Scientific Reports

Table 11 Percentage relative efficiency using Population 1, 2, 3 when \(\left\{x={Q}_{3}(x),y={Q}_{3}(y)\right\}\).

From: Construction of improved comprehensive classes of estimators for population distribution function

Estimators			Population 1			Population 2			Population 3
Estimators			\({\widehat{\mathcal{F}}}_{GK}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{GK}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{GK}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}(\mathcal{Y})\)
\({\widehat{\mathcal{F}}}_{GK}^{(1)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(1)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(1)}(\mathcal{Y})\)	494.04	510.24	523.86	621.69	647.49	651.53	245.26	288.33	270.29
\({\widehat{\mathcal{F}}}_{GK}^{(2)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(2)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(2)}(\mathcal{Y})\)	493.99	510.12	523.81	621.62	647.33	651.36	244.78	287.10	269.15
\({\widehat{\mathcal{F}}}_{GK}^{(3)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(3)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(3)}(\mathcal{Y})\)	494.10	510.39	524.09	621.77	647.70	651.74	245.76	289.70	271.56
\({\widehat{\mathcal{F}}}_{GK}^{(4)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(4)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(4)}(\mathcal{Y})\)	493.98	510.10	523.79	621.61	647.32	651.35	244.72	286.97	269.02
\({\widehat{\mathcal{F}}}_{GK}^{(5)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(5)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(5)}(\mathcal{Y})\)	494.02	210.18	523.88	621.66	647.42	651.45	245.12	287.96	269.95
\({\widehat{\mathcal{F}}}_{GK}^{(6)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(6)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(6)}(\mathcal{Y})\)	493.99	510.14	523.83	621.63	647.36	651.39	244.91	287.43	269.45
\({\widehat{\mathcal{F}}}_{GK}^{(7)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(7)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(7)}(\mathcal{Y})\)	494.04	510.22	523.92	621.68	647.48	651.51	245.13	287.98	269.97
\({\widehat{\mathcal{F}}}_{GK}^{(8)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(8)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(8)}(\mathcal{Y})\)	494.07	510.30	524.00	621.72	647.58	651.62	245.59	289.24	271.13
\({\widehat{\mathcal{F}}}_{GK}^{(9)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(9)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(9)}(\mathcal{Y})\)	493.99	510.11	523.80	621.62	647.33	651.36	244.74	287.03	269.08
\({\widehat{\mathcal{F}}}_{GK}^{(10)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(10)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(10)}(\mathcal{Y})\)	493.98	510.09	523.79	621.6	647.30	651.33	244.68	286.87	268.93
	\(\widehat{\mathcal{F}}(y)\)			100.00			100.00				100.00
	\({\widehat{\mathcal{F}}}_{R}(\mathcal{Y})\)			468.75			598.06				213.53
	\({\widehat{\mathcal{F}}}_{P}(\mathcal{Y})\)			26.04			25.730				27.780
	\({\widehat{\mathcal{F}}}_{BT,R}(\mathcal{Y})\)			279.25			298.47				206.54
	\({\widehat{\mathcal{F}}}_{BT,P}(\mathcal{Y})\)			46.750			46.250				49.600
	\({\widehat{\mathcal{F}}}_{Reg}(\mathcal{Y})\)			493.83			621.46				241.84
	\({\widehat{\mathcal{F}}}_{R,D}(\mathcal{Y})\)			493.98			621.60				244.68

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