The median of a jittered Poisson distribution

Abstract

Let $N_\lambda$ and U be two independent random variables respectively distributed as a Poisson distribution with parameter $\lambda >0$ and a uniform distribution on (0, 1). This paper establishes that the median, say M, of $N_\lambda +U$ is close to $\lambda +1/3$ and more precisely that $M-\lambda -1/3=o(\lambda ^{-1})$ as $\lambda \rightarrow \infty$. This result is used to construct a very simple robust estimator of $\lambda$ which is consistent and asymptotically normal. Compared to known robust estimates, this one can still be used with large datasets ($n\simeq 10^9$).

Appendix

Appendix gathers technical lemmas, used in the proof of Proposition 1.

Lemma 1

Let $k\in \mathbb {R}$, $x\in [0,1)$ and $r_n(x,k) = k/(n+x)$. Then the sequence $w_n(x,k)$ reads as follows:

• (i) If $x+r_n(x,k) \in [-1/3,2/3)$

  $$\begin{aligned} w_n(x,k)= \mathrm {P}\left( N_{n+x}\le n \right) +\left( x-\frac{2}{3}+\frac{k}{n+x}\right) \mathrm {P}\left( N_{n+x}=n\right) . \end{aligned}$$

  (14)

• (ii) If $x+r_n(x,k)\in [2/3,5/3)$

  $$\begin{aligned} w_n(x,k)= \mathrm {P}\left( N_{n+x}\le n\right) +\left( x-\frac{2}{3}+\frac{k}{n+x}\right) \mathrm {P}(N_{n+x}=n+1). \end{aligned}$$

  (15)

Let $k\in \mathbb {R}$, then for n sufficiently large, if $x\in [0,2/3)$ or $x=2/3$ and $k<0$, then $w_n(x,k)$ reads as in (14). In the same way, if $x\in (2/3,1)$ or $x=2/3$ and $k\ge 0$, then $w_n(x,k)$ reads as in (15). The proof of Lemma 1 is omitted as it derives easily from (5).

Lemma 2

Let $k\in \mathbb {R}$, $x\in [0,1)$ and $r_n(x,k) = k/(n+x)$. Let $\varDelta _n(x,k)$ be given by

$$\begin{aligned} \varDelta _n(x,k) = \frac{(n+1)!}{g_{n+1}(n+1+x)} \bigg ( w_{n+1}(x,k) - w_{n}(x,k) \bigg ). \end{aligned}$$

There exists $n_0 \in \mathbb {N}$ such that for all $n\ge n_0$, we have the two following cases:

• (i) If $x+r_{n}(x,k) \in [-1/3,2/3)$

  $$\begin{aligned} \varDelta _n(x,k) =&\;c_n(0,x)-\int _0^1 c_n(v,x)\mathrm {d}v + \left( x-\frac{2}{3} \right) \left( 1-c_n(0,x) \frac{n+1}{n+x}\right) \nonumber \\&+ r_{n+1}(x,k)- \frac{n+1}{n+x} c_n(0,x)r_n(x,k) \end{aligned}$$

  (16)

  where $c_n(\cdot ,x):[0,1]\rightarrow \mathbb {R}$ is defined by

  $$\begin{aligned} c_n(v,x)= \left( \frac{n+v+x}{n+1+x} \right) ^{n+1} \exp (1-v). \end{aligned}$$

• (ii) If $x+r_{n}(x,k) \in [2/3,5/3)$

  $$\begin{aligned} \varDelta _n(x,k) =&\,c_n(0,x) -\int _0^1 c_n(v,x)\mathrm {d}v +\left( x-\frac{2}{3} \right) \left( \frac{n+1+x}{n+2}-c_n(0,x) \right) \nonumber \\&+ \frac{n+1+x}{n+2}r_{n+1}(x,k)- c_n(0,x)r_n(x,k). \end{aligned}$$

  (17)

