Automatic data-based bin width selection for rose diagram

Abstract

A rose diagram is a representation that circularly organizes data with the bin width as the central angle. This diagram is widely used to display and summarize circular data. Some studies have proposed the selector of bin width based on data. However, only a few papers have discussed the property of these selectors from a statistical perspective. Thus, this study aims to provide a data-based bin width selector for rose diagrams using a statistical approach. We consider that the radius of the rose diagram is a nonparametric estimator of the square root of two times the circular density. We derive the mean integrated square error of the rose diagram and its optimal bin width and propose two new selectors: normal reference rule and biased cross-validation. We show that biased cross-validation converges to its optimizer. Additionally, we propose a polygon rose diagram to enhance the rose diagram.

Appendices

Appendix A

Proof of Theorem 1

Recalling that $W_{k}=\hat{f}(\theta ; h_{k}) - \textrm{E}[\hat{f}(\theta ; h_{k}) ]$ for $\theta$ in $B_{k}$, the Taylor expansion of the rose diagram estimator $r_{f}(\theta ;h)$ around $\textrm{E}[\hat{f}(\theta ; h_{k})]$ is:

$$\begin{aligned} \hat{r}_{f}(\theta ; h) = \sqrt{2\textrm{E}[\hat{f}(\theta ; h_{k})]} + \sum ^{\infty }_{t=1}(-1)^{t-1}\frac{\sqrt{2} (2t-3)!!}{2^{t}t!}\textrm{E}[\hat{f}(\theta ; h_{k})]^{1/2-t} W_{k}^{t}, \end{aligned}$$

(17)

for $\theta \in B_{k}$, where $(2t-1)!!:=1\cdot 3\cdot 5\cdots (2t - 3)(2t-1)$ and (-1)!!:=1. Combining Lemma 1 and (17), the expectation of the rose diagram estimator approximates to:

$$\begin{aligned} \textrm{E}[\hat{r}_{f}(\theta ; h)] =\sqrt{2p_{k}/h_{k}} +O((nh)^{-1}),\ \theta \in B_{k}. \end{aligned}$$

(18)

Note that $B_{k}=[t_{k}, t_{k}+h_{k})$. From the Taylor expansion of $f(\cdot )$ around $\theta \in B_{k}$, we obtain

$$\begin{aligned} p_{k}&= \int ^{t_{k}+h_{k}}_{t_{k}}f(y)dy \nonumber \\&= \int ^{t_{k}+h_{k}}_{t_{k}}[f(\theta ) + f^{\prime }(\theta )(y-\theta ) + O((y-\theta )^{2})]dy\nonumber \\&= (t_{k} + h_{k})f(\theta ) + f^{\prime }(\theta )(t_{k} + h_{k}-\theta )^{2}/2 + O((t_{k}+h_{k}-\theta )^{3})\nonumber \\&\,\quad -[ t_{k}f(\theta ) + f^{\prime }(\theta )(t_{k}-\theta )^{2}/2 + O((t_{k}-\theta )^{3})]\nonumber \\&=h_{k}f(\theta ) +f^{\prime }(\theta )[h_{k}^{2}/2 +(t_{k} -\theta )h_{k}] +O(h^{3}). \end{aligned}$$

(19)

Approximating $p_{k}$ in the RHS in Eq. (18) by Eq. (19) leads to:

$$\begin{aligned} \textrm{E}[\hat{r}_{f}(\theta ; h)]&=\sqrt{2}\sqrt{f(\theta ) +f^{\prime }(\theta )(h_{k}/2 +t_{k} -\theta )+O(h^{2})} +O((nh^{-1}))\nonumber \\&= \sqrt{2f(\theta ) }+(2f(\theta ))^{-1/2}f^{\prime }(\theta )(h_{k}/2 + t_{k} -\theta ) +O(h^{2} + (nh)^{-1}) \nonumber \\&=r_{f}(\theta ) + r_{f}^{\prime }(\theta )(h_{k}/2 + t_{k} -\theta ) +O(h^{2} + (nh)^{-1}), \end{aligned}$$

(20)

for $\theta \in B_{k}$. From (20), we obtain:

$$\begin{aligned} \textrm{Bias}[\hat{r}_{f}(\theta ; h)] = r_{f}^{\prime }(\theta )(h_{k}/2 + t_{k} -\theta ) +O(h^{2} + (nh)^{-1}), \end{aligned}$$

(21)

for $\theta \in B_{k}$. By applying the mean theorem to the first term in the RHS in Eq. (21), we can obtain $\eta _{k} \in (t_{k}, t_{k}+h_{k} )$, which satisfies $\int _{B_{k}} r_{f}^{\prime }(\theta )^{2}(h_{k}/2 + t_{k} -\theta )^{2}d\theta = r_{f}^{\prime }(\eta _{k})^{2}\int _{B_{k}}(h_{k}/2 + t_{k} -\theta )^{2}d\theta$. Thus, we obtain:

$$\begin{aligned} \int _{B_{k}} \textrm{Bias}[\hat{r}_{f}(\theta ; h)] ^{2}d\theta&=r_{f}^{\prime }(\eta _{k})^{2} \int _{B_{k}}(\theta - t_{k} - h_{k}/2)^{2}d\theta [1+o(1)]\nonumber \\&= r_{f}^{\prime }(\eta _{k})^{2}(h_{k}^{3}/12)[1+o(1)]. \end{aligned}$$

(22)

Let $\textrm{ISB}: = \int ^{2\pi }_{0} \textrm{Bias}[\hat{r}_{f}(\theta ; h)] ^{2}d\theta$. We find that $\textrm{ISB}$ is the sum of the integrations in $B_{k}$ in (22) for $k=1,2,\dots , m$ with $m: =m_{n}$, such as $m_{n}\rightarrow \infty$ as $n \rightarrow \infty$; that is,

$$\begin{aligned} \textrm{ISB}&= \frac{h^{2}}{12} \sum _{k=0}^{m-1}r_{f}^{\prime }(\eta _{k})^{2}h[1+o(1)] + \frac{c^{3}h^{3}}{12}r_{f}^{\prime }(\eta _{m})^{2}[1+o(1)] \nonumber \\&=\frac{h^{2}}{12} \left[ \int ^{2\pi }_{0} r_{f}^{\prime }(\theta )^{2}d \theta - \int ^{2\pi }_{2\pi - ch} r_{f}^{\prime }(\theta )^{2}d \theta + c^{3}hr_{f}^{\prime }(\eta _{m})^{2}\right] [1+o(1)]\nonumber \\&= \frac{h^{2}}{12} \int ^{2\pi }_{0} r_{f}^{\prime }(\theta )^{2}d \theta [1+o(1)] , \end{aligned}$$

