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The selection of the number of terms in an orthogonal series cumulative function estimator

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Abstract

Let h(.) be a continuous, strictly positive probability density function over an interval [ab] and H(.) its associated cumulative distribution function (cdf). Given a sample set \(X_{1},\ldots ,X_{n}\) of independent identically distributed variables, we want to estimate H(.) from this sample set. The present work has two goals. The first one is to propose an estimator of a cdf based on an orthogonal trigonometric series and to give its statistical and asymptotic proprieties (bias, variance, mean square error, mean integrated squared error, convergence of the bias, convergence of variance, convergence of the mean squared error, convergence of the mean integrated squared error, uniform convergence in probability and the rate of convergence of the mean integrated squared error). The second is to introduce a new method for the selection of a “smoothing parameter”. The comparison by simulation between this method and Kronmal–Tarter’s method, shows that the new method is more performant in the sense of the mean integrated square error.

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Acknowledgments

We are very much thankful to Editor, an Editorial Board member and the referees for their instructive comments and suggestions.

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Correspondence to Nora Saadi.

Appendix

Appendix

Proof of Proposition 1

  1. (i)

    For \(k=0\), \(\mathbb {E}(\hat{A}_{0})=\mathbb {E}\big (\frac{\pi -\overline{X}}{\sqrt{2\pi }}\big )= \frac{\pi -\mu }{\sqrt{2\pi }}.\) However, \(A_{0} =\frac{1}{\sqrt{2\pi }}\int _{-\pi }^{\pi } H(x)dx.\) Applying integration by parts, we get

    $$\begin{aligned} A_{0}= \frac{1}{\sqrt{2\pi }}\left[ [xH(x)]_{-\pi }^{\pi }-\int _{-\pi }^{\pi } x h(x)dx\right] = \frac{\pi -\mu }{\sqrt{2\pi }}=\mathbb {E}(\hat{A}_{0}). \end{aligned}$$

    For \(k \ne 0\)

    $$\begin{aligned} \mathbb {E}(\hat{A}_{k})= & {} \mathbb {E}\left[ \frac{\hat{\gamma }_{k}}{k}-\frac{\hat{\beta }_{k}}{k}+\,\frac{(-1)^{k+1}}{k\sqrt{2\pi }}\right] =\frac{1}{k\sqrt{2\pi }}\mathbb {E}(\cos (kX))\\&-\,\frac{1}{k\sqrt{2\pi }}\mathbb {E}(\sin (kX))+\,\frac{(-1)^{k+1}}{k\sqrt{2\pi }} =\frac{\gamma _{k}}{k}-\frac{\beta _{k}}{k}+\,\frac{(-1)^{k+1}}{k\sqrt{2\pi }}, \end{aligned}$$

    where

    $$\begin{aligned} \frac{\gamma _{k}}{k}=\frac{1}{k\sqrt{2\pi }}\int _{-\pi }^{\pi }\cos (kx) h(x)dx \; \; \hbox {and} \; \; \frac{\beta _{k}}{k}=\frac{1}{k\sqrt{2\pi }}\int _{-\pi }^{\pi }\sin (kx) h(x)dx. \end{aligned}$$

    Applying integration by parts, we get

    $$\begin{aligned} \frac{\gamma _{k}}{k}= & {} \frac{1}{\sqrt{2\pi }}\left[ \frac{\cos (kx)}{k}H(x)\right] _{-\pi }^{\pi }+\,\frac{1}{\sqrt{2\pi }}\int _{-\pi }^{\pi }\sin (kx)H(x)dx,\\ \frac{-\beta _{k}}{k}= & {} \frac{1}{\sqrt{2\pi }}\left[ -\frac{\sin (kx)}{k}H(x)\right] _{\pi }^{-\pi }+\,\frac{1}{\sqrt{2\pi }}\int _{-\pi }^{\pi }\sin (kx)H(x)dx. \end{aligned}$$

    Consequently,

    $$\begin{aligned} \frac{\gamma _{k}}{k}=\frac{(-1)^{k}}{k\sqrt{2\pi }}+\,\frac{1}{\sqrt{2\pi }}\int _{-\pi }^{\pi }\sin (kx)H(x)dx \; \; \hbox {and} \; \; \frac{-\beta _{k}}{k}=\frac{1}{\sqrt{2\pi }}\int _{-\pi }^{\pi }\cos (kx)H(x)dx. \end{aligned}$$

