Approximation by Normal Distribution for a Sample Sum in Sampling Without Replacement from a Finite Population

Mohamed, Ibrahim; Mirakhmedov, Sherzod M.

doi:10.1007/s13171-016-0088-9

Cited by 2 publications

(3 citation statements)

References 22 publications

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“…According to approximation distribution for a sample sum in sampling without replacement from a finite population in Mohamed and Mirakhmedov (2016) [ 26 ], the asymptotic properties of

and

would be obtained as follows.…”

Section: Sufficient Screening Utilitymentioning

confidence: 99%

“…

has the following properties:

In addition, for all

, where

For continuous variables subject to arbitrary distribution

, let

. It is easy to find that the random variable

is a special case in Mohamed and Mirakhmedov (2016) [ 26 ], where

. According to the definition of

and

, it is obviously established that

and

.…”

Section: Appendix A1 Proof Of Remarkmentioning

confidence: 99%

“…In other words,

and

have the same distribution, which is

. If

, we have the following relationship by using Theorem 3.4, Corollary 3.4, Corollary 3.5, and Corollary 3.6 of Mohamed and Mirakhmedov (2016) [ 26 ], and we can easily obtain that For all

, we obtain that

where

is an odd power series that, for all sufficiently large N , is majorized by a power series with coefficients not depending on N , and is convergent in some discussions, and

converges uniformly in n for sufficiently small values of v , where

. For all

, we obtain that

For all

, we obtain that

For all

, we have

Based on the above cases, using Taylor’s expansion of the Equations ( A2 )–( A5 ), the following conclusions can be obtained through the order correlation of u and n : if

is true, then

where

and

.…”

Section: Appendix A1 Proof Of Remarkmentioning

confidence: 99%

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Quantile-Adaptive Sufficient Variable Screening by Controlling False Discovery

Yuan

Chen

Qiu

et al. 2023

Entropy

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Sufficient variable screening rapidly reduces dimensionality with high probability in ultra-high dimensional modeling. To rapidly screen out the null predictors, a quantile-adaptive sufficient variable screening framework is developed by controlling the false discovery. Without any specification of an actual model, we first introduce a compound testing procedure based on the conditionally imputing marginal rank correlation at different quantile levels of response to select active predictors in high dimensionality. The testing statistic can capture sufficient dependence through two paths: one is to control false discovery adaptively and the other is to control the false discovery rate by giving a prespecified threshold. It is computationally efficient and easy to implement. We establish the theoretical properties under mild conditions. Numerical studies including simulation studies and real data analysis contain supporting evidence that the proposal performs reasonably well in practical settings.

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“…According to approximation distribution for a sample sum in sampling without replacement from a finite population in Mohamed and Mirakhmedov (2016) [ 26 ], the asymptotic properties of

and

would be obtained as follows.…”

Section: Sufficient Screening Utilitymentioning

confidence: 99%

“…

has the following properties:

In addition, for all

, where

For continuous variables subject to arbitrary distribution

, let

. It is easy to find that the random variable

is a special case in Mohamed and Mirakhmedov (2016) [ 26 ], where

. According to the definition of

and

, it is obviously established that

and

.…”

Section: Appendix A1 Proof Of Remarkmentioning

confidence: 99%

“…In other words,

and

have the same distribution, which is

. If

, we have the following relationship by using Theorem 3.4, Corollary 3.4, Corollary 3.5, and Corollary 3.6 of Mohamed and Mirakhmedov (2016) [ 26 ], and we can easily obtain that For all

, we obtain that

where

is an odd power series that, for all sufficiently large N , is majorized by a power series with coefficients not depending on N , and is convergent in some discussions, and

converges uniformly in n for sufficiently small values of v , where

. For all

, we obtain that

For all

, we obtain that

For all

, we have

Based on the above cases, using Taylor’s expansion of the Equations ( A2 )–( A5 ), the following conclusions can be obtained through the order correlation of u and n : if

is true, then

where

and

.…”

Section: Appendix A1 Proof Of Remarkmentioning

confidence: 99%

See 1 more Smart Citation

Quantile-Adaptive Sufficient Variable Screening by Controlling False Discovery

Yuan

Chen

Qiu

et al. 2023

Entropy

View full text Add to dashboard Cite

show abstract

On normal approximations to symmetric hypergeometric laws

Mattner¹,

Schulz²

2017

Trans. Amer. Math. Soc.

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The Kolmogorov distances between a symmetric hypergeometric law with standard deviation σ \sigma and its usual normal approximations are computed and shown to be less than 1 / ( 8 π σ ) 1/(\sqrt {8\pi }\,\sigma ) , with the order 1 / σ 1/\sigma and the constant 1 / 8 π 1/\sqrt {8\pi } being optimal. The results of Hipp and Mattner (2007) for symmetric binomial laws are obtained as special cases. Connections to Berry-Esseen type results in more general situations concerning sums of simple random samples or Bernoulli convolutions are explained. Auxiliary results of independent interest include rather sharp normal distribution function inequalities, a simple identifiability result for hypergeometric laws, and some remarks related to Lévy’s concentration-variance inequality.

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Approximation by Normal Distribution for a Sample Sum in Sampling Without Replacement from a Finite Population

Cited by 2 publications

References 22 publications

Quantile-Adaptive Sufficient Variable Screening by Controlling False Discovery

Quantile-Adaptive Sufficient Variable Screening by Controlling False Discovery

On normal approximations to symmetric hypergeometric laws

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