Numerical comparison of averages

In the next table numerical results are displayed. An exact random-Poisson generator has been used to generate Poisson deviates of given average value $\lambda$ , with $\lambda=1,10,100,\ldots,1000000$ . For each value $\lambda$ $N=10^8$ deviates have been generated. Then averages have been taken for each value $\lambda$ and compared with the true value. For each value $\lambda$ - in order to have a scale for comparison - we evaluate the expected absolute s.d. of averages as $\xi_\lambda=\sqrt{\lambda/N}$ , and the relative s.d. of averages as $\epsilon_\lambda=\sqrt{\lambda/N}/\lambda=1/\sqrt{N\lambda}$ . Then - for each averaging method - we evaluate the error $E_\lambda$ (average minus $\lambda$ ), the relative error $e_\lambda=E_\lambda/\lambda$ , and finally the comparison criterion $e_\lambda/\epsilon_\lambda$ (bold). The comparison criterion is expected to be close to 1 in absolute value. Values much larger than one mean that we are introducing a systematic error.

	$\lambda =$ 1. ; $\xi_\lambda =$ 0.0001 ; $\epsilon_\lambda$ = 0.0001
	${\langle x \rangle_{\!\mathrm{w(1)}}}$	${\langle x \rangle_{\!\mathrm{w(2)}}}$	$\langle x\rangle^*$	$\langle x \rangle$
Averages	1.303772380383934	0.9999155361216990	1.581941754994651	0.9999283300000000
$E_\lambda$	0.3037723803839338	-0.8446387830096658E-04	0.5819417549946508	-0.7166999999996815E-04
$e_\lambda$	0.3037723803839338	-0.8446387830096658E-04	0.5819417549946508	-0.7166999999996815E-04
$e_\lambda/\epsilon_\lambda$	3037.723803839338	-0.8446387830096658	5819.417549946508	-0.7166999999996815
	$\lambda =$ 10.000000000000002 ; $\xi_\lambda =$ 0.00031622776601683794 ; $\epsilon_\lambda$ = 0.00003162277660168379
	${\langle x \rangle_{\!\mathrm{w(1)}}}$	${\langle x \rangle_{\!\mathrm{w(2)}}}$	$\langle x\rangle^*$	$\langle x \rangle$
Averages	8.848248847530357	10.00025732384808	10.00052232372917	10.00006800000000
$E_\lambda$	-1.151751152469645	0.2573238480785278E-03	0.5223237291644978E-03	0.6799999999884676E-04
$e_\lambda$	-0.1151751152469645	0.2573238480785278E-04	0.5223237291644977E-04	0.6799999999884675E-05
$e_\lambda/\epsilon_\lambda$	-3642.156939527943	0.8137294562072904	1.651732660112730	0.2150348808878029
	$\lambda =$ 100.00000000000004 ; $\xi_\lambda =$ 0.0010000000000000002 ; $\epsilon_\lambda$ = 0.000009999999999999997
	${\langle x \rangle_{\!\mathrm{w(1)}}}$	${\langle x \rangle_{\!\mathrm{w(2)}}}$	$\langle x\rangle^*$	$\langle x \rangle$
Averages	98.98978896904168	100.0001037814804	100.0002153600000	100.0002153600000
$E_\lambda$	-1.010211030958359	0.1037814803765968E-03	0.2153599999559219E-03	0.2153599999559219E-03
$e_\lambda$	-0.1010211030958359E-01	0.1037814803765968E-05	0.2153599999559218E-05	0.2153599999559218E-05
$e_\lambda/\epsilon_\lambda$	-1010.211030958359	0.1037814803765968	0.2153599999559219	0.2153599999559219
	$\lambda =$ 1000.0000000000007 ; $\xi_\lambda =$ 0.0031622776601683803 ; $\epsilon_\lambda$ = 0.000003162277660168378
	${\langle x \rangle_{\!\mathrm{w(1)}}}$	${\langle x \rangle_{\!\mathrm{w(2)}}}$	$\langle x\rangle^*$	$\langle x \rangle$
Averages	999.0029754507847	1000.003978305674	1000.003836760000	1000.003836760000
$E_\lambda$	-0.9970245492160075	0.3978305673513205E-02	0.3836759999330752E-02	0.3836759999330752E-02