(23)

where we use the Riemann integral approximation in the second RHS and $\int ^{2\pi }_{2\pi - ch} r_{f}^{\prime }(\theta )^{2}d \theta - chr_{f}^{\prime }(\eta _{m})^{2} = o(1)$ in the last RHS. From Lemma 1 and (17), we obtain:

$$\begin{aligned} \textrm{Var}[\hat{r}_{f}(\theta ;h)]&= \textrm{Var}[ 2^{-1/2} \textrm{E}[\hat{f}(\theta ;h_{k})]^{-1/2}W_{k} +O_{p}(W_{k}^{2})]\nonumber \\&= (2 \textrm{E}[\hat{f}(\theta ;h_{k})] )^{-1}\textrm{E}[W_{k}^{2}] + O(\textrm{E}[W_{k}^{4}])\nonumber \\&= (2f(\xi _{k}))^{-1}[f(\xi _{k})/(nh_{k})-f(\xi _{k})^{2}/n]+O((nh)^{-2})\nonumber \\&= 1/(2nh_{k}) - f(\xi _{k})/(2n) +O((nh)^{-2}). \end{aligned}$$

(24)

Let $\textrm{IV}:= \int ^{2\pi }_{0}\textrm{Var}[\hat{r}_{f}(\theta ;h)]d\theta$. Integrating (24) provides:

$$\begin{aligned} \textrm{IV}&= \int ^{2\pi }_{0} \frac{1}{2nh}d\theta - \int ^{2\pi }_{2\pi - ch} \frac{1}{2nh}d\theta + \int ^{2\pi }_{2\pi - ch} \frac{1}{2cnh}d\theta +O(n^{-1} + (nh)^{-2})\nonumber \\&= \pi /(nh) + o((nh)^{-1}). \end{aligned}$$

(25)

Combining (23) and (25) leads to:

$$\begin{aligned} \textrm{MISE}[\hat{r}_{f}(\theta ;h)] = h^{2}R(r_{f}^{\prime })/12 +\pi /(nh) +o(h^{2} +(nh)^{-1}). \end{aligned}$$

(26)

Theorem 1 completes the proof from combining (21), (24), and (26). $\square$

Appendix B

We describe the expectation, variance, and covariance of the histogram estimator. We base these proofs on the results in Scott (1979, 1985). We use these to derive the properties of the rose diagram estimator.

The Taylor expansion of f yields:

$$\begin{aligned} p_{k}&= \int ^{t_{k} + h_{k}}_{t_{k}} f(\theta )d\theta \nonumber \\&= \int ^{t_{k} + h_{k} }_{t_{k}} \left[ f(t_{k}) + f^{\prime }(t_{k})(\theta - t_{k}) + \frac{1}{2} f^{\prime \prime }(t_{k})(\theta -t_{k})^{2} + O((\theta - t_{k})^{3} ) \right] d\theta \nonumber \\&= f(t_{k})h_{k} + \frac{1}{2}f^{\prime }(t_{k})h_{k}^{2} + \frac{1}{6} f^{\prime \prime }(t_{k})h_{k}^{3} + O(h^{4}). \end{aligned}$$

(27)

Note that $p_{k-1} = \int ^{t_{k} }_{t_{k}-h_{k-1}} f(\theta )d\theta$ and $p_{0} = \int ^{t_{m+1} }_{t_{m+1}-h_{m}}f(\theta )d\theta$. The Taylor expansion of f also yields:

$$\begin{aligned} p_{k-1}&= \left\{ \begin{array}{ll} f(t_{k})h_{k-1} - \frac{1}{2}f^{\prime }(t_{k})h_{k-1}^{2} + \frac{1}{6} f^{\prime \prime }(t_{k})h_{k-1}^{3} + O(h^{4})&{}\text {if}\quad k \in \{2, 3, \dots , m\},\\ f(t_{1})h_{m} - \frac{1}{2}f^{\prime }(t_{1})h_{m}^{2} + \frac{1}{6} f^{\prime \prime }(t_{1})h_{m}^{3} + O(h^{4}),&{}\text {if}\quad k = 1, \end{array}\right. \end{aligned}$$

(28)

where use $f(t_{m +1}) = f(t_{1})$. From (27), we obtain:

$$\begin{aligned} \textrm{E}[\hat{f}_{k}]&= f(t_{k}) + \frac{1}{2}f^{\prime }(t_{k})h_{k} + \frac{1}{6} f^{\prime \prime }(t_{k})h_{k}^{2} + O(h^{3}), \end{aligned}$$

(29)

and

$$\begin{aligned} \textrm{E}[W_{k}^{2}]&= \left[ f(t_{k})h_{k} + \frac{f^{\prime }(t_{k})h_{k}^{2}}{2} + O(h^{3})\right] \frac{1 - f(t_{k})h_{k}+O(h^{2})}{nh_{k}^{2}} \nonumber \\&=\frac{f(t_{k})}{nh_{k}} + \frac{f^{\prime }(t_{k})}{2n} - \frac{f(t_{k})^{2}}{ n} +O(n^{-1}h). \end{aligned}$$

(30)

From (28), we obtain:

$$\begin{aligned} \textrm{E}[\hat{f}_{k-1}]&= f(t_{k}) - \frac{1}{2}f^{\prime }(t_{k})h_{k-1} + \frac{1}{6} f^{\prime \prime }(t_{k})h_{k-1}^{2} + O(h^{3}), \end{aligned}$$

