2009; 119(4):1270-1297.

Stochastic processes and their applications. 2009; 119(4):1270-1297. Minimize

Document Type:

Original research ; Academic journal article ; Journal article

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http://www.manchester.ac.uk/escholar/uk-ac-man-scw:51976

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Title:

THE EFFECT OF TRIMMING ON THE STRONG LAW OF LARGE NUMBERS

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'Trimmed ' sample sums may be defined for r = 1, 2,., by and We _c _v(l)_y(2) _ _yW where Sn = X1 + X-, +. + Xn is the sum of independent and identically distributed random variables Xit M^s=. 25 M 1 " ' denote Xx,., Xn arranged in decreasing order, and X ^ is the observation with i ^ x n g g ^ the yth largest modulus. We investigate the effects...

'Trimmed ' sample sums may be defined for r = 1, 2,., by and We _c _v(l)_y(2) _ _yW where Sn = X1 + X-, +. + Xn is the sum of independent and identically distributed random variables Xit M^s=. 25 M 1 " ' denote Xx,., Xn arranged in decreasing order, and X ^ is the observation with i ^ x n g g ^ the yth largest modulus. We investigate the effects of these kinds of trimming on various forms of convergence and divergence of the sample sum. In particular, we provide integral tests for <- r) Sn/n-*±», and analytical criteria for almost sure relative stability when the number of points trimmed, r, is fixed, but n—> °°. Some surprising results occur. For example, when r = 0,1, 2,., {r) Sn may be almost surely negatively relatively stable ( (r) 5n/5n — *-1 a.s. as n — * «> for some non-stochastic sequence Bn f o°) only if- « <EX ^ =£(), and a striking corollary of this is an example of a random walk Sn which is recurrent (even has mean 0), but for which (r) 5n and (r) 5n are transient when r s * 1. 1. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-02-03

Source:

http://plms.oxfordjournals.org/cgi/reprint/s3-71/2/441.pdf

http://plms.oxfordjournals.org/cgi/reprint/s3-71/2/441.pdf Minimize

Document Type:

text

Language:

en

DDC:

519 Probabilities & applied mathematics *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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and

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We compare the largest jump in a Lévy process Xt up till time t, i.e, Yt = sup{|Xs − Xs− | : s ≤ t}, to the two-sided maximal value of the process, Let Mt = sup{|Xs | : s ≤ t}. T(r) = inf{t> 0: |Xt |> r}, r> 0, be the two-sided passage time out of the two-sided strip [−r, r]. Then we show that Yt is negligible with respect to Mt for small times,...

We compare the largest jump in a Lévy process Xt up till time t, i.e, Yt = sup{|Xs − Xs− | : s ≤ t}, to the two-sided maximal value of the process, Let Mt = sup{|Xs | : s ≤ t}. T(r) = inf{t> 0: |Xt |> r}, r> 0, be the two-sided passage time out of the two-sided strip [−r, r]. Then we show that Yt is negligible with respect to Mt for small times, i.e., lim t↓0 Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-10-06

Source:

http://www.math.ku.dk/%7Emikosch/maphysto_extremes_2005/Slides/Maller.pdf

http://www.math.ku.dk/%7Emikosch/maphysto_extremes_2005/Slides/Maller.pdf Minimize

Document Type:

text

Language:

en

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Some aspects of a mixture model for estimating the boundary of a set of data

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This paper discusses a method for estimating the parameters of the boundary of a set of points for which the y variable is bounded above. The boundary may be linear, for example, in which case a scatterplot of the data may have a triangular appearance. An example is obtained by plotting cube root of food volume in an animal's incompletely filled...

This paper discusses a method for estimating the parameters of the boundary of a set of points for which the y variable is bounded above. The boundary may be linear, for example, in which case a scatterplot of the data may have a triangular appearance. An example is obtained by plotting cube root of food volume in an animal's incompletely filled stomach against the animal's length; full stomachs constitute the upper boundary, while other volumes fall between 0 and the boundary, depending on how full or empty a stomach may be. Our method is to fit a mixture of an ordinary regression model and a variate having an unknown mixing distribution, which represents the variation of another unobserved factor such as the proportionate fullness of the stomach, for example. The mixing distribution is discretized and the number of mixing classes is estimated by Akaike's procedure. The method is found to be stable and reliable for simulated sets of data, and is illustrated by application to three data sets. Examples of occurrence of the problem in various areas of ecology, and in other areas, are given. Minimize

Publisher:

Oxford University Press

Year of Publication:

1990-01-01 00:00:00.0

Document Type:

TEXT

Language:

en

Subjects:

Articles

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DDC:

310 Collections of general statistics *(computed)*

Rights:

Copyright (C) 1990, International Council for the Exploration of the Sea/Conseil International pour l'Exploration de la Mer

Copyright (C) 1990, International Council for the Exploration of the Sea/Conseil International pour l'Exploration de la Mer Minimize

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Title:

Small-time versions of Strassen's law for Levy processes

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We study aspects of the ‘small-time’ behaviour (as t ↓ 0) of a Lévy process X ( t ), obtaining a very general small-time version of Strassen's almost sure (a.s.) functional law of the iterated logarithm (LIL) for random walks. The class of Lévy processes for which this holds is characterised by an explicit analytic condition on the Lévy measu...