    Consequently, Then, for \(k \ne 0\), we deduce that

    $$\begin{aligned} \mathbb {E}(\hat{A}_{k})= & {} \frac{(-1)^{k}}{k\sqrt{2\pi }}+\,\frac{1}{\sqrt{2\pi }}\int _{-\pi }^{\pi }\sin (kx)H(x)dx +\,\frac{1}{\sqrt{2\pi }}\int _{-\pi }^{\pi }\cos (kx)H(x)dx\\&+\,\frac{(-1)^{k+1}}{k\sqrt{2\pi }}=A_{k}. \end{aligned}$$
  2. (ii)

    For \(k=0\), \(\mathbb {V}ar(\hat{A}_{k})= \mathbb {V}ar(\hat{A}_{0}) = \mathbb {V}ar(\frac{\pi -\overline{X}}{\sqrt{2\pi }})= \frac{\alpha }{2\pi n}.\) For \(k\ne 0\), we have

    $$\begin{aligned} \mathbb {V}ar(\hat{A}_{k})= & {} \frac{1}{2\pi nk^{2}}\mathbb {V}ar\left[ \cos (kX) -\sin (kX)+(-1)^{k+1}\right] \\= & {} \frac{1}{2\pi nk^{2}}-\frac{\beta _{2k}}{nk^{2}\sqrt{2\pi } }-\frac{(\gamma _{k}-\beta _{k})^{2}}{ nk^{2}}. \end{aligned}$$
  1. (iii)

    We have

    $$\begin{aligned} \mathbb {C}ov(\hat{A}_{k},\hat{A}_{j})=\mathbb {E}(\hat{A}_{k}\hat{A}_{j})-\mathbb {E}(\hat{A}_{k})\mathbb {E}(\hat{A}_{j}). \end{aligned}$$
    $$\begin{aligned} \mathbb {E}(\hat{A}_{k}\hat{A}_{j})= & {} \frac{1}{kj}\left[ \mathbb {E}(\hat{\gamma }_{k}\hat{\gamma }_{j})-\mathbb {E}(\hat{\gamma }_{k}\hat{\beta }_{j})+\,\frac{(-1)^{j+1}}{\sqrt{2\pi }}\mathbb {E}(\hat{\gamma }_{k}) -\mathbb {E}(\hat{\beta }_{k}\hat{\gamma }_{j})+\mathbb {E}(\hat{\beta }_{k}\hat{\beta }_{j})\nonumber \right. \\&\left. -\frac{(-1)^{j+1}}{\sqrt{2\pi }}\mathbb {E}(\hat{\beta }_{k})+ \frac{(-1)^{k+1}}{\sqrt{2\pi }}\mathbb {E}(\hat{\gamma }_{j})-\frac{(-1)^{k+1}}{\sqrt{2\pi }}\mathbb {E}(\hat{\beta }_{j})+\frac{(-1)^{j+k+2}}{2\pi }\right] ,\nonumber \\ \end{aligned}$$
    (26)

    and

    $$\begin{aligned} \mathbb {E}(\hat{A}_{k})\mathbb {E}(\hat{A}_{j})= & {} \frac{1}{kj}\left[ \mathbb {E}(\hat{\gamma }_{k})-\mathbb {E}(\hat{\beta }_{k})+\frac{(-1)^{k+1}}{\sqrt{2\pi }}\right] \left[ \mathbb {E}(\hat{\gamma }_{j})-\mathbb {E}(\hat{\beta }_{j})+\frac{(-1)^{j+1}}{\sqrt{2\pi }}\right] .\nonumber \\ \end{aligned}$$
    (27)

Calculation of the \(\mathbb {E}(\hat{\gamma }_{k} \hat{\gamma }_{j})\), \(\mathbb {E}(\hat{\gamma }_{k} \hat{\beta }_{j})\), \(\mathbb {E}(\hat{\beta }_{k} \hat{\gamma }_{j})\) and \(\mathbb {E}(\hat{\beta }_{k} \hat{\beta }_{j})\)

We have the following properties:

$$\begin{aligned} \cos (kx)\cos (jx)=\frac{1}{2}[\cos (k+j)x+ \cos (k-j)x]. \end{aligned}$$

