Scientific Reports

Table 13 Percentage relative efficiency using Population 4, 5, 6 when \(\left\{x={Q}_{1}(x),y={Q}_{1}(y)\right\}\).

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

Estimators			Population 4			Population 5			Population 6
Estimators			\({\widehat{\mathcal{F}}}_{GK}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{GK}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{GK}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}(\mathcal{Y})\)
\({\widehat{\mathcal{F}}}_{GK}^{(1)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(1)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(1)}(\mathcal{Y})\)	216.53	221.15	251.90	158.54	179.63	173.36	240.55	241.12	260.94
\({\widehat{\mathcal{F}}}_{GK}^{(2)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(2)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(2)}(\mathcal{Y})\)	216.44	221.97	251.69	158.41	179.33	173.07	240.54	241.11	260.93
\({\widehat{\mathcal{F}}}_{GK}^{(3)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(3)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(3)}(\mathcal{Y})\)	217.22	223.59	253.54	159.50	181.82	175.48	240.60	241.23	261.06
\({\widehat{\mathcal{F}}}_{GK}^{(4)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(4)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(4)}(\mathcal{Y})\)	216.65	221.41	251.19	158.76	180.13	173.85	240.56	241.14	260.96
\({\widehat{\mathcal{F}}}_{GK}^{(5)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(5)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(5)}(\mathcal{Y})\)	217.32	223.81	253.78	186.88	267.89	257.67	240.60	241.22	261.05
\({\widehat{\mathcal{F}}}_{GK}^{(6)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(6)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(6)}(\mathcal{Y})\)	218.52	226.37	256.71	185.40	261.80	251.90	240.68	241.39	261.23
\({\widehat{\mathcal{F}}}_{GK}^{(7)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(7)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(7)}(\mathcal{Y})\)	216.42	221.94	251.65	158.18	178.80	171.56	240.54	241.11	260.93
\({\widehat{\mathcal{F}}}_{GK}^{(8)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(8)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(8)}(\mathcal{Y})\)	219.48	228.48	259.13	184.96	260.00	250.20	240.74	241.52	261.37
\({\widehat{\mathcal{F}}}_{GK}^{(9)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(9)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(9)}(\mathcal{Y})\)	216.37	221.84	251.54	158.18	178.80	171.56	240.54	241.10	260.92
\({\widehat{\mathcal{F}}}_{GK}^{(10)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{1}}^{(10)}(\mathcal{Y})\)	\({\widehat{\mathcal{F}}}_{P{r}_{2}}^{(10)}(\mathcal{Y})\)	216.31	221.72	251.40	158.19	178.80	171.58	240.53	241.09	260.91
	\(\widehat{\mathcal{F}}(y)\)			100.00			100.00				100.00
	\({\widehat{\mathcal{F}}}_{R}(\mathcal{Y})\)			160.14			51.000				19.560
	\({\widehat{\mathcal{F}}}_{P}(\mathcal{Y})\)			19.810			26.780				210.36
	\({\widehat{\mathcal{F}}}_{BT,R}(\mathcal{Y})\)			177.87			80.850				238.73
	\({\widehat{\mathcal{F}}}_{BT,P}(\mathcal{Y})\)			51.60			81.470				240.53
	\({\widehat{\mathcal{F}}}_{Reg}(\mathcal{Y})\)			189.77			100.02				205.05
	\({\widehat{\mathcal{F}}}_{R,D}(\mathcal{Y})\)			216.30			158.18				49.690

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