$e_\lambda$	-0.9970245492160069E-03	0.3978305673513202E-05	0.3836759999330750E-05	0.3836759999330750E-05
$e_\lambda/\epsilon_\lambda$	-315.2868458625229	1.258050715667192	1.213290043331128	1.213290043331128
	$\lambda =$ 10000.00000000001 ; $\xi_\lambda =$ 0.010000000000000005 ; $\epsilon_\lambda$ = 9.999999999999995E-7
	${\langle x \rangle_{\!\mathrm{w(1)}}}$	${\langle x \rangle_{\!\mathrm{w(2)}}}$	$\langle x\rangle^*$	$\langle x \rangle$
Averages	9998.995728116572	9999.995828163173	9999.995919900000	9999.995919900000
$E_\lambda$	-1.004271883437468	-0.4171836835666909E-02	-0.4080100008650334E-02	-0.4080100008650334E-02
$e_\lambda$	-0.1004271883437467E-03	-0.4171836835666905E-06	-0.4080100008650330E-06	-0.4080100008650330E-06
$e_\lambda/\epsilon_\lambda$	-100.4271883437468	-0.4171836835666907	-0.4080100008650331	-0.4080100008650331
	$\lambda =$ 100000.0000000002 ; $\xi_\lambda =$ 0.031622776601683826 ; $\epsilon_\lambda$ = 3.162277660168376E-7
	${\langle x \rangle_{\!\mathrm{w(1)}}}$	${\langle x \rangle_{\!\mathrm{w(2)}}}$	$\langle x\rangle^*$	$\langle x \rangle$
Averages	99999.01275394148	100000.0127639189	100000.0125627100	100000.0125627100
$E_\lambda$	-0.9872460587212117	0.1276391866849735E-01	0.1256270980229601E-01	0.1256270980229601E-01
$e_\lambda$	-0.9872460587212097E-05	0.1276391866849733E-06	0.1256270980229599E-06	0.1256270980229599E-06
$e_\lambda/\epsilon_\lambda$	-31.21946156583365	0.4036305486159527	0.3972677655897895	0.3972677655897895
	$\lambda =$ 1000000.0000000013 ; $\xi_\lambda =$ 0.10000000000000006 ; $\epsilon_\lambda$ = 9.999999999999993E-8
	${\langle x \rangle_{\!\mathrm{w(1)}}}$	${\langle x \rangle_{\!\mathrm{w(2)}}}$	$\langle x\rangle^*$	$\langle x \rangle$
Averages	999999.1188353101	1000000.118835812	1000000.118809340	1000000.118809340
$E_\lambda$	-0.8811646911781281	0.1188358106883243	0.1188093387754634	0.1188093387754634
$e_\lambda$	-0.8811646911781270E-06	0.1188358106883241E-06	0.1188093387754633E-06	0.1188093387754633E-06
$e_\lambda/\epsilon_\lambda$	-8.811646911781276	1.188358106883242	1.188093387754633	1.188093387754633

As it is visible from the table:

1.    

${\langle x \rangle_{\!\mathrm{w(1)}}}\ :$ the weighted average using straight Poisson statistics is consistenty bad at all values of $\lambda$ , that means at all counting levels;

2.    

${\langle x \rangle^*}\$ : the normal average skipping zero count data is bad for $\lambda<100$ , that means at low counting levels (of course at higher counting levels zeroes are not happening);

3.    

${\langle x \rangle}\$   and${\langle x \rangle_{\!\mathrm{w(2)}}}$ : the normal average including zero count data and the Mighell-Poisson weighted average are consistently and equivalently good at all counting levels.

Therefore there is no bias when using the Mighell-Poisson weighted method to average data w.r.t. the usual average. The former, however, has already accomplished the passage to normal statistics, therefore all operations on data that are not simple averaging can be done in the framework of normal statistics, where everything is known and clear. In the next section, on the opposite, it is shown that even simple operations as scaling data lead to the necessity of abandoning Poisson statistics in order to estimate correctly the standard deviations.