Then,

$$\begin{aligned} \mathbb {E}(\hat{\gamma }_{k} \hat{\gamma }_{j})= \frac{1}{4 \pi n}\mathbb {E}[\cos (k+j)X+ \cos (k-j)X]+\left[ \frac{n-1}{ 2 \pi n}\right] \mathbb {E}(\cos (kX) \mathbb {E}(\cos (jX)). \end{aligned}$$

In addition

$$\begin{aligned} \mathbb {E}[\cos (kX)]= \int _{-\pi }^{\pi } h(x) \cos (kx) dx=\sqrt{2 \pi }\gamma _{k}. \end{aligned}$$

And

$$\begin{aligned}&\mathbb {E} [\cos (k+j)X]=\sqrt{2 \pi }\gamma _{k+j}.\\&\mathbb {E}[\cos (k-j)X]=\sqrt{2 \pi }\gamma _{k-j}. \end{aligned}$$

Finally,

$$\begin{aligned} \mathbb {E}(\hat{\gamma }_{k} \hat{\gamma }_{j})=\frac{1}{4 \pi n}\left[ \sqrt{2 \pi }(\gamma _{k+j}+ \gamma _{k-j})\right] +\left[ \frac{n-1}{n}\right] \gamma _{k}\gamma _{j}. \end{aligned}$$
(28)

Finally:

$$\begin{aligned} \mathbb {E}(\hat{\gamma }_{k} \hat{\beta }_{j})=\frac{1}{4 \pi n}\left[ \sqrt{2 \pi }(\beta _{k+j}+ \beta _{j-k})\right] +\left[ \frac{n-1}{n}\right] \gamma _{k}\beta _{j}. \end{aligned}$$
(29)

Using the similar calculus we obtain the following results

$$\begin{aligned} \mathbb {E}(\hat{\beta }_{k} \hat{\beta }_{j})=\frac{1}{4 \pi n}\left[ \sqrt{2 \pi }(\gamma _{j-k}+ \gamma _{j+k})\right] +\left[ \frac{n-1}{n}\right] \beta _{k}\beta _{j}. \end{aligned}$$
(30)

and

$$\begin{aligned} \mathbb {E}(\hat{\beta }_{k} \hat{\gamma }_{j})=\frac{1}{4 \pi n}\left[ \sqrt{2 \pi }(\beta _{k+j}+ \beta _{j-k})\right] +\left[ \frac{n-1}{n}\right] \gamma _{j}\beta _{k}. \end{aligned}$$
(31)

Substituting (29), (30) and (31) in (26) and (27) respectively, we deduce that

\(\square \)

Proof of Property 1

  1. (i)

    For \(k=0\),

    $$\begin{aligned} \lim _{n \longrightarrow \infty }\mathbb {V}ar(\hat{A}_{k})=\lim _{n \longrightarrow \infty }\frac{\alpha }{2\pi n} =0. \end{aligned}$$

    For \(k\ne 0\), we have

    $$\begin{aligned} \lim _{n \longrightarrow \infty }\mathbb {V}ar(\hat{A}_{k})=\lim _{n \longrightarrow \infty }\left[ \frac{1}{2\pi nk^{2}}-\frac{\beta _{2k}}{nk^{2}\sqrt{2\pi }}-\frac{(\gamma _{k}-\beta _{k})^{2}}{ nk^{2}}\right] =0. \end{aligned}$$
  2. (ii)

    We have

    $$\begin{aligned} \lim _{n \longrightarrow \infty }\mathbb {E} |\hat{A}_{k}-A_{k}|^{2}=\lim _{n \longrightarrow \infty }\mathbb {E} |\hat{A}_{k}-E[\hat{A}_{k}]|^{2}=\lim _{n \longrightarrow \infty }\mathbb {V}ar(\hat{A}_{k}). \end{aligned}$$

    according to (i) we have \(\lim _{n \longrightarrow \infty }\mathbb {V}ar(\hat{A}_{k})=0,\quad \forall k=0,1,\ldots \). Then we deduce that:

    $$\begin{aligned} \lim _{n \longrightarrow \infty }\mathbb {E} |\hat{A}_{k}-A_{k}|^{2}=0. \end{aligned}$$

\(\square \)

