#
Non-Asymptotic Confidence Sets for Circular Means^{ †}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

**C**is a real vector space, so the Euclidean sample mean ${\overline{Z}}_{n}=\frac{1}{n}{\sum}_{k=1}^{n}{Z}_{k}\in \mathbf{C}$ is well-defined. However, unless all ${Z}_{k}$ take identical values, it will (by the strict convexity of the closed unit disc) lie inside the circle, i.e., its modulus $|{\overline{Z}}_{n}|$ will be less than 1. Though ${\overline{Z}}_{n}$ cannot be taken as a mean on the circle, if ${\overline{Z}}_{n}\ne 0$, one might say that it specifies a direction; this leads to the idea of calling ${\overline{Z}}_{n}/|{\overline{Z}}_{n}|$ the circular sample mean of the data.

**C**; therefore, μ is also called the set of extrinsic population means. If we measured distances intrinsically along the circle, i.e., using arc-length instead of chordal distance, we would obtain what is called the set of intrinsic population means. We will not consider the latter in the following, see e.g., [7] for a comparison and [8,9] for generalizations of these concepts.

**C**with ${\mathbf{R}}^{2}$. Now, assume $\mathbf{E}Z\ne 0$ so μ is unique. Then, the orthogonal projection is differentiable in a neighbourhood of $\mathbf{E}Z$, so the δ-method (see e.g., [1] (p. 111) or [4] (Lemma 3.1)) can be applied and one easily obtains

## 2. Construction Using Hoeffding’s Inequality

**Proposition**

**1.**

- (i)
- ${C}_{H}$ is a $(1-\alpha )$-confidence set for the circular population mean set. In particular, if $\mathbf{E}Z=0$, i.e., the circular population mean set equals ${\mathsf{S}}^{1}$, then $|{\overline{Z}}_{n}|>\sqrt{2}{s}_{0}$ with probability at most $\alpha ,$ so indeed ${C}_{H}={\mathsf{S}}^{1}$ with probability at least $1-\alpha .$
- (ii)
- ${s}_{0}$ and ${s}_{p}$ are of order ${n}^{-{\textstyle \frac{1}{2}}}$.
- (iii)
- If $\mathbf{E}Z\ne 0,$ then $\sqrt{n}{\delta}_{H}\to 0$ in probability and the probability of obtaining the trivial confidence set, i.e., $\mathbf{P}({C}_{H}={\mathsf{S}}^{1})=\mathbf{P}(|{\overline{Z}}_{n}|\le \sqrt{2}{s}_{0})$, goes to 0 exponentially fast.

**Proof.**

## 3. Estimating the Variance

Algorithm 1: Algorithm for computation of ${C}_{V}$. |

**Proposition**

**2.**

- (i)
- The set ${C}_{V}$ resulting from Algorithm 1 is a $(1-\alpha )$-confidence set for the circular population mean set. In particular, if $\mathbf{E}Z=0$, i.e., the circular population mean set equals ${\mathsf{S}}^{1}$, then $|{\overline{Z}}_{n}|>\sqrt{2}{s}_{0}$ with probability at most $\alpha ,$ so indeed ${C}_{V}={\mathsf{S}}^{1}$ with probability of at least $1-\alpha .$
- (ii)
- ${s}_{V}$ is of order ${n}^{-{\textstyle \frac{1}{2}}}$.
- (iii)
- If $\mathbf{E}Z\ne 0,$ i.e., if the circular population mean is unique, then $\sqrt{n}{\delta}_{V}\to 0$ in probability, and the probability of obtaining a trivial confidence set, i.e., $\mathbf{P}({C}_{H}={\mathsf{S}}^{1})=\mathbf{P}(|{\overline{Z}}_{n}|\le \sqrt{2}{s}_{0})$, goes to 0 exponentially fast.
- (iv)
- If $\mathbf{E}Z\ne 0$, then$$\underset{n\to \infty}{\mathrm{lim\; sup}}\frac{{\delta}_{V}}{{\delta}_{A}}\le {\textstyle \frac{\sqrt{-2\mathrm{ln}{\textstyle \frac{\alpha}{4}}}}{{q}_{1-{\textstyle \frac{\alpha}{2}}}}}\phantom{\rule{1.em}{0ex}}a.s.$$

**Proof.**

## 4. Simulation and Application to Real Data

`R`(version 2.15.3) [14] .