(31)

and

$$\begin{aligned} \textrm{E}[W_{k-1}^{2}]&= \left\{ \begin{array}{ll} \frac{f(t_{k})}{nh_{k-1}} - \frac{ f^{\prime }(t_{k})}{2n} + \frac{ f(t_{k})^{2}}{n} +O(n^{-1}h)&{} \text {if}\quad k \in \{2, 3, \dots , m\},\\ \frac{f(t_{1})}{nh_{m}} - \frac{ f^{\prime }(t_{1})}{2n} + \frac{ f(t_{1})^{2}}{n} +O(n^{-1}h)&{} \text {if}\quad k = 1. \end{array}\right. \end{aligned}$$

(32)

We know that polynomial distribution’s covariance is $\textrm{Cov}[v_{k}, v_{k-1}] = -np_{k}p_{k-1}$. Thus, combining (27) and (28) yields

$$\begin{aligned} \textrm{E}[W_{k}W_{k-1} ]&=-\frac{np_{k}p_{k-1}}{n^{2}h_{k}h_{k-1}}\nonumber \\&= -\frac{[f(t_{k})h_{k} + O(h^{2})][f(t_{k})h_{k-1} + O(h^{2})]}{nh_{k}h_{k-1}}\nonumber \\&= - \frac{f(t_{k})^{2}}{n} +O(n^{-1}h), \end{aligned}$$

(33)

where use $h_{0} = h_{m}$.

Appendix C

Proof of Theorem 2

First, we derive the expectation $\hat{R}_{1}$. Let the histogram estimator $\hat{f}(\cdot )$ in bin $B_{k}$ be $\hat{f}_{k}$. We obtain:

$$\begin{aligned} \textrm{E}[\hat{R}_{1}]&= 2\sum _{k=1}^{m}\tilde{h}_{k}^{-1}( \textrm{E}[\hat{f}_{k}] + \textrm{E}[\hat{f}_{k-1}] - \textrm{E}[\hat{r}_{k} \hat{r}_{k-1}] ). \end{aligned}$$

(34)

We obtain the following Taylor expansion of $\hat{r}_{k}\hat{r}_{k-1}$:

$$\begin{aligned} \hat{r}_{k} \hat{r}_{k-1}&= 2(\textrm{E}[\hat{f}_{k}]\textrm{E}[\hat{f}_{k-1}])^{1/2} + \textrm{E}[\hat{f}_{k}]^{-1/2}\textrm{E}[\hat{f}_{k-1}]^{1/2}W_{k} + \textrm{E}[\hat{f}_{k}]^{1/2}\textrm{E}[\hat{f}_{k-1}]^{-1/2}W_{k-1} \nonumber \\&\quad -\frac{1}{4} \textrm{E}[\hat{f}_{k}]^{-3/2}\textrm{E}[\hat{f}_{k-1}]^{1/2}W_{k}^{2} - \frac{1}{4} \textrm{E}[\hat{f}_{k}]^{1/2}\textrm{E}[\hat{f}_{k-1}]^{-3/2}W_{k-1}^{2}\nonumber \\&\quad + \frac{1}{2}\textrm{E}[\hat{f}_{k}]^{-1/2}\textrm{E}[\hat{f}_{k-1}]^{-1/2}W_{k}W_{k-1} + O_{p}((W_{k} + W_{k-1})^{3} + (W_{k} + W_{k-1})^{4}). \end{aligned}$$

(35)

Combining Lemma 1 and (35) yields $\textrm{E}[\hat{r}_{k} \hat{r}_{k-1} ]$, which is equal to

$$\begin{aligned} \textrm{E}[\hat{r}_{k} \hat{r}_{k-1} ]&= 2(\textrm{E}[\hat{f}_{k}]\textrm{E}[\hat{f}_{k-1}])^{1/2} -\frac{1}{4} \textrm{E}[\hat{f}_{k}]^{-3/2}\textrm{E}[\hat{f}_{k-1}]^{1/2}\textrm{E}[W_{k}^{2}]\nonumber \\&\quad - \frac{1}{4} \textrm{E}[\hat{f}_{k}]^{1/2}\textrm{E}[\hat{f}_{k-1}]^{-3/2}\textrm{E}[W_{k-1}^{2}] + \frac{1}{2}\textrm{E}[\hat{f}_{k}]^{-1/2}\textrm{E}[\hat{f}_{k-1}]^{-1/2}\textrm{E}[W_{k}W_{k-1}]\nonumber \\&\quad + O((nh)^{-2}). \end{aligned}$$

(36)

Combining (34) and (36) provides:

$$\begin{aligned} \textrm{E}[\hat{R}_{1}]&= 2\sum ^{m}_{k=1}\tilde{h}_{k}^{-1}\Bigl [(\textrm{E}[\hat{f}_{k}]^{1/2} - \textrm{E}[\hat{f}_{k-1}]^{1/2})^{2} + \frac{1}{4} \textrm{E}[\hat{f}_{k}]^{-3/2}\textrm{E}[\hat{f}_{k-1}]^{1/2}\textrm{E}[W_{k}^{2}]\nonumber \\&\quad + \frac{1}{4} \textrm{E}[\hat{f}_{k}]^{1/2}\textrm{E}[\hat{f}_{k-1}]^{-3/2}\textrm{E}[W_{k-1}^{2}] - \frac{1}{2}\textrm{E}[\hat{f}_{k}]^{-1/2}\textrm{E}[\hat{f}_{k-1}]^{-1/2}\textrm{E}[W_{k}W_{k-1}]\nonumber \\&\quad + O((nh)^{-2})\Bigr ]. \end{aligned}$$

(37)