Proof of Theorem 1

$$\begin{aligned} \mathbb {V}ar[\hat{H}_{d_{n}}(x)]= & {} \mathbb {V}ar\left[ \sum _{k=0}^{d_{n}}\hat{A}_{k}e_{k}(x)\right] \\= & {} \frac{1}{ n^{2}}\sum _{i=1}^{n}\mathbb {V}ar\left[ \sum _{k=0}^{d_{n}}\left( \int ^{\pi }_{X_{i}} e_{k}(y)dy\right) e_{k}(x)\right] \\ {}= & {} \frac{1}{ n}\mathbb {V}ar\left[ \sum _{k=0}^{d_{n}}\left( \int ^{\pi }_{X} e_{k}(y)dy\right) e_{k}(x)\right] \end{aligned}$$

It is known that \(\mathbb {V}ar(X)\le \mathbb {E}(X^{2})\), one obtains

$$\begin{aligned} \mathbb {V}ar[\hat{H}_{d_{n}}(x)]\le & {} \frac{1}{n}\int _{-\pi }^{\pi }\left[ \int ^{\pi }_{z}\sum _{k=0}^{d_{n}}e_{k}(y) e_{k}(x)dy\right] ^{2}h(z)dz\\= & {} \frac{1}{n}\int ^{\pi }_{-\pi }\left[ \int ^{\pi }_{z}\left[ \sum _{k=0}^{d_{n}}\frac{1}{2\pi }\left[ \cos (k(y-x))+\sin (k(y+x))\right] dy\right] ^{2} h(z)dz\right] \end{aligned}$$

Poses \(s=\sum _{k=0}^{d_{n}}\frac{1}{2\pi }[\cos k(y-x)+\sin k(y+x)]\). We deduce that

$$\begin{aligned} S= & {} \left[ \frac{1 }{2 \pi }+\frac{1}{2 \pi }[\cos (y-x)+\sin (y+x)+\cdots +\cos (d_{n}(y-x))+\sin (d_{n}(y+x))]\right] \\= & {} \frac{1}{2\pi }[1+\cos (y-x)+\sin (y+x)+\cdots +\cos (d_{n}(y-x))+\sin (d_{n}(y+x))]\\= & {} \frac{1}{2\pi }\left[ \frac{1}{2}+\cos (y-x)+\cos 2(y-x)+\cdots +\cos (d_{n}(y-x)) \right. \\&\left. +\,\frac{1}{2}+\sin (y+x)+\sin 2(y+x)+\cdots +\sin (d_{n}(y+x))\right] \\= & {} \frac{1 }{2\pi }\left[ \frac{1}{2}+\cos (y-x)+\cos 2(y-x)+\cdots +\cos (d_{n}(y-x))\right. \\&\left. +\,\frac{1}{2}+\cos \Big (\frac{\pi }{2}-(y+x)\Big )+\cos 2\Big (\frac{\pi }{2}-(y+x)\Big )+\cdots + \cos d_{n}\Big (\frac{\pi }{2}-(y+x)\Big )\right] \\= & {} \frac{1}{2\pi }\left[ \frac{1}{2}+\sum _{k=1}^{d_{n}}\cos (k(y-x))+\,\frac{1}{2}+\sum _{k=1}^{d_{n}} \cos \left( k\Big (\frac{\pi }{2}-(y+x)\Big )\right) \right] \\= & {} \dfrac{1}{2 \pi }\left[ \frac{\sin \Big [\frac{(2d_{n}+1)(y-x)}{2}\Big ]}{2\sin \Big [\frac{y-x}{2}\Big ]}+\dfrac{\sin \Big [\frac{(2d_{n}+1)(\frac{\pi }{2}-(y+x))}{2}\Big ]}{2\sin \Big [\frac{\frac{\pi }{2}-(y+x)}{2}\Big ]}\right] \\= & {} \dfrac{1}{4 \pi }\left[ \frac{\sin \Big [\frac{(2d_{n}+1)(y-x)}{2}\Big ]}{\sin \Big [\frac{y-x}{2}\Big ]}+\dfrac{\sin \Big [\frac{(2d_{n}+1)(\frac{\pi }{2}-(y+x))}{2}\Big ]}{\sin \Big [\frac{\frac{\pi }{2}-(y+x)}{2}\Big ]}\right] . \end{aligned}$$

