On the MLE of the Waring distribution

Figures & data

Table 1. Relative biases and relative standard errors of estimated parameters, for $\text{β}=0.5$ and 1.

			$\text{β}=0.5$				$\text{β}=1$
			rBias (%)		rStd (%)		rBias (%)		rStd (%)
	n	method	α	β	α	β	α	β	α	β
$\alpha =2$	100	EFF	63.0	69.1	230.3	271.3	41.9	46.9	137.3	157.6
		MLE	63.0	67.1	309.8	274.9	41.9	48.4	180.4	194.1
	200	EFF	24.5	25.6	69.7	85.2	15.0	15.4	34.3	41.8
		MLE	18.2	19.5	57.7	68.2	11.6	12.7	35.8	43.7
	400	EFF	9.5	9.0	25.0	29.5	6.8	6.4	19.8	24.8
		MLE	6.2	5.9	24.3	28.9	4.4	4.3	19.3	24.3
	1000	EFF	3.7	3.3	14.6	18.1	2.9	3.0	12.5	15.6
		MLE	1.6	1.4	13.5	17.2	1.5	1.6	11.4	14.6
$\alpha =1.5$	100	EFF	37.3	40.3	64.7	78.1	27.4	31.4	48.1	62.4
		MLE	26.3	30.4	66.4	79.8	18.4	23.5	50.9	64.5
	200	EFF	19.0	19.2	30.7	38.2	14.5	14.8	24.7	33.4
		MLE	10.2	11.0	32.3	39.6	7.1	8.0	25.1	33.1
	400	EFF	10.3	9.8	18.5	23.3	8.2	7.8	15.8	21.0
		MLE	4.0	3.7	17.8	22.6	2.9	2.7	15.0	20.4
	1000	EFF	5.6	5.5	12.2	15.2	4.5	4.3	10.7	13.1
		MLE	1.3	1.2	10.4	13.6	0.9	0.7	8.9	11.8
$\alpha =1.2$	100	EFF	33.9	36.7	41.6	54.8	27.4	30.9	31.5	45.6
		MLE	16.8	20.4	45.0	57.0	12.4	16.8	34.8	47.1
	200	EFF	21.2	21.1	22.8	30.7	18.2	18.8	19.3	27.5
		MLE	6.9	7.3	24.4	31.4	5.5	6.5	20.2	27.6
	400	EFF	14.3	13.8	15.0	20.4	12.6	12.3	13.2	18.7
		MLE	2.9	2.6	14.7	19.7	2.5	2.4	12.8	18.2
	1000	EFF	9.8	9.8	10.4	13.6	8.7	8.5	9.2	11.9
		MLE	1.0	1.0	8.9	12.1	0.7	0.6	7.7	10.6
$\alpha =1.1$	100	EFF	34.6	36.7	35.4	46.8	29.6	33.2	28.5	42.5
		MLE	14.3	17.4	40.0	50.4	11.3	15.6	32.4	45.1
	200	EFF	23.5	23.2	20.6	28.8	20.8	21.3	17.7	26.2
		MLE	6.0	6.4	22.7	29.7	4.9	5.8	19.1	26.5
	400	EFF	17.0	16.3	13.7	19.2	15.4	15.0	12.1	17.9
		MLE	2.6	2.2	13.9	18.7	2.2	2.0	12.1	17.6
	1000	EFF	12.7	12.6	9.5	12.7	11.7	11.4	8.4	11.3
		MLE	1.0	0.9	8.4	11.5	0.7	0.5	7.3	10.3
$\alpha =1.05$	100	EFF	35.7	37.8	33.3	45.1	31.0	34.5	27.1	41.3
		MLE	13.4	16.5	37.6	48.5	10.5	14.6	30.9	43.0
	200	EFF	25.0	24.6	19.3	27.7	22.5	22.9	16.7	25.4
		MLE	5.7	6.0	21.7	28.5	4.6	5.4	18.3	25.6
	400	EFF	18.8	18.1	12.9	18.4	17.3	16.9	11.5	17.6
		MLE	2.5	2.0	13.3	18.1	2.1	2.0	11.8	17.3
	1000	EFF	14.6	14.6	8.9	12.4	13.6	13.4	7.9	11.1
		MLE	1.0	0.9	8.1	11.3	0.7	0.6	7.2	10.3

Table 2. Relative biases and relative standard errors of estimated parameters, for $\text{β}=1.5$ and 2.

