# A Non-Parametric Sequential Procedure for the Generalized Partition Problem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Normal Populations Case

## 3. Non-Parametric Partition Problem

- ${n}^{1/2}\left({L}_{i}\left(n\right)-{\Delta}_{i}\right)={A}^{-1}{Z}_{i}\left(n\right)+o\left(1\right)$ a.s. as $n\to \infty $ where ${Z}_{i}\left(n\right)$ is a standardized average of independent and identically distributed random variables having a finite second moment and $0<A=A\left(F\right)<\infty $.
- For an estimator ${S}_{n}^{2}$ of A, as $n\to \infty $, we have $lim{S}_{n}^{2}={A}^{-2}$ a.s.
- The set $\left\{{\delta}^{2}N\left(\delta \right):\delta >0\right\}$ is uniformly integral.

**Theorem**

**1.**

- (i)
- $N\left({\delta}^{*}\right)\to \infty $ monotonically as ${\delta}^{*}\to 0$ a.s.
- (ii)
- $E\left(N\left({\delta}^{*}\right)\right)\to \infty $ as ${\delta}^{*}\to 0$.
- (iii)
- $lim{{\delta}^{*}}^{2}N\left({\delta}^{*}\right)=2{b}^{2}/\phantom{2{b}^{2}{A}^{2}}\phantom{\rule{0.0pt}{0ex}}{A}^{2}$ a.s.
- (iv)
- $liminfP\left(CD\right)\ge {P}^{*}$ as ${\delta}^{*}\to 0$.

**Proof.**

## 4. Monte Carlo Simulation Results

## 5. An Example

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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$\mathit{\alpha}$ | $\mathit{\delta}$ | ${\mathit{n}}^{*}$ | $\overline{\mathit{n}}$ | $\mathit{std}\left(\overline{\mathit{n}}\right)$ | $\overline{\mathit{P}}$ | $\mathit{std}\left(\overline{\mathit{P}}\right)$ |
---|---|---|---|---|---|---|

0.75 | 0.499 | 50 | 52.050 | 0.143 | 0.867 | 0.011 |

0.75 | 0.353 | 100 | 102.298 | 0.189 | 0.870 | 0.011 |

0.75 | 0.250 | 200 | 202.597 | 0.263 | 0.870 | 0.011 |

0.75 | 0.177 | 400 | 402.507 | 0.376 | 0.877 | 0.010 |

0.75 | 0.125 | 800 | 803.636 | 0.492 | 0.847 | 0.011 |

0.85 | 0.499 | 50 | 52.958 | 0.122 | 0.865 | 0.011 |

0.85 | 0.353 | 100 | 103.046 | 0.180 | 0.865 | 0.011 |

0.85 | 0.250 | 200 | 203.638 | 0.255 | 0.855 | 0.011 |

0.85 | 0.177 | 400 | 403.382 | 0.365 | 0.857 | 0.011 |

$\mathit{\alpha}$ | $\mathit{\delta}$ | ${\mathit{n}}^{*}$ | $\overline{\mathit{n}}$ | $\mathit{std}\left(\overline{\mathit{n}}\right)$ | $\overline{\mathit{P}}$ | $\mathit{std}\left(\overline{\mathit{P}}\right)$ |
---|---|---|---|---|---|---|

0.75 | 0.399 | 50 | 55.570 | 0.183 | 0.970 | 0.005 |

0.75 | 0.282 | 100 | 106.486 | 0.264 | 0.978 | 0.005 |

0.75 | 0.199 | 200 | 206.231 | 0.351 | 0.969 | 0.005 |

0.75 | 0.141 | 400 | 408.060 | 0.514 | 0.975 | 0.005 |

0.75 | 0.099 | 800 | 808.374 | 0.687 | 0.975 | 0.005 |

0.85 | 0.399 | 50 | 56.872 | 0.175 | 0.976 | 0.005 |

0.85 | 0.282 | 100 | 107.685 | 0.244 | 0.975 | 0.005 |

0.85 | 0.199 | 200 | 207.481 | 0.347 | 0.978 | 0.005 |

0.85 | 0.141 | 400 | 409.598 | 0.505 | 0.969 | 0.006 |

$\mathit{\alpha}$ | $\mathit{\delta}$ | ${\mathit{n}}^{*}$ | $\overline{\mathit{n}}$ | $\mathit{std}\left(\overline{\mathit{n}}\right)$ | $\overline{\mathit{P}}$ | $\mathit{std}\left(\overline{\mathit{P}}\right)$ |
---|---|---|---|---|---|---|