Proof

• (i) Using the Poisson-Gamma relation $\mathrm {P}(N_\lambda \le n)=\frac{1}{n!}\int _\lambda ^\infty g_n(u)\mathrm {d}u$ with $g_n(u)=\mathrm {e}^{-u}u^n$ and Lemma 1 (i), we can rearrange the difference $w_{n+1}(x,k)-w_n(x,k)$ as

  $$\begin{aligned} w_{n+1}(x,k)-w_n(x,k)&=\frac{1}{(n+1)!}\int _{n+x}^{n+1+x}-g_{n+1}(u)du+\frac{g_{n+1}(n+x)}{(n+1)!}\\&\qquad +\left( x-\frac{2}{3}+r_{n+1}(x,k)\right) \mathrm {P}\left( N_{n+1+x}=n+1\right) \\&\qquad -\left( x-\frac{2}{3}+r_n(x,k)\right) \mathrm {P}\left( N_{n+x}=n\right) , \end{aligned}$$

  which leads to the result after little algebra by noticing that

  $$\begin{aligned} \mathrm {P}(N_{n+x}=n)=\frac{(n+1)(n+x)^n\mathrm {e}^{-(n+x)}}{(n+1)!}=\frac{(n+1)g_n(n+x)}{(n+1)!}. \end{aligned}$$

• (ii) Using the Poisson-Gamma relation and Lemma 1 (ii), we can rearrange the difference $w_{n+1}(x,k)-w_n(x,k)$ as

  $$\begin{aligned} w_{n+1}(x,k)-w_n(x,k) =&\,\frac{1}{(n+1)!}\int _{n+1+x}^{\infty }\mathrm {e}^{-u}u^{n+1}du\\&-\frac{1}{(n+1)!}\int _{n+x}^{\infty }\mathrm {e}^{-u}u^{n+1}du\\&+\left( x-\frac{2}{3}+r_{n+1}(x,k)\right) \mathrm {P}\left( N_{n+1+x}=n+2\right) \\&-\left( x-\frac{2}{3}+r_n(x,k)\right) \mathrm {P}\left( N_{n+x}=n+1\right) , \end{aligned}$$

  which leads to the result after little algebra by noticing that

  $$\begin{aligned} \mathrm {P}(N_{n+1+x}=n+2)=\frac{1}{(n+1)!}g_{n+1}(n+1+x)\frac{n+1+x}{(n+2)} \end{aligned}$$

  and

  $$\begin{aligned} \mathrm {P}(N_{n+x}=n+1)=\frac{1}{(n+1)!}g_{n+1}(n+1+x)c_n(0,x). \end{aligned}$$

$\square$

Lemma 3

Let $k\in \mathbb {R}$, $x\in [0,1)$ and $r_n(x,k) = k/(n+x)$, for any $k \in \mathbb {R}$ and $x\in [0,1)$, then for n sufficiently large we have

$$\begin{aligned} \varDelta _n(x,k) = \frac{3}{2(n+1+x)^2} \big ( {\mathcal {H}}(x) - k \big ) \; + \; o\left( \frac{1}{n^2}\right) \end{aligned}$$

where ${\mathcal {H}}$ is the function given by (7).

Proof

Let $k\in \mathbb {R}$, then for any $x\in [0,1)$, there exists $n_0\in \mathbb {N}$ such that for $n\ge n_0$, either (i) $x+r_{n}(x,k)\in [-1/3,2/3)$ or (ii) $x+r_n(x,k)\in [2/3,5/3)$. In the sequel, we consider both cases and expand the expression of $\varDelta _n(x,k)$ given by Lemma 2. The following expansions extensively make use of auxiliary results gathered in Lemma 4.

• (i) Case $x+r_{n}(x,k)\in [-1/3,2/3)$.

  $$\begin{aligned} \varDelta _n(x,k)&=\frac{1}{n+1+x}\left( -\frac{x}{2}+\frac{1}{6}\right) +\frac{1}{(n+1+x)^2}\left( -\frac{x^2}{6}-\frac{x}{24}+\frac{7}{120}\right) \\&\qquad +\frac{1}{n+1+x}\left( x-\frac{1}{2}\right) +\frac{1}{(n+1+x)^2}\left( \frac{x^2}{2}-\frac{5}{24}\right) \\&\qquad +\left( x-\frac{2}{3}\right) \left[ \frac{-1/2}{n+1+x}+\frac{1}{(n+1+x)^2}\left( \frac{x^2}{2}-\frac{x}{2}-\frac{7}{24}\right) \right] \\&\qquad -\frac{3k}{2}\frac{1}{(n+1+x)^2}+o\left( \frac{1}{n^2}\right) \\&=\frac{3}{2(n+1+x)^2}\left( \frac{x^2(x-1)}{3}+\frac{4}{135}-k\right) +o\left( \frac{1}{n^2}\right) \\&=\frac{3}{2(n+1+x)^2} \big ( {\mathcal {H}}(x) - k \big ) \; + \; o\left( \frac{1}{n^2}\right) . \end{aligned}$$

• (ii) Case $x+r_{n}(x,k)\in [2/3,5/3)$.