			$\text{β}=1.5$				$\text{β}=2$
			rBias (%)		rStd (%)		rBias (%)		rStd (%)
	n	method	α	β	α	β	α	β	α	β
$\alpha =2$	100	EFF	35.3	41.6	96.7	119.1	42.8	51.5	223.7	260.9
		MLE	35.5	43.6	144.4	169.4	39.1	48.3	259.9	299.2
	200	EFF	13.7	14.8	34.3	44.4	12.8	14.2	32.0	42.2
		MLE	10.3	11.7	32.6	40.8	9.7	11.3	30.6	39.2
	400	EFF	6.3	6.3	19.1	25.2	6.1	6.4	18.4	24.7
		MLE	4.1	4.3	18.4	24.1	4.0	4.3	17.6	23.3
	1000	EFF	2.5	2.4	11.6	14.6	2.5	2.5	11.7	15.0
		MLE	1.3	1.2	10.5	13.5	1.3	1.3	10.3	13.3
$\alpha =1.5$	100	EFF	24.9	29.4	44.2	59.7	24.7	30.5	44.5	64.2
		MLE	16.6	22.0	47.2	61.7	15.9	21.6	43.0	57.1
	200	EFF	13.2	13.9	22.7	31.1	12.8	13.9	22.1	31.3
		MLE	6.5	7.7	23.0	30.8	6.3	7.6	22.5	30.4
	400	EFF	7.8	7.8	15.1	20.7	7.3	7.3	14.3	19.6
		MLE	2.9	3.1	14.3	19.8	2.7	2.8	13.4	18.7
	1000	EFF	4.2	4.2	10.4	13.3	4.0	3.9	10.2	13.4
		MLE	1.0	0.9	8.6	11.6	0.8	0.7	8.4	11.5
$\alpha =1.2$	100	EFF	26.0	30.1	30.7	45.9	25.3	30.4	30.5	48.0
		MLE	11.4	16.0	33.1	45.6	11.0	15.9	31.8	44.5
	200	EFF	17.1	17.7	18.3	27.0	16.6	17.7	17.8	27.8
		MLE	4.9	6.0	19.3	27.0	4.8	6.0	18.5	26.6
	400	EFF	11.9	11.7	12.4	17.6	11.6	11.9	12.0	17.8
		MLE	2.2	2.2	11.8	17.0	2.3	2.5	11.5	17.0
	1000	EFF	8.4	8.3	8.9	12.1	8.3	8.3	8.6	12.0
		MLE	0.7	0.6	7.5	10.7	0.7	0.7	7.2	10.6
$\alpha =1.1$	100	EFF	28.1	32.3	27.3	43.4	27.4	32.8	26.2	43.3
		MLE	10.2	14.7	30.3	42.8	10.1	15.2	29.2	42.4
	200	EFF	19.7	20.3	16.5	25.8	19.4	20.9	16.2	26.8
		MLE	4.4	5.3	17.9	25.7	4.5	5.9	17.4	25.6
	400	EFF	14.7	14.5	11.3	16.7	14.5	14.7	11.0	17.1
		MLE	2.0	2.0	11.1	16.4	2.1	2.3	10.9	16.6
	1000	EFF	11.3	11.1	8.0	11.4	11.2	11.1	7.6	11.1
		MLE	0.6	0.5	7.1	10.3	0.6	0.6	6.8	10.1
$\alpha =1.05$	100	EFF	29.4	33.5	25.0	41.1	29.3	35.7	25.6	44.0
		MLE	9.6	14.1	28.8	41.4	9.8	15.1	28.2	41.2
	200	EFF	21.5	22.2	15.7	25.1	21.4	23.3	15.6	27.2
		MLE	4.2	5.2	17.3	24.9	4.4	5.8	16.9	25.3
	400	EFF	16.7	16.7	10.7	16.5	16.5	17.0	10.4	16.9
		MLE	2.0	2.1	10.9	16.2	2.1	2.4	10.7	16.5
	1000	EFF	13.4	13.3	7.5	11.1	13.3	13.2	7.0	10.7
		MLE	0.7	0.6	6.9	10.2	0.6	0.6	6.6	10.0

Figure 1. Box-plots of $T_n$ corresponding to $\text{β}=1$ (first row), 1.5 (second row), and 2 (third row), respectively, where the dashed line indicates the critical value 3.84, and the number at the right side of the figure is the proportion that $T_n>3.84$. In the last piece, all $T_n$'s are much larger than 3.84, and thus the dashed line and the rejection proportion are not shown in the figure.

Table 3. Proportion of replicates that the Yule-Simon distribution is rejected at nominal level 0.05.

	n = 100	n = 200	n = 400	n = 1000
$\text{β}=1$	0.066	0.042	0.056	0.054
$\text{β}=1.5$	0.214	0.348	0.582	0.944
$\text{β}=2$	0.528	0.804	0.986	1.000

Table 4. Comparison of actual distribution (A) with discrete Pareto law fitting (P), Waring fitting with EFF (E) and MLE (M).