0.75 | 0.566 | 50 | 52.981 | 0.159 | 0.896 | 0.010 |

0.75 | 0.400 | 100 | 103.358 | 0.224 | 0.898 | 0.010 |

0.75 | 0.283 | 200 | 202.923 | 0.269 | 0.893 | 0.010 |

0.75 | 0.200 | 400 | 403.129 | 0.423 | 0.901 | 0.009 |

0.85 | 0.566 | 50 | 54.494 | 0.147 | 0.901 | 0.009 |

0.85 | 0.400 | 100 | 104.488 | 0.209 | 0.909 | 0.009 |

0.85 | 0.283 | 200 | 204.676 | 0.293 | 0.913 | 0.009 |

0.85 | 0.200 | 400 | 404.660 | 0.413 | 0.918 | 0.009 |

0.90 | 0.566 | 50 | 54.605 | 0.144 | 0.928 | 0.008 |

0.90 | 0.400 | 100 | 105.242 | 0.213 | 0.893 | 0.010 |

0.90 | 0.283 | 200 | 204.816 | 0.280 | 0.913 | 0.009 |

0.95 | 0.566 | 50 | 55.769 | 0.135 | 0.929 | 0.008 |

0.95 | 0.400 | 100 | 105.988 | 0.208 | 0.912 | 0.009 |

0.95 | 0.283 | 200 | 205.799 | 0.279 | 0.926 | 0.008 |

$\mathit{\alpha}$ | $\mathit{\delta}$ | ${\mathit{n}}^{*}$ | $\overline{\mathit{n}}$ | $\mathit{std}\left(\overline{\mathit{n}}\right)$ | $\overline{\mathit{P}}$ | $\mathit{std}\left(\overline{\mathit{P}}\right)$ |
---|---|---|---|---|---|---|

0.60 | 0.282 | 50 | 42.792 | 0.564 | 0.487 | 0.016 |

0.60 | 0.199 | 100 | 104.732 | 0.409 | 0.599 | 0.016 |

0.60 | 0.141 | 200 | 210.747 | 0.236 | 0.621 | 0.015 |

0.75 | 0.282 | 50 | 56.769 | 0.117 | 0.641 | 0.015 |

0.75 | 0.199 | 100 | 110.106 | 0.129 | 0.64 | 0.015 |

0.75 | 0.141 | 200 | 214.045 | 0.175 | 0.62 | 0.015 |

0.85 | 0.282 | 50 | 58.122 | 0.094 | 0.653 | 0.015 |

0.85 | 0.199 | 100 | 111.698 | 0.114 | 0.610 | 0.015 |

0.85 | 0.141 | 200 | 216.071 | 0.146 | 0.604 | 0.015 |

0.99 | 0.282 | 50 | 63.737 | 0.070 | 0.719 | 0.014 |

0.99 | 0.199 | 100 | 118.374 | 0.089 | 0.648 | 0.015 |

0.99 | 0.141 | 200 | 224.796 | 0.119 | 0.654 | 0.015 |

**Table 5.**Simulation results for mixture of two normal distributions: $X=0.35N\left({x}_{1};0,1\right)+0.65N\left({x}_{2};0,2\right)$.