Then,

$$\begin{aligned} \mathbb {V}ar[\hat{H}_{d_{n}}(x)]\le & {} \frac{1}{16\pi ^{2}n}\int ^{\pi }_{-\pi } \left[ \int ^{\pi }_{z}\left[ \frac{\sin \Big [\frac{(2d_{n}+1)(y-x)}{2}\Big ]}{\sin \Big [\frac{y-x}{2}\Big ]}+\frac{\sin \Big [\frac{(2d_{n}+1)(\frac{\pi }{2}-(y+x))}{2}\Big ]}{\sin \Big [\frac{\frac{\pi }{2}-(y+x)}{2}\Big ]}\right] dy\right] ^{2}h(z)dz. \end{aligned}$$

We have \(\frac{\sin (kx)}{\sin x}\le k\), then

$$\begin{aligned} \mathbb {V}ar[\hat{H}_{d_{n}}(x)]\le \frac{(2d_{n}+1)^{2}}{4\pi ^{2} n}(\alpha +(\pi -\mu )^{2}). \end{aligned}$$

We deduce that

$$\begin{aligned} \lim _{n\longrightarrow \infty }\mathbb {V}ar[\hat{H}_{d_{n}}(x)]=0. \end{aligned}$$

\(\square \)

Proof of Theorem 4

$$\begin{aligned} |\hat{H}_{d_{n}}(x)-\mathbb {E}(\hat{H}_{d_{n}}(x))|=\left| \sum _{k=0}^{d_{n}}\hat{A}_{k}e_{k}(x)-\sum _{k=0}^{d_{n}}A_{k}e_{k}(x)\right| \end{aligned}$$

Then,

$$\begin{aligned} \sup _{x\in [-\pi ,\pi ]}| \hat{H}_{d_{n}}(x)-\mathbb {E}(\hat{H}_{d_{n}}(x)|\le \frac{2}{\sqrt{2\pi }}\sum _{k=0}^{d_{n}}|\hat{A}_{k}-A_{k}|. \end{aligned}$$

In addition,

$$\begin{aligned} \mathbb {E}\left[ \left( \sup \nolimits _{x\in [-\pi ,\pi ]}|\hat{H}_{d_{n}}(x)-\mathbb {E}(\hat{H}_{d_{n}}(x))|\right) ^{2}\right] \le \tfrac{2}{\pi }\mathbb {E}\left( {\sum \nolimits _{k=0}^{d_{n}}}|\hat{A}_{k}-A_{k}|\right) ^{2}\!. \end{aligned}$$

According to the Property 3 and the Property 1 hence,

$$\begin{aligned} \lim \nolimits _{n\longrightarrow \infty } \mathbb {E}\left[ \left( \sup \nolimits _{x\in [-\pi ,\pi ]}|\hat{H}_{d_{n}}(x)-\mathbb {E}(\hat{H}_{d_{n}}(x))|\right) ^{2}\right] =0. \end{aligned}$$

Consequently,

$$\begin{aligned} \lim \nolimits _{n\longrightarrow \infty } \left\{ \mathbb {E}\left[ \left( \sup \nolimits _{x\in [-\pi ,\pi ]}|\hat{H}_{d_{n}}(x)-\mathbb {E}(\hat{H}_{d_{n}}(x))|\right) ^{2}\right] \right\} ^{\frac{1}{2}}=0. \end{aligned}$$

However,

$$\begin{aligned} \lim _{n\longrightarrow \infty }\hat{H}_{d_{n}}(x)=H(x). \end{aligned}$$

It follow that:

$$\begin{aligned} \mathbb {E}\left[ \left( \sup \nolimits _{x\in [-\pi ,\pi ]}|\hat{H}_{d_{n}}(x)-H(x)|\right) ^{2}\right] =0. \end{aligned}$$

Then, we deduce that

$$\begin{aligned} \lim _{n\longrightarrow \infty } P\left[ \sup _{x\in [-\pi ,\pi ]}|\hat{H}_{d_{n}}(x)-H(x)|<\epsilon \right] =1,\quad \forall \epsilon >0. \end{aligned}$$

\(\square \)

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Saadi, N., Adjabi, S. & Gannoun, A. The selection of the number of terms in an orthogonal series cumulative function estimator. Stat Papers 59, 127–152 (2018). https://doi.org/10.1007/s00362-016-0756-9

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