We now derive each term in the RHS in Eq. (37). Combining (29) and (31) (in Appendix B) leads to:

$$\begin{aligned}&(\textrm{E}[\hat{f}_{k}]^{1/2} - \textrm{E}[\hat{f}_{k-1}]^{1/2})^{2} \nonumber \\&\quad =\Bigl \{ \Bigl [ f(t_{k}) + \frac{1}{2}f^{\prime }(t_{k})h_{k} + O(h^{2})\Bigr ]^{1/2} - \Bigl [ f(t_{k}) - \frac{1}{2}f^{\prime }(t_{k})h_{k-1}+ O(h^{2})\Bigr ]^{1/2} \Bigr \}^{2}\nonumber \\&\quad = \left\{ f(t_{k})^{1/2} + \frac{1}{4}f(t_{k})^{-1/2}f^{\prime }(t_{k})h_{k} - \left[ f(t_{k})^{1/2} - \frac{1}{4}f(t_{k})^{-1/2}f^{\prime }(t_{k})h_{k-1}\right] + O(h^{2})\right\} ^{2}\nonumber \\&\quad = \left[ \frac{1}{2}f(t_{k})^{-1/2}f^{\prime }(t_{k})\tilde{h}_{k} +O(h^{2}) \right] ^{2} \nonumber \\&\quad = \frac{1}{4}f(t_{k})^{-1}f^{\prime }(t_{k})^{2}\tilde{h}_{k}^{2} + O(h^{3})\nonumber \\&\quad = \frac{1}{2} r_{f}^{\prime }(t_{k})^{2}\tilde{h}_{k}^{2} + O(h^{3}). \end{aligned}$$

(38)

Combining (29), (30), and (31) provides:

$$\begin{aligned} \textrm{E}[\hat{f}_{k}]^{-3/2}\textrm{E}[\hat{f}_{k-1}]^{1/2}\textrm{E}[W_{k}^{2}]&= \frac{[f(t_{k}) + O(h)]^{1/2}}{[f(t_{k}) + O(h)]^{3/2}}\left[ \frac{f(t_{k})}{nh_{k}} + O(n^{-1})\right] \nonumber \\&= \frac{1}{nh_{k}} + O(n^{-1}). \end{aligned}$$

(39)

Combining (29), (31), and (32) provides:

$$\begin{aligned} \textrm{E}[\hat{f}_{k}]^{1/2}\textrm{E}[\hat{f}_{k-1}]^{-3/2}\textrm{E}[W_{k-1}^{2}]&= \frac{1}{nh_{k-1}} + O(n^{-1}). \end{aligned}$$

(40)

Combining (29), (31), and (33) provides:

$$\begin{aligned} \textrm{E}[\hat{f}_{k}]^{-1/2}\textrm{E}[\hat{f}_{k-1}]^{-1/2}\textrm{E}[W_{k}W_{k-1}]&= -\frac{f(t_{k})}{n} + O(n^{-1}h). \end{aligned}$$

(41)

By combining (37), (38), (39), (40), and (41), we obtain:

$$\begin{aligned} \textrm{E}[\hat{R}_{1}]&= 2\sum _{k=1}^{m}\tilde{h}_{k}^{-1}\left[ \frac{1}{2}r_{f}^{\prime }(t_{k})^{2}\tilde{h}_{k}^{2} +\frac{1}{4nh_{k}} + \frac{1}{4nh_{k-1}}+ O(h^{3} + n^{-1} + (nh)^{-2})\right] \nonumber \\&= \left[ \sum _{k=1}^{m}r_{f}^{\prime }(t_{k})^{2}\tilde{h}_{k} + \sum _{k=1}^{m}\frac{1}{2nh_{k}\tilde{h}_{k}} + \sum _{k=1}^{m}\frac{1}{2nh_{k-1}\tilde{h}_{k}} \right] [1 + o(1)] \nonumber \\&= \left[ \int ^{2\pi }_{0}r_{f}^{\prime }(\theta )^{2}d\theta + \sum _{k=2}^{m-1} \frac{1}{nh^{2}} + \frac{2}{n(h^{2} + ch^{2})} + \frac{2}{n(ch^{2} + c^{2}h^{2})}\right] [1 + o(1)] \nonumber \\&= R(r_{f}^{\prime })[1+o(1)] + \left[ \frac{2\pi - (1 + c)h}{nh^{3}} + \frac{2}{n(h^{2} + ch^{2})} + \frac{2}{n(ch^{2} + c^{2}h^{2})} \right] [1+o(1)]\nonumber \\&= R(r_{f}^{\prime }) + \frac{2\pi }{nh^{3}} + o(1 + n^{-1}h^{-3}). \end{aligned}$$

(42)

From (42), we obtain:

$$\begin{aligned} \textrm{E}[\textrm{BCV}(h)]&=\frac{h^{2}}{12}\ \left\{ \textrm{E}[\hat{R}_{1}]-\frac{2\pi }{nh^{3}} \right\} + \frac{\pi }{nh} \nonumber \\&= \frac{h^{2}R(r^{\prime }_{f}) }{12} + \frac{\pi }{nh} + o(h^{2} + (nh)^{-1})\nonumber \\&= \textrm{AMISE}[\hat{r}_{f}(\theta ;h)] + o(h^{2} + (nh)^{-1}). \end{aligned}$$

(43)

We reduce $\hat{R}_{1}$ to the simpler form. Recalling that $B_{0} =B_{m}$ implies $v_{0} = v_{m}$ and $h_{0} = h_{m}$, we obtain:

$$\begin{aligned} \hat{R}_{1}&= \sum _{k=1}^{m}(2\hat{f}_{k} + 2\hat{f}_{k-1} -2\hat{r}_{k}\hat{r}_{k-1})\tilde{h}_{k}^{-1}\nonumber \\&= 2\left[ \sum _{k=2}^{m-1}\frac{v_{k}}{nh^{2}} + \sum _{k=2}^{m-1}\frac{v_{k-1}}{nh^{2}} + \frac{v_{1}}{nh\tilde{h}_{1}} + \frac{v_{m}}{nh_{m}\tilde{h}_{m}} + \frac{v_{0}}{nh_{0}\tilde{h}_{1}} - \frac{v_{m-1}}{nh\tilde{h}_{m}} - \sum _{k=1}^{m}\tilde{h}_{k}^{-1}\hat{r}_{k}\hat{r}_{k-1} \right] \nonumber \\&= 2 \Bigl \{2 \sum _{k=1}^{m}\frac{v_{k}}{nh^{2}} + \left[ \frac{2}{(1+c)h}-\frac{1}{h}\right] (\hat{f}_{1}+\hat{f}_{m-1}) + 2\left[ \frac{2}{c(1+c)h}-\frac{1}{h}\right] \hat{f}_{m}\nonumber \\&\quad - \sum _{k=1}^{m}\tilde{h}_{k}^{-1}\hat{r}_{k}\hat{r}_{k-1} \Bigr \}\nonumber \\&= \frac{4}{h^{2}} + \frac{2a(c)}{h}(\hat{f}_{1}+\hat{f}_{m-1}) + \frac{4b(c)}{h}\hat{f}_{m} - 2 \sum _{k=1}^{m}\tilde{h}_{k}^{-1}\hat{r}_{k}\hat{r}_{k-1}, \end{aligned}$$