$\mathit{\alpha}$ | $\mathit{\delta}$ | ${\mathit{n}}^{*}$ | $\overline{\mathit{n}}$ | $\mathit{std}\left(\overline{\mathit{n}}\right)$ | $\overline{\mathit{P}}$ | $\mathit{std}\left(\overline{\mathit{P}}\right)$ |
---|---|---|---|---|---|---|

0.75 | 0.804 | 50 | 52.859 | 0.162 | 0.903 | 0.009 |

0.75 | 0.569 | 100 | 103.243 | 0.213 | 0.905 | 0.009 |

0.75 | 0.402 | 200 | 203.962 | 0.303 | 0.911 | 0.009 |

0.85 | 0.804 | 50 | 53.685 | 0.140 | 0.916 | 0.007 |

0.85 | 0.569 | 100 | 104.216 | 0.216 | 0.926 | 0.008 |

0.85 | 0.402 | 200 | 204.205 | 0.285 | 0.912 | 0.009 |

0.90 | 0.804 | 50 | 54.817 | 0.143 | 0.909 | 0.009 |

0.90 | 0.569 | 100 | 104.823 | 0.203 | 0.902 | 0.009 |

0.90 | 0.402 | 200 | 204.928 | 0.290 | 0.900 | 0.009 |

0.95 | 0.804 | 50 | 55.676 | 0.142 | 0.928 | 0.008 |

0.95 | 0.569 | 100 | 105.801 | 0.202 | 0.918 | 0.009 |

0.95 | 0.402 | 200 | 206.601 | 0.271 | 0.913 | 0.009 |

**Table 6.**Simulation results for mixture of two normal distributions: $X=0.8N\left({x}_{1};0,1\right)+0.2N\left({x}_{2};0,5\right)$.

$\mathit{\alpha}$ | $\mathit{\delta}$ | ${\mathit{n}}^{*}$ | $\overline{\mathit{n}}$ | $\mathit{std}\left(\overline{\mathit{n}}\right)$ | $\overline{\mathit{P}}$ | $\mathit{std}\left(\overline{\mathit{P}}\right)$ |
---|---|---|---|---|---|---|

0.75 | 0.678 | 50 | 54.424 | 0.187 | 0.952 | 0.007 |

0.75 | 0.480 | 100 | 104.593 | 0.259 | 0.935 | 0.008 |

0.75 | 0.339 | 200 | 205.031 | 0.351 | 0.932 | 0.008 |

0.85 | 0.678 | 50 | 55.826 | 0.169 | 0.934 | 0.008 |

0.85 | 0.450 | 100 | 106.334 | 0.254 | 0.926 | 0.008 |

0.85 | 0.339 | 200 | 206.534 | 0.332 | 0.933 | 0.008 |

0.85 | 0.240 | 400 | 406.497 | 0.486 | 0.942 | 0.007 |

0.90 | 0.678 | 50 | 56.762 | 0.177 | 0.955 | 0.007 |

0.90 | 0.480 | 100 | 106.746 | 0.244 | 0.924 | 0.008 |

0.90 | 0.339 | 200 | 207.888 | 0.351 | 0.935 | 0.008 |

0.95 | 0.678 | 50 | 58.671 | 0.173 | 0.959 | 0.006 |

0.95 | 0.480 | 100 | 108.742 | 0.235 | 0.947 | 0.007 |

0.95 | 0.339 | 200 | 208.019 | 0.332 | 0.931 | 0.008 |

Procedure | Sample Size |
---|---|

Two-stage | 71 |

Purely Sequential | 66 |

Non-Parametric Sequential | 42 $\left(\alpha =0.75\right)$ |

52 $\left(\alpha =0.80\right)$ | |

53 $\left(\alpha =0.85\right)$ | |

60 $\left(\alpha =0.90\right)$ | |

67 $\left(\alpha =0.95\right)$ |

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**MDPI and ACS Style**

Solanky, T.K.S.; Zhou, J.
A Non-Parametric Sequential Procedure for the Generalized Partition Problem. *Mathematics* **2024**, *12*, 591.
https://doi.org/10.3390/math12040591

**AMA Style**

Solanky TKS, Zhou J.
A Non-Parametric Sequential Procedure for the Generalized Partition Problem. *Mathematics*. 2024; 12(4):591.
https://doi.org/10.3390/math12040591

**Chicago/Turabian Style**

Solanky, Tumulesh K. S., and Jie Zhou.
2024. "A Non-Parametric Sequential Procedure for the Generalized Partition Problem" *Mathematics* 12, no. 4: 591.
https://doi.org/10.3390/math12040591