Central age	27.5				32.5				37.5				42.5				47.5
j	P	E	M	A	P	E	M	A	P	E	M	A	P	E	M	A	P	E	M	A
1	101.9	101	100.8	101	242.7	241	241.2	241	285.3	283	283.8	283	305.2	307	306.6	307	235.9	233	233.6	233
2	6.8	8.5	8.7	8	22.3	24.6	24.4	26	28.6	30.9	30.3	35	35.5	33.8	33.4	31	28.8	31.9	31.4	35
3	1.4	1.2	1.2	2	5.5	5.8	5.8	3	7.5	8	7.9	4	10.1	10	10.1	12	8.4	9.2	9.1	4
4	0.5	0.2	0.2	0	2	1.9	2	3	2.9	2.9	2.9	2	4.1	4.2	4.3	6	3.5	3.6	3.6	5
5	0.2	0.1	0	0	0.9	0.8	0.8	0	1.4	1.3	1.3	0	2.1	2.1	2.2	0	1.8	1.7	1.7	3
6	0.1	0	0	0	0.5	0.4	0.4	2	0.8	0.7	0.7	2	1.2	1.2	1.3	1	1	0.9	0.9	0
7	0.1	0	0	0	0.3	0.2	0.2	0	0.4	0.4	0.4	0	0.7	0.7	0.8	2	0.6	0.5	0.5	1
8	0	0	0	0	0.2	0.1	0.1	0	0.3	0.2	0.2	1	0.5	0.5	0.5	0	0.4	0.3	0.3	0
9	0	0	0	0	0.1	0.1	0.1	0	0.2	0.1	0.1	1	0.3	0.3	0.4	0	0.3	0.2	0.2	0
10	0	0	0	0	0.1	0	0	0	0.1	0.1	0.1	0	0.2	0.2	0.3	0	0.2	0.1	0.1	0
11	0	0	0	0	0.1	0	0	0	0.1	0.1	0.1	0	0.2	0.2	0.2	2	0.2	0.1	0.1	0
12	0	0	0	0	0	0	0	0	0.1	0	0	0	0.1	0.1	0.2	0	0.1	0.1	0.1	0
13	0	0	0	0	0	0	0	0	0.1	0	0	0	0.1	0.1	0.1	0	0.1	0.1	0.1	1
14	0	0	0	0	0	0	0	0	0	0	0	0	0.1	0.1	0.1	0	0.1	0	0	0
15	0	0	0	0	0	0	0	0	0	0	0	0	0.1	0.1	0.1	0	0.1	0	0	0
16	0	0	0	0	0	0	0	0	0	0	0	0	0.1	0.1	0.1	0	0.1	0	0	0
17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0.1	0	0	0	0	0
18	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0.1	0	0	0	0	0
Totals	111	111	110.9	111	274.7	274.9	275	275	327.8	327.7	327.8	328	360.6	360.7	360.9	361	281.6	281.7	281.7	282
Central age	52.5				57.5				62.5				67.5				72.5
j	P	E	M	A	P	E	M	A	P	E	M	A	P	E	M	A	P	E	M	A
1	199.3	200	200	200	110.8	108	107.7	108	72	69	69.5	69	35.1	33	/	33	27.4	26	24.9	26
2	24.7	24.1	23.6	24	16.6	20.1	20.1	20	6.3	7.1	7.1	10	5.5	8.3	/	7	5.2	6.9	7.7	5
3	7.3	7.3	7.3	7	5.5	6.3	6.4	5	1.5	2.2	2.1	1	1.9	2.4	/	4	2	2.5	2.9	4
4	3	3.1	3.2	2	2.5	2.6	2.6	2	0.6	1	0.9	0	0.9	0.8	/	1	1	1.1	1.2	1
5	1.6	1.6	1.6	0	1.4	1.2	1.3	4	0.3	0.5	0.5	0	0.5	0.3	/	0	0.6	0.6	0.6	2
6	0.9	0.9	1	3	0.8	0.7	0.7	0	0.1	0.3	0.3	0	0.3	0.1	/	0	0.4	0.3	0.3	0
7	0.6	0.6	0.6	1	0.5	0.4	0.4	0	0.1	0.2	0.2	0	0.2	0.1	/	0	0.3	0.2	0.2	0
8	0.4	0.4	0.4	1	0.4	0.2	0.2	1	0	0.1	0.1	0	0.1	0	/	0	0.2	0.1	0.1	0
9	0.3	0.3	0.3	0	0.3	0.2	0.2	0	0	0.1	0.1	0	0.1	0	/	0	0.1	0.1	0.1	0
10	0.2	0.2	0.2	1	0.2	0.1	0.1	0	0	0.1	0.1	0	0.1	0	/	0	0.1	0	0	0
11	0.1	0.1	0.2	0	0.2	0.1	0.1	0	0	0.1	0	0	0.1	0	/	0	0.1	0	0	0
12	0.1	0.1	0.1	0	0.1	0.1	0.1	0	0	0	0	0	0	0	/	0	0.1	0	0	0
13	0.1	0.1	0.1	0	0.1	0	0	0	0	0	0	0	0	0	/	0	0.1	0	0	0
14	0.1	0.1	0.1	0	0.1	0	0	0	0	0	0	0	0	0	/	0	0	0	0	0
15	0.1	0	0.1	0	0.1	0	0	0	0	0	0	0	0	0	/	0	0	0	0	0
16	0	0	0	0	0.1	0	0	0	0	0	0	0	0	0	/	0	0	0	0	0
17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	/	0	0	0	0	0
18	0	0	0	0	0	0	0	0	0	0	0	1	0	0	/	0	0	0	0	0
Totals	238.8	238.9	238.8	239	139.7	140	139.9	140	80.9	80.7	80.9	81	44.8	45	/	45	37.6	37.8	38	38