(44)

where $a(c): = (1 - c) / (1+c)$ and $b(c): = [2 -(c+c^{2})] /(c+c^{2})$. Next, we provide the variance of $\textrm{BCV}(h)$. By combining (11) and (44), we reduce $\textrm{BCV}(h)$ to

$$\begin{aligned} \textrm{BCV}(h)&= \frac{1}{3} + \frac{5\pi }{6nh} +\frac{ha(c)}{6}(\hat{f}_{1}+\hat{f}_{m-1}) + \frac{hb(c)}{3}\hat{f}_{m} - \frac{h^{2}}{6}\sum _{k=1}^{m}\tilde{h}_{k}^{-1}\hat{r}_{k}\hat{r}_{k-1}. \end{aligned}$$

(45)

Let $Y_{k, k-1}:= \hat{r}_{k}\hat{r}_{k-1} -\textrm{E}[\hat{r}_{ k}\hat{r}_{k-1}]$. Then, from (45), we obtain:

$$\begin{aligned} \textrm{Var}[\textrm{BCV}(h) ]&= \frac{h^{4}}{36}\sum _{k = 1}^{m}\tilde{h}_{k}^{-2}\textrm{E}[Y_{k, k-1}^{2}] + \frac{h^{2}a(c)^{2}}{36}(\textrm{E}[W_{1}^{2}] + \textrm{E}[W_{m-1}^{2}]) + \frac{h^{2}b(c)^{2}}{9}\textrm{E}[W_{m}^{2}]\nonumber \\&\quad -\frac{h^{3}a(c)}{18}\sum _{k = 1}^{m}\tilde{h}_{k}^{-1}(\textrm{E}[W_{1}Y_{k, k-1}] + \textrm{E}[W_{m-1}Y_{k, k-1}] ) -\frac{h^{3}b(c)}{9}\sum _{k=1}^{m}\tilde{h}_{k}^{-1}\textrm{E}[W_{m}Y_{k, k-1}]\nonumber \\&\quad + \frac{h^{4}}{18}\sum _{k < l}\tilde{h}_{k}^{-1}\tilde{h}_{l}^{-1}\textrm{E}[Y_{k, k-1}Y_{l, l-1}] + \frac{h^{2}a(c)^{2}}{18}\textrm{E}[W_{1}W_{m-1}]\nonumber \\&\quad +\frac{h^{2}a(c)b(c)}{9}(\textrm{E}[W_{1}W_{m}] + \textrm{E}[W_{m-1}W_{m}]). \end{aligned}$$

(46)

Combining (35) and (36) provides the following approximations of $Y_{k, k-1}$:

$$\begin{aligned} Y_{k, k-1}&= \textrm{E}[\hat{f}_{k}]^{-1/2}\textrm{E}[\hat{f}_{k-1}]^{1/2}W_{k} + \textrm{E}[\hat{f}_{k}]^{1/2}\textrm{E}[\hat{f}_{k-1}]^{-1/2}W_{k-1} \nonumber \\&\quad -\frac{1}{4} \textrm{E}[\hat{f}_{k}]^{-3/2}\textrm{E}[\hat{f}_{k-1}]^{1/2}(W_{k}^{2} -\textrm{E}[ W_{k}^{2}] )- \frac{1}{4} \textrm{E}[\hat{f}_{k}]^{1/2}\textrm{E}[\hat{f}_{k-1}]^{-3/2}(W_{k-1}^{2} - \textrm{E}[ W_{k-1}^{2}])\nonumber \\&\quad + \frac{1}{2}\textrm{E}[\hat{f}_{k}]^{-1/2}\textrm{E}[\hat{f}_{k-1}]^{-1/2}(W_{k}W_{k-1} -\textrm{E}[W_{k}W_{k-1}])\nonumber \\&\quad + O_{p}((W_{k} + W_{k-1})^{3} + (W_{k} + W_{k-1})^{4}) + O((nh)^{-2}). \end{aligned}$$

(47)

By combining, (29), (30), (31), (32) (33), and (47) we obtain the following simple form:

$$\begin{aligned} Y_{k, k-1}&= [1 + O(h)](W_{k} + W_{k-1}) + O_{p}( (W_{k} + W_{k-1} )^{2}) + O( (nh)^{-1}). \end{aligned}$$

(48)

Combining, (30), (32) (33), and (48) yields:

$$\begin{aligned} \textrm{E}[Y_{k, k-1}^{2} ]&=[1 + O(h)]^{2}( \textrm{E}[W_{k}^{2}] + \textrm{E}[W_{k-1}^{2}] + \textrm{E}[W_{k}W_{k-1}]) + O((nh)^{-2})\nonumber \\&=\frac{f(t_{k})}{nh_{k}} + \frac{f(t_{k})}{nh_{k-1}} + O(n^{-1} + (nh)^{-2}), \end{aligned}$$