#### 4.1. Simulation 1: Two Points of Equal Mass at $\pm {10}^{\circ}$

#### 4.2. Simulation 2: Three Points Placed Asymmetrically

#### 4.3. Real Data: Movements of Ants

`Ants_radians`within the

`R`package

`CircNNTSR`[16].

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proofs of Monotonicity

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

## References

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**Figure 1.**The construction for the test of the hypothesis $\mu ={\mathsf{S}}^{1},$ or equivalently $\mathbf{E}Z=0.$

**Figure 2.**The construction for the test of the hypothesis $\mathbf{E}Z=\lambda \zeta $ with $\lambda >0$.

**Figure 3.**The critical ${\overline{Z}}_{n}$ regarding the rejection of ζ. ${\delta}_{H}$ bounds the angle between ${\widehat{\mu}}_{n}$ and any accepted $\zeta .$

**Figure 6.**Ant data ( ) placed at increasing radii to visually resolve ties; in addition, the circular mean direction ( ) as well as confidence sets ${C}_{H}$ ( ), ${C}_{V}$ ( ), and ${C}_{A}$ ( ) are shown.

**Table 1.**Results for simulation 1 (two points of equal mass at $\pm {10}^{\circ}$) based on 10,000 repetitions with $n=400$ observations each: average observed ${\delta}_{H}$, ${\delta}_{V}$, and ${\delta}_{A}$ (with corresponding standard deviation), as well as frequency (with corresponding standard error) with which $\mu =1$ was covered by ${C}_{H}$, ${C}_{V}$, and ${C}_{A}$, respectively; the nominal coverage probability was $1-\alpha =95\%$.

Confidence Set | Mean δ (±s.d.) | Coverage Frequency (±s.e.) |
---|---|---|

${C}_{H}$ | ${8.2}^{\circ}$ ($\pm {0.0005}^{\circ}$) | $100.0\%$ ($\pm 0.0\%$) |

${C}_{V}$ | ${2.4}^{\circ}$ ($\pm {0.0025}^{\circ}$) | $100.0\%$ ($\pm 0.0\%$) |

${C}_{A}$ | ${1.0}^{\circ}$ ($\pm {0.0019}^{\circ}$) | $94.8\%$ ($\pm 0.2\%$) |

**Table 2.**Results for simulation 2 (three points placed asymmetrically) based on 10,000 repetitions with $n=100$ observations each: average observed ${\delta}_{H}$, ${\delta}_{V}$, and ${\delta}_{A}$ (with corresponding standard deviation), as well as frequency (with corresponding standard error) with which $\mu =1$ was covered by ${C}_{H}$, ${C}_{V}$, and ${C}_{A}$, respectively; the nominal coverage probability was $1-\alpha =90\%$.

Confidence Set | Mean δ (±s.d.) | Coverage Frequency (±s.e.) |
---|---|---|

${C}_{H}$ | ${16.5}^{\circ}$ ($\pm {0.85}^{\circ}$) | $100.0\%$ ($\pm 0.0\%$) |

${C}_{V}$ | ${5.0}^{\circ}$ ($\pm {0.38}^{\circ}$) | $100.0\%$ ($\pm 0.0\%$) |

${C}_{A}$ | ${0.4}^{\circ}$ ($\pm {0.28}^{\circ}$) | $62.8\%$ ($\pm 0.5\%$) |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hotz, T.; Kelma, F.; Wieditz, J.
Non-Asymptotic Confidence Sets for Circular Means. *Entropy* **2016**, *18*, 375.
https://doi.org/10.3390/e18100375

**AMA Style**

Hotz T, Kelma F, Wieditz J.
Non-Asymptotic Confidence Sets for Circular Means. *Entropy*. 2016; 18(10):375.
https://doi.org/10.3390/e18100375

**Chicago/Turabian Style**

Hotz, Thomas, Florian Kelma, and Johannes Wieditz.
2016. "Non-Asymptotic Confidence Sets for Circular Means" *Entropy* 18, no. 10: 375.
https://doi.org/10.3390/e18100375