  $$\begin{aligned} \varDelta _n(x,k)&=\frac{1}{n+1+x}\left( -\frac{x}{2}+\frac{1}{6}\right) +\frac{1}{(n+1+x)^2}\left( -\frac{x^2}{6}-\frac{x}{24}+\frac{7}{120}\right) \\&\qquad +\frac{1}{n+1+x}\left( x-\frac{1}{2}\right) +\frac{1}{(n+1+x)^2}\left( \frac{x^2}{2}-\frac{5}{24}\right) \\&\qquad +\left( x-\frac{2}{3}+r_{n+1}(x,k)\right) \left( \frac{n+1+x}{n+2}\right) \\&\qquad -\left( x-\frac{2}{3}+r_n(x,k)\right) c_n(0,x)\\&=\frac{1}{(n+1+x)^2}\left( \frac{x^3}{2}-2x^2+\frac{5x}{2}-\frac{43}{45}-\frac{3k}{2}\right) +o\left( \frac{1}{n^2}\right) \\&=\frac{3}{2(n+1+x)^2}\left( \frac{x^3}{3}-\frac{4x^2}{3}+\frac{5x}{3}-\frac{86}{135}-k\right) +o\left( \frac{1}{n^2}\right) \\&=\frac{3}{2(n+1+x)^2} \big ( {\mathcal {H}}(x) - k \big ) \; + \; o\left( \frac{1}{n^2}\right) . \end{aligned}$$

$\square$

Lemma 4

Let $v,x \in [0,1]$, then we have the following expansions as $n \rightarrow \infty$:

• (i)
  $$\begin{aligned} c_n(v,x)=&\,1+\frac{1}{n+1+x}\left( x(1-v)-\frac{(1-v)^2}{2}\right) \nonumber \\&+\frac{1}{(n+1+x)^2}\left( \frac{(1-v)^2}{2}(x^2+x)\right. \nonumber \\&\left. -(1-v)^3\left( \frac{x}{2}+\frac{1}{3}\right) +\frac{(1-v)^4}{8}\right) \nonumber \\&+o\left( \frac{1}{n^2}\right) . \end{aligned}$$

  (18)
• (ii)
  $$\begin{aligned}&c_n(0,x) =1+\frac{1}{n+1+x}\left( x-\frac{1}{2}\right) +\frac{1}{(n+1+x)^2}\left( \frac{x^2}{2}-\frac{5}{24}\right) +o\left( \frac{1}{n^2}\right) . \end{aligned}$$

  (19)
• (iii)
  $$\begin{aligned} \int _0^1c_n(v,x)dv=&\,1+\frac{1}{n+1+x}\left( \frac{x}{2}-\frac{1}{6}\right) \nonumber \\&+\frac{1}{(n+1+x)^2}\left( \frac{x^2}{6}+\frac{x}{24}-\frac{7}{120}\right) +o\left( \frac{1}{n^2}\right) . \end{aligned}$$

  (20)
• (iv)
  $$\begin{aligned} 1-c_n(0,x)\frac{n+1}{n+x}= & {} \frac{-1/2}{n+1+x}+\frac{1}{(n+1+x)^2}\left[ \frac{x^2}{2}-\frac{x}{2}-\frac{7}{24}\right] \nonumber \\&+o\left( \frac{1}{n^2}\right) . \end{aligned}$$

  (21)
• (v)
  $$\begin{aligned} r_{n+1}(x,k)- \frac{n+1}{n+x} c_n(0,x)r_n(x,k)=\frac{-3k}{2}\frac{1}{(n+1+x)^2}+o\left( \frac{1}{n^2}\right) . \end{aligned}$$

  (22)
• (vi)
  $$\begin{aligned} \frac{n+1+x}{n+2}-c_n(0,x)= & {} \frac{1/2}{n+1+x} + \frac{1}{(n+1+x)^2} \left( \frac{x^2}{2}-2x -\frac{5}{24}\right) \nonumber \\&+o\left( \frac{1}{n^2} \right) . \end{aligned}$$

  (23)
• (vii)
  $$\begin{aligned} \frac{n+1+x}{n+2}r_{n+1}(x,k)- c_n(0,x)r_n(x,k) = \frac{-3k}{2}\frac{1}{(n+1+x)^2}+o\left( \frac{1}{n^2}\right) .\nonumber \\ \end{aligned}$$

  (24)

The proof which relies upon simple Taylor expansions, is omitted to save space. Details can be provided upon request from the authors.