(49)

where $h_{0} = h_{m}$. Using the same method as (33), we show that $\textrm{E}[W_{k}W_{l}] =O(n^{-1})$ for any $k \ne l$. Therefore, combining this, (30), and (48) yields the following covariances:

$$\begin{aligned} \textrm{E}[W_{j}Y_{k,k-1}]&= \textrm{E}\left[ [1 + O(h)](W_{j}W_{k} + W_{j}W_{k-1} ) + O_{p}(W_{j}(W_{k} + W_{k-1})^{2} ) \right] \nonumber \\&= [1 + O(h)]( \textrm{E}[W_{j}W_{k} ] + \textrm{E}[W_{j}W_{k-1} ] )\nonumber \\&={\left\{ \begin{array}{ll} f(t_{j})(nh_{j})^{-1} + O(n^{-1}) &{} \text {if}\, k = j\,\text { or}\, k = j + 1,\\ O(n^{-1}) &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

(50)

for $j \in \{1, 2, \dots , m\}$,

$$\begin{aligned} \textrm{E}[W_{m}Y_{k,k-1}]&={\left\{ \begin{array}{ll} f(t_{m})(nh_{m})^{-1} + O(n^{-1}) &{} \text {if}\, k = m\,\text { or}\, k =1,\\ O(n^{-1}) &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

(51)

and

$$\begin{aligned} \textrm{E}[Y_{k, k-1}, Y_{l, l-1} ]&=[1 + O(h)]^{2}( \textrm{E}[W_{k}W_{l}] + \textrm{E}[W_{k}W_{l-1}] + \textrm{E}[W_{k-1}W_{l}] + \textrm{E}[W_{k-1}W_{l-1}])\nonumber \\&\quad + O( (nh)^{-2}) \nonumber \\&= O(n^{-1} + (nh)^{-2}). \end{aligned}$$

(52)

Combining (33) and (52) demonstrates that the magnitude of the sixth, seventh, and eighth terms in the RHS in (46) is $O(n^{-1}h^{2})$. This indicates that these terms are ignored. Therefore, by combining (30), (32), (46), (49), (50), and (51), we obtain:

$$\begin{aligned} \textrm{Var}[\textrm{BCV}(h) ]&= \frac{h^{4}}{36} \left[ 2\sum _{k=2}^{m-1}\frac{f(t_{k})}{nh^{3}} + \frac{f(t_{1})}{nh\tilde{h}_{1}^{2}} + \frac{f(t_{m})}{nh_{m}\tilde{h}_{m}^{2}} +\frac{f(t_{m})}{nh\tilde{h}_{m}^{2}} + \frac{f(t_{1})}{nh^{2}\tilde{h}_{m}} +O(n^{-1} + (nh)^{-2}) \right] \nonumber \\&\quad + \frac{h^{2}a(c)^{2}}{36}\left[ \frac{f(t_{1})}{nh} + \frac{f(t_{m-1})}{nh}\right] + \frac{h^{2}b(c)^{2}}{9} \frac{f(t_{m})}{nh_{m}} \nonumber \\&\quad - \frac{h^{2}a(c)}{18}\left[ 2\frac{f(t_{1})}{nh} + \frac{f(t_{m-1})}{nh} + \frac{f(t_{m-1})}{nch}\right] \nonumber \\&\quad - \frac{h^{3}b(c)}{9}\left[ \frac{f(t_{m})}{nh_{m}\tilde{h}_{m}} + \frac{f(t_{m})}{nh\tilde{h}_{m}}\right] + O(n^{-1}h^{2})\nonumber \\&= \frac{1}{18n}\int ^{2\pi - ch}_{h}f(\theta )d\theta [1+o(1)]\nonumber \\&= \frac{1}{18n} + o(n^{-1}). \end{aligned}$$

(53)

Theorem 2 completes the proof from (43) and (53). $\square$

Appendix D

Proof of Corollary 2

We denote the MISE and the asymptotic MISE of the rose diagram estimator as $\textrm{MISE}(h)$ and $\textrm{AMISE}(h)$, respectively. From (5), we obtain:

$$\begin{aligned} \textrm{AMISE}(h)/\textrm{MISE}(h) \xrightarrow {p} 1. \end{aligned}$$

(54)

From combining Theorem 2 and the law of large numbers, we obtain:

$$\begin{aligned} \textrm{BCV}(h)/\textrm{AMISE}(h) \xrightarrow {p} 1. \end{aligned}$$

(55)

Let $h_{*} = h_{\textrm{opt}}$ and $\gamma := \hat{h}_{\textrm{BCV}}/h_{*}$ We find that (54) and (55) hold for all h. Therefore, we obtain:

$$\begin{aligned} \textrm{AMISE}(\gamma h_{*})/\textrm{MISE}(\gamma h_{*}) \xrightarrow {p} 1, \end{aligned}$$

(56)

and

$$\begin{aligned} \textrm{BCV}(\gamma h_{*})/\textrm{MISE}(\gamma h_{*}) \xrightarrow {p} 1. \end{aligned}$$

(57)

The ratio of the AMISE of bin width $\gamma h_{*}$ to that of bin width $h_{*}$ is a function expressed in the following equation:

$$\begin{aligned} \textrm{AMISE}(\gamma h_{*})/\textrm{AMISE}(h_{*}) = \frac{\gamma ^{2}}{3} + \frac{2}{3\gamma } . \end{aligned}$$

(58)

The ratio in (58) is a convex function with minimum value at $\gamma =1$.Thus, if $\gamma \ne 1$ and n is sufficiently large, from (56), we obtain:

$$\begin{aligned} \textrm{MISE}(\gamma h_{*})>\textrm{MISE}(h_{*}). \end{aligned}$$

(59)

Suppose that $\gamma$ does not converge to 1.If n is sufficiently large, then $\textrm{BCV}(h)$ is a convex function with a minimum at $h =\gamma h_{*}$, as $\textrm{BCV}(h)$ closes to $\textrm{AMISE}(h)$. Therefore, we obtain:

$$\begin{aligned} \textrm{P}(\textrm{BCV}(\gamma h_{*})< \textrm{BCV}(h_{*}) ) \xrightarrow {p} 1, \end{aligned}$$

(60)

as $n\rightarrow \infty$. We combine (57) and (60), and find that

$$\begin{aligned} \textrm{MISE}(\gamma h_{*}) < \textrm{MISE}(h_{*}), \end{aligned}$$

