A bootstrap algorithm for the isotropic random sphere

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993a)) on the sphere, define to be a realization of the random process and to be the cardinality of . A bootstrap algorithm is presented and conditions for strong uniform consistency of the bootstrap cumulative distribution function of the standardized sample mean, , are given. We illustrate the bootstrap algorithm with global land-area data.

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Strong uniform consistency rates of conditional quantile estimation in the single functional index model under random censorship

Dependence Modeling ◽

10.1515/demo-2018-0013 ◽

2018 ◽

Vol 6 (1) ◽

pp. 197-227 ◽

Cited By ~ 1

Author(s):

Nadia Kadiri ◽

Abbes Rabhi ◽

Amina Angelika Bouchentouf

Keyword(s):

Cumulative Distribution Function ◽

Complete Convergence ◽

Asymptotic Properties ◽

Cumulative Distribution ◽

Index Structure ◽

Functional Index ◽

Conditional Quantile ◽

Index Model ◽

Strong Uniform Consistency ◽

Mixing Sequence

AbstractThe main objective of this paper is to non-parametrically estimate the quantiles of a conditional distribution in the censorship model when the sample is considered as an -mixing sequence. First of all, a kernel type estimator for the conditional cumulative distribution function (cond-cdf) is introduced. Afterwards, we estimate the quantiles by inverting this estimated cond-cdf and state the asymptotic properties when the observations are linked with a single-index structure. The pointwise almost complete convergence and the uniform almost complete convergence (with rate) of the kernel estimate of this model are established. This approach can be applied in time series analysis.

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A central limit theorem for the isotropic random sphere

Advances in Applied Probability ◽

10.1017/s0001867800027099 ◽

1995 ◽

Vol 27 (03) ◽

pp. 642-651

Author(s):

Jason J. Brown

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Asymptotic Normality ◽

Random Field ◽

Random Process ◽

Positive Constant ◽

Marginal Distribution ◽

Dependence Structure ◽

Sample Mean ◽

Isotropic Random

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993)) on the sphere, define to be a realization of the random process and to be the cardinality of . Without specifying the dependence structure of nor the marginal distribution of the , conditions for asymptotic normality of the standardized sample mean, , are given. The conditions on and are motivated by the ideas and results for dependent stationary sequences.

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Saddlepoint approximations for bivariate distributions

Journal of Applied Probability ◽

10.1017/s0021900200039139 ◽

1990 ◽

Vol 27 (03) ◽

pp. 586-597 ◽

Cited By ~ 2

Author(s):

Suojin Wang

Keyword(s):

Monte Carlo ◽

Distribution Function ◽

Normal Distribution ◽

Cumulative Distribution Function ◽

Saddlepoint Approximation ◽

Monte Carlo Study ◽

Cumulative Distribution ◽

Normal Distribution Function ◽

Sample Mean ◽

Saddlepoint Approximations

A saddlepoint approximation is derived for the cumulative distribution function of the sample mean of n independent bivariate random vectors. The derivations use Lugannani and Rice's saddlepoint formula and the standard bivariated normal distribution function. The separate versions of the approximation for the discrete cases are also given. A Monte Carlo study shows that the new approximation is very accurate.

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Saddlepoint approximations in conditional inference

Journal of Applied Probability ◽

10.2307/3214848 ◽

1993 ◽

Vol 30 (2) ◽

pp. 397-404 ◽

Cited By ~ 3

Author(s):

Suojin Wang

Keyword(s):

Distribution Function ◽

Linear Function ◽

Relative Error ◽

Cumulative Distribution Function ◽

Conditional Inference ◽

Cumulative Distribution ◽

Important Work ◽

Sample Mean ◽

Saddlepoint Approximations ◽

Higher Dimensional

Saddlepoint approximations are derived for the conditional cumulative distribution function and density of where is the sample mean of n i.i.d. bivariate random variables and g(x, y) is a non-linear function. The relative error of order O(n–1) is retained. The results extend the important work of Skovgaard (1987), and are useful in conditional inference, especially in the case of small or moderate sample sizes. Generalizations to higher-dimensional random vectors are also discussed. Some examples are demonstrated.

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Saddlepoint approximations in conditional inference

Journal of Applied Probability ◽

10.1017/s0021900200117401 ◽

1993 ◽

Vol 30 (02) ◽

pp. 397-404

Author(s):

Suojin Wang

Keyword(s):

Distribution Function ◽

Linear Function ◽

Relative Error ◽

Cumulative Distribution Function ◽

Conditional Inference ◽

Cumulative Distribution ◽

Important Work ◽

Sample Mean ◽

Saddlepoint Approximations ◽

Higher Dimensional

Saddlepoint approximations are derived for the conditional cumulative distribution function and density of where is the sample mean of n i.i.d. bivariate random variables and g(x, y) is a non-linear function. The relative error of order O(n –1) is retained. The results extend the important work of Skovgaard (1987), and are useful in conditional inference, especially in the case of small or moderate sample sizes. Generalizations to higher-dimensional random vectors are also discussed. Some examples are demonstrated.

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A central limit theorem for the isotropic random sphere

Advances in Applied Probability ◽

10.2307/1428128 ◽

1995 ◽

Vol 27 (3) ◽

pp. 642-651

Author(s):

Jason J. Brown

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Asymptotic Normality ◽

Random Field ◽

Random Process ◽

Positive Constant ◽

Marginal Distribution ◽

Dependence Structure ◽

Sample Mean ◽

Isotropic Random

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993)) on the sphere, define to be a realization of the random process and to be the cardinality of . Without specifying the dependence structure of nor the marginal distribution of the , conditions for asymptotic normality of the standardized sample mean, , are given. The conditions on and are motivated by the ideas and results for dependent stationary sequences.

Download Full-text

Saddlepoint approximations for bivariate distributions

Journal of Applied Probability ◽

10.2307/3214543 ◽

1990 ◽

Vol 27 (3) ◽

pp. 586-597 ◽

Cited By ~ 12

Author(s):

Suojin Wang

Keyword(s):

Monte Carlo ◽

Distribution Function ◽

Normal Distribution ◽

Cumulative Distribution Function ◽

Saddlepoint Approximation ◽

Monte Carlo Study ◽

Cumulative Distribution ◽

Normal Distribution Function ◽

Sample Mean ◽

Saddlepoint Approximations

A saddlepoint approximation is derived for the cumulative distribution function of the sample mean of n independent bivariate random vectors. The derivations use Lugannani and Rice's saddlepoint formula and the standard bivariated normal distribution function. The separate versions of the approximation for the discrete cases are also given. A Monte Carlo study shows that the new approximation is very accurate.

Download Full-text