We study aspects of the ‘small-time’ behaviour (as t ↓ 0) of a Lévy process X ( t ), obtaining a very general small-time version of Strassen's almost sure (a.s.) functional law of the iterated logarithm (LIL) for random walks. The class of Lévy processes for which this holds is characterised by an explicit analytic condition on the Lévy measure of X , related to an analogous condition of Kesten for a generalised (large-time) random walk LIL. Both centred and uncentred versions of the small-time result are proved. Subsidiary results concerning functional weak convergence of X ( t ) to Brownian motion as t ↓ 0 are shown to be equivalent to the main a.s. results. The quadratic variation process of X is considered, and applications via continuous functionals are suggested. Minimize

Publisher:

Oxford University Press

Year of Publication:

2009-03-01 00:00:00.0

Document Type:

TEXT

Language:

en

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Articles

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Copyright (C) 2009, London Mathematical Society

Copyright (C) 2009, London Mathematical Society Minimize

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Title:

On the exponential model for survival

Description:

A simple sufficient condition for the existence, consistency and asymptotic normality of the regression coefficient in the exponential model for survial is given. The intuition that consistent estimation can only be guaranteed when censoring variable are not too small, depending on the values of the covariates, can be quantified in some examples...

A simple sufficient condition for the existence, consistency and asymptotic normality of the regression coefficient in the exponential model for survial is given. The intuition that consistent estimation can only be guaranteed when censoring variable are not too small, depending on the values of the covariates, can be quantified in some examples with the use of this condition. Minimize

Publisher:

Oxford University Press

Year of Publication:

1988-09-01 00:00:00.0

Document Type:

TEXT

Language:

en

Subjects:

Articles

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Rights:

Copyright (C) 1988, Biometrika Trust

Copyright (C) 1988, Biometrika Trust Minimize

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Title:

Estimating the proportion of immunes in a censored sample

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<sec> An estimator for the proportion of ‘immunes’ in a population is proposed, when a sample of censored failure times is available. Our suggestion is to use one minus the maximum observed value of the Kaplan-Meier empirical distribution function. This estimator is shown to be consistent and asymptotically normal, under modest conditions on the...

<sec> An estimator for the proportion of ‘immunes’ in a population is proposed, when a sample of censored failure times is available. Our suggestion is to use one minus the maximum observed value of the Kaplan-Meier empirical distribution function. This estimator is shown to be consistent and asymptotically normal, under modest conditions on the censoring mechanism. Simulations suggest that the estimator is approximately normal for quite small sample sizes, provided the immune proportion is not too close to zero. A simple nonparametric statistic is proposed to test whether the assumptions of the analysis are likely to be valid. </sec> Minimize

Publisher:

Oxford University Press

Year of Publication:

1992-12-01 00:00:00.0

Document Type:

TEXT

Language:

en

Subjects:

Articles

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Rights:

Copyright (C) 1992, Biometrika Trust

Copyright (C) 1992, Biometrika Trust Minimize

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Title:

The Effect of Trimming on the Strong Law of Large Numbers

Description:

‘Trimmed’ sample sums may be defined for r = 1, 2, …, by<fd>$${}^{\left(r\right)}{S}_{n}={S}_{n}-{M}_{n}^{\left(1\right)}-{M}_{n}^{\left(2\right)}-\mathrm{.}-{M}_{n}^{\left(r\right)}$$</fd>and<fd>$${}^{\left(r\right)}{S}_{n}={S}_{n}-{X}_{n}^{\left(1\right)}-{X}_{n}^{\left(2\right)}-\mathrm{.}-{X}_{n}^{\left(r\right)}$$</fd>where S n = X 1...

‘Trimmed’ sample sums may be defined for r = 1, 2, …, by<fd>$${}^{\left(r\right)}{S}_{n}={S}_{n}-{M}_{n}^{\left(1\right)}-{M}_{n}^{\left(2\right)}-\mathrm{.}-{M}_{n}^{\left(r\right)}$$</fd>and<fd>$${}^{\left(r\right)}{S}_{n}={S}_{n}-{X}_{n}^{\left(1\right)}-{X}_{n}^{\left(2\right)}-\mathrm{.}-{X}_{n}^{\left(r\right)}$$</fd>where S n = X 1 + X 2 + … + X n is the sum of independent and identically distributed random variables X i , <f>$${M}_{n}^{\left(1\right)}\ge \dots \hbox{ \hspace{0.17em} }\ge {M}_{n}^{\left(n\right)}$$</f> denote X l , …, X n arranged in decreasing order, and <f>$${X}_{n}^{\left(j\right)}$$</f> is the observation with the j th largest modulus. We investigate the effects of these kinds of trimming on various forms of convergence and divergence of the sample sum. In particular, we provide integral tests for ( r ) S n / n → ±∞, and analytical criteria for almost sure relative stability when the number of points trimmed, r , is fixed, but n → ∞. Some surprising results occur. For example, when r = 0, 1, 2, …, ( r ) S n may be almost surely negatively relatively stable (( r ) S n / B n → −1 a.s. as n → ∞ for some non-stochastic sequence B n ↑ ∞) only if −∞ < EX 1 ≤ 0, and a striking corollary of this is an example of a random walk S n which is recurrent (even has mean 0), but for which ( r ) S n and <f>$${\tilde{S}}_{n}$$</f> are transient when r ≥ 1. Minimize

Publisher:

Oxford University Press

Year of Publication:

1995-09-01 00:00:00.0

Document Type:

TEXT

Language:

en

Subjects:

Articles

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Rights:

Copyright (C) 1995, London Mathematical Society

Copyright (C) 1995, London Mathematical Society Minimize

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