(61)

as $n\rightarrow \infty$. The contradiction between (59) and (61) completes the proof. $\square$

Appendix E

Proof of Theorem 3

We now calculate the bias. Note that $\hat{r}_{k}(\theta ; h) = \hat{r}_{k}$. Combining (18) and (29) provides the approximation of $\textrm{E}[\hat{r}_{k}]$:

$$\begin{aligned} \textrm{E}[\hat{r}_{k}]&= \sqrt{2}\left( f(t_{k}) + \frac{1}{2}f^{\prime }(t_{k})h_{k} + \frac{1}{6}f^{\prime \prime }(t_{k})h_{k}^{2}+ O(h^{3})\right) ^{1/2} + O((nh)^{-1}) \nonumber \\&=\sqrt{ 2f(t_{k})} + \frac{ f^{\prime }(t_{k})h_{k}}{2\sqrt{ 2f(t_{k})}} + \frac{ f^{\prime \prime }(t_{k})h_{k}^{2} }{6\sqrt{ 2f(t_{k})}}- \frac{f^{\prime }(t_{k})^{2}h_{k}^{2}}{16\sqrt{2}f(t_{k})^{3/2}} + O(h^{3} + (nh)^{-1})\nonumber \\&= r_{f}(t_{k}) + \frac{1}{2}r_{f}^{\prime }(t_{k})h_{k} +\frac{4r^{\prime \prime }_{f}(t_{k}) + r^{\prime }_{f}(t_{k})^{2}r_{f}(t_{k})^{-1}}{24}h_{k}^{2} + O(h^{3} + (nh)^{-1}). \end{aligned}$$

(62)

Similarly, combining (18) and (31) provides the approximation of $\textrm{E}[\hat{r}_{k-1}]$:

$$\begin{aligned} \textrm{E}[\hat{r}_{k-1}]&= r_{f}(t_{k}) - \frac{1}{2}r_{f}^{\prime }(t_{k})h_{k-1} + \frac{4r^{\prime \prime }_{f}(t_{k}) + r^{\prime }_{f}(t_{k})^{2}r_{f}(t_{k})^{-1}}{24}h_{k-1}^{2}+ O(h^{3} + (nh)^{-1}). \end{aligned}$$

(63)

By combining (62) and (63), we obtain:

$$\begin{aligned} \textrm{E}[\tilde{r}_{f}(\theta ;h)]&= r_{f}(t_{k}) + r_{f}^{\prime }(t_{k})(\theta -t_{k})\nonumber \\&\quad +\frac{4r^{\prime \prime }_{f}(t_{k}) + r^{\prime }_{f}(t_{k})^{2}r_{f}(t_{k})^{-1}}{24}[h_{k}h_{k-1} + 2(\theta - t_{k})(h_{k} - h_{k-1})]\nonumber \\&\quad + O(h^{3} + (nh)^{-1}), \end{aligned}$$

(64)

for $\theta \in [t_{k} -h_{k-1}/2, t_{k} +h_{k}/2)$. The Taylor expansion of $r_{f}(\theta )$ for $\theta \in [t_{k} -h_{k -1 }/2, t_{k} +h_{k}/2)$ is

$$\begin{aligned} r_{f}(\theta ) = r_{f}(t_{k}) + r^{\prime }(t_{k})(\theta - t_{k}) + \frac{1}{2}r^{\prime \prime }_{f}(t_{k})(\theta -t_{k})^{2} + O( (\theta -t_{k})^{3}). \end{aligned}$$

(65)

Combining (64) and (65) leads to:

$$\begin{aligned} \textrm{Bias}[\tilde{r}(\theta ;h)]&= \frac{r^{\prime \prime }_{f}(t_{k})}{6} [h_{k}h_{k-1} + 2(\theta - t_{k})(h_{k} -h_{k-1}) - 3(\theta -t_{k})^{2}]\nonumber \\&\quad + \frac{1}{24}r^{\prime }_{f}(t_{k})^{2}r_{f}(t_{k})^{-1}[h_{k}h_{k-1} + 2(\theta - t_{k})(h_{k} -h_{k-1})]\nonumber \\&\quad + O(h^{3} + (nh)^{-1}). \end{aligned}$$

(66)

Let $\textrm{ISB}: = \int ^{2\pi }_{0} \textrm{Bias}[\tilde{r}(\theta ;h)]^{2}d\theta$. Note that $\textrm{ISB}= \int ^{2\pi - (c + 1/2)h}_{h/2} \textrm{Bias}[\tilde{r}(\theta ;h)]^{2}d\theta + o(1) = \sum _{k=2}^{m-1} \int ^{t_{k} + h/2}_{t_{k} - h/2}\textrm{Bias}[\tilde{r}(\theta ;h)]^{2}d\theta [ 1 +o(1)]$. For $\theta \in [t_{k} -h/2, t_{k} +h/2)$, $k= 2, 3,\dots , m-1$, $\textrm{Bias}[\tilde{r}(\theta ;h)]$ in (66) is

$$\begin{aligned} \textrm{Bias}[\tilde{r}(\theta ;h)] = \frac{r^{\prime \prime }_{f}(t_{k})}{6}[h^{2} - 3(\theta -t_{k})^{2}]+ \frac{1}{24}r^{\prime }_{f}(t_{k})^{2}r_{f}(t_{k})^{-1}h^{2} + O(h^{3} + (nh)^{-1}). \end{aligned}$$

(67)

Integration of $\textrm{Bias}[\tilde{r}(\theta ;h)]$ in (67) for $\theta \in [t_{k} -h/2, t_{k} +h/2)$, $k = 2, 3, \dots , m-1$ is

$$\begin{aligned}&\int ^{t_{k} + h/2}_{t_{k} - h/2}\textrm{Bias}[\tilde{r}(\theta ;h)]^{2}d\theta \nonumber \\&\quad =\int ^{t_{k} + h/2}_{t_{k} - h/2}\Biggl \{\frac{r^{\prime \prime }_{f}(t_{k})}{36}[h^{4} - 6(\theta -t_{k})^{2} + 9(\theta -t_{k})^{4}] +\frac{1}{72} r^{\prime \prime }_{f}(t_{k})r^{\prime }_{f}(t_{k})^{2}r_{f}(t_{k})^{-1}[h^{4}-3(\theta -t_{k})^{2}h^{2}] \nonumber \\&\qquad + \frac{1}{576}r^{\prime }_{f}(t_{k})^{4}r_{f}(t_{k})^{-2}h^{4}\Biggr \}d\theta [1 + o(1)] \nonumber \\&\quad = \left[ \frac{49}{2880}r_{f}^{\prime \prime }(t_{k})^{2}+ \frac{1}{96}r_{f}^{\prime \prime }(t_{k})r_{f}^{\prime }(t_{k})^{2}r_{f}(t_{k})^{-1} + \frac{1}{576}r^{\prime }_{f}(t_{k})^{4}r_{f}(t_{k})^{-2}\right] h^{5}[1 + o(1)] . \end{aligned}$$

(68)

We find that $\sum _{k=2}^{m-1}r_{f}^{\prime \prime }(t_{k})^{2}h = \int _{h/2}^{2\pi - (c+1/2)h}r_{f}^{\prime \prime }(\theta )^{2}d\theta + o(1)= \int ^{2\pi }_{0}r_{f}^{\prime \prime }(\theta )^{2}d\theta$ + o(1). Calculating each term in (68) for $k= 2, 3,\dots , m-1$ yields the following form:

$$\begin{aligned} \textrm{ISB}&= \left\{ \frac{49}{2880}R(r_{f}^{\prime \prime }) + \frac{1}{96}R((r_{f}^{\prime \prime })^{1/2} r_{f}^{\prime }r_{f}^{-1/2})+ \frac{1}{576}R((r^{\prime }_{f})^{2}r_{f}^{-1})\right\} h^{4} + o(h^{4}). \end{aligned}$$

(69)

Next, we derive $\textrm{IV}: = \int ^{2\pi }_{0} \textrm{Var}[\tilde{r}(\theta ;h)]d\theta$. We calculate the variances of $\hat{r}_{k}$ and $\hat{r}_{k-1}$ by applying the same method as in (24). By combining (17), (29), (30), (31), and (32), we obtain:

$$\begin{aligned} \textrm{Var}[\hat{r}_{k} ]&= (2 \textrm{E}[\hat{f}_{k}])^{-1}\textrm{E}[W_{k}^{2}] + O((nh)^{-2})\nonumber \\&= \frac{1}{2nh_{k}} + O(n^{-1} + (nh)^{-2}) , \end{aligned}$$

(70)

and

$$\begin{aligned} \textrm{Var}[\hat{r}_{k-1} ]&= \frac{1}{2nh_{k-1}} + O(n^{-1}+ (nh)^{-2}). \end{aligned}$$

(71)

Combining (17), (29), (31), and (33) leads to:

$$\begin{aligned} \textrm{Cov}[\hat{r}_{k-1}, \hat{r}_{k} ]&= 2^{-1}(E[\hat{f}_{k}]E[\hat{f}_{k-1}])^{-1/2}\textrm{E}[W_{k} W_{k-1} ] +O((nh)^{-2})\nonumber \\&= \frac{1}{2[f(t_{k}) + O(h)]}\left[ -\frac{f(t_{k})^{2}}{n} + O(n^{-1}h)\right] + O((nh)^{-2})\nonumber \\&= -\frac{f(t_{k})}{2n} + O(n^{-1}h +(nh)^{-2}) . \end{aligned}$$

(72)

By combining (70), (71), and (72), we obtain:

$$\begin{aligned} \textrm{Var}[\tilde{r}_{f}(\theta ; h)]&= \left( \frac{h_{k}}{2\tilde{h}_{k}} - \frac{\theta - t_{k}}{\tilde{h}_{k}} \right) ^{2} \frac{1}{2nh_{k-1}} + \left( \frac{h_{k-1}}{2\tilde{h}_{k}} + \frac{\theta - t_{k}}{\tilde{h}_{k}} \right) ^{2} \frac{1}{2nh_{k}} + o( (nh)^{-1} ). \end{aligned}$$

(73)

Note that $\textrm{IV} = \sum _{k=2}^{m-1} \int ^{t_{k} + h/2}_{t_{k} - h/2}\textrm{Var}[\tilde{r}(\theta ;h)]d\theta [ 1 +o(1)]$. For $\theta \in [t_{k} -h/2, t_{k} +h/2)$, $k= 2, 3, \dots , m-1$, from (73), we obtain:

$$\begin{aligned} \textrm{Var}[\tilde{r}_{f}(\theta ; h)]&=\frac{1}{2nh}\left[ \frac{1}{2} + 2\frac{(\theta - t_{k})^{2}}{h^{2}} \right] + o( (nh)^{-1}). \end{aligned}$$

(74)

Integration of $\textrm{Var}[\tilde{r}_{f}(\theta ; h)]$ in (74) for $\theta \in [t_{k} -h/2, t_{k} +h/2)$, $k= 2, 3, \dots , m-1$ is

$$\begin{aligned} \int ^{t_{k} + h/2}_{t_{k} - h/2}\textrm{Var}[\tilde{r}(\theta ;h)]d\theta&= \frac{1}{3n}[1+o(1)]. \end{aligned}$$

(75)

From (75), we obtain:

$$\begin{aligned} \textrm{IV}&= \sum _{k=2}^{m-1} \int ^{t_{k} + h/2}_{t_{k} - h/2}\textrm{Var}[\tilde{r}(\theta ;h)]^{2}d\theta [ 1 +o(1)] \nonumber \\&= \sum _{k =2}^{m-1} \frac{1}{3nh} [ 1 + o(1) ]h\nonumber \\&= \frac{2\pi - (1 + c)h}{3nh} +o( (nh)^{-1} )\nonumber \\&= \frac{2\pi }{3nh} + o( (nh)^{-1} ). \end{aligned}$$

(76)

Theorem 3 completes the proof from combining (69) and (76). $\square$