Enhanced estimation of population mean using simple random sampling

Figures & data

Table 1 Several individuals from the proposed estimators.

Unknown members	u	v
$T_{(1)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left(\frac{\overline{X}-\overline{X}}{\overline{X}+\overline{X}}\right)$	1	0
$T_{(2)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(\overline{X}+C_x)-(\overline{X}+C_x)}{(\overline{X}+C_x)+(\overline{X}+C_x)}\right\}$	1	Cx
$T_{(3)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{({\beta }_2(x)\overline{X}+C_x)-({\beta }_2(x)\overline{X}+C_x)}{({\beta }_2(x)\overline{X}+C_x)+({\beta }_2(x)\overline{X}+C_x)}\right\}$	${\beta }_2(x)$	Cx
$T_{(4)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(C_x\overline{X}+{\beta }_2(x))-(C_x\overline{X}+{\beta }_2(x))}{(C_x\overline{X}+{\beta }_2(x))+(C_x\overline{X}+{\beta }_2(x))}\right\}$	Cx	${\beta }_2(x)$
$T_{(5)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(\overline{X}+{\rho }_{\mathit{\text{xy}}})-(\overline{X}+{\rho }_{\mathit{\text{xy}}})}{(\overline{X}+{\rho }_{\mathit{\text{xy}}})+(\overline{X}+{\rho }_{\mathit{\text{xy}}})}\right\}$	1	ρxy
$T_{(6)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(\overline{X}+{\beta }_2(x))-(\overline{X}+{\beta }_2(x))}{(\overline{X}+{\beta }_2(x))+(\overline{X}+{\beta }_2(x))}\right\}$	1	${\beta }_2(x)$
$T_{(7)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(C_x\overline{X}+{\rho }_{\mathit{\text{xy}}})-(C_x\overline{X}+{\rho }_{\mathit{\text{xy}}})}{(C_x\overline{X}+{\rho }_{\mathit{\text{xy}}})+(C_x\overline{X}+{\rho }_{\mathit{\text{xy}}})}\right\}$	Cx	ρxy
$T_{(8)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{({\beta }_2(x)\overline{X}+{\rho }_{\mathit{\text{xy}}})-({\beta }_2(x)\overline{X}+{\rho }_{\mathit{\text{xy}}})}{({\beta }_2(x)\overline{X}+{\rho }_{\mathit{\text{xy}}})+({\beta }_2(x)\overline{X}+{\rho }_{\mathit{\text{xy}}})}\right\}$	${\beta }_2(x)$	ρxy

Table 2 Descriptive statistics of different populations.

Parameters	Population 1: Source: Sarndal et al. (2003), 652–659), $Y=$ Total number of seats (S82) in municipal council in 1982 and $X=$ Number of conservative seats (CS82) in municipal council in 1982.	Population 2: Source: Singh (2003b, 1115), $Y=$ Seasonal average price per pound during 1996 and $X=$ Seasonal average price per pound during 1995.	Population 3: Source: Alomair and Shahzad (2023), $Y=$ Humidity (%) in Karachi during 2022 and $X=$ Humidity (%) in Karachi during 2021.	Population 4: Source: Kadilar and Cingi (2003), $Y=\text{Quantity}$ of apples produced in the Black Sea area of Turkey and $X=$ Apple trees’ quantity in the Black Sea area of Turkey.
N	284	36	365	204
n	50	14	100	40
$\overline{Y}$	46.07	0.20	90.09	966.95
$\overline{X}$	9.09	0.18	90.82	26441.72
Mx	8.00	0.18	91	13650
Cy	0.27	0.39	0.08	2.47
Cx	0.54	0.40	0.08	1.71
${\rho }_{\mathit{\text{xy}}}$	0.68	0.87	0.75	0.71
${\beta }_1(x)$	1.24	0.77	–0.48	4.62
${\beta }_2(x)$	5.38	3.34	3.21	29.77

Table 3 Optimum values of $(k_1,k_2)$ of the members of the proposed estimator for different populations.

Members of	Population 1	Population 2	Population 3	Population 4
estimator T
T(1)	(0.981,–1.653)	(0.964,–1.143)	(0.999,–1.220)	(0.805,–0.973)
T(2)	(0.983,–1.540)	(0.995, 0.228)	(0.999,–1.218)	(0.805,–0.973)
T(3)	(0.981,–1.631)	(0.986,–0.353)	(0.999,–1.219)	(0.805,–0.973)
T(4)	(0.996,–0.609)	(0.996, 0.812)	(0.999,–0.607)	(0.805,–0.972)
T(5)	(0.984,–1.513)	(0.997, 0.507)	(0.999,–1.203)	(0.805,–0.973)
T(6)	(0.993,–0.909)	(0.997, 0.751)	(0.999,–1.151)	(0.805,–0.971)
T(7)	(0.985,–1.409)	(0.997, 0.698)	(0.999,–1.031)	(0.805,–0.973)
T(8)	(0.982,–1.625)	(0.993, 0.027)	(0.999,–1.214)	(0.805,–0.973)

Table 4 Range of k₁ at the fixed value of k₂ for different populations.

Estimators	Population 1	Population 2	Population 3	Population 4
	($k_2=-1.65)$	($k_2=-1.14)$	($k_2=-1.22)$	($k_2=-0.97)$
tm	(0.95, 1.01)	(0.87, 1.06)	(0.994, 1.000)	(0.51, 1.100)
tr	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.000)	(0.63, 1.000)
treg	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
ts	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
tw	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
tsd	(0.94, 1.03)	(0.90, 1.03)	(0.998, 1.001)	(0.62, 0.999)
tre	(0.96, 1.00)	(0.92, 1.01)	(0.998, 1.002)	(0.58, 1.030)
tsk	(0.96, 1.00)	(0.88, 1.05)	(0.999, 1.001)	(0.62, 1.000)
$t_{u{p}_1}$	(0.94, 1.02)	(0.88, 1.05)	(0.998, 1.006)	(0.59, 1.020)
$t_{u{p}_2}$	(0.96, 1.00)	(0.89, 1.04)	(0.997, 1.003)	(0.52, 1.090)
$t_{s_1}$	(0.96, 1.01)	(0.94, 1.00)	(0.999, 1.000)	(0.57, 1.040)
tsn	(0.96, 1.00)	(0.92, 1.01)	(0.998, 1.002)	(0.58, 1.030)
tyk	(0.96, 1.00)	(0.92, 1.01)	(0.998, 1.002)	(0.60, 1.010)
tgk	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
tia	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
$t_{p_1}$	(0.93, 1.03)	(0.88, 1.04)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_2}$	(0.93, 1.03)	(0.90, 1.03)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_3}$	(0.93, 1.03)	(0.88, 1.05)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_4}$	(0.93, 1.03)	(0.88, 1.05)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_5}$	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_6}$	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_7}$	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_8}$	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_9}$	(0.93, 1.03)	(0.92, 1.01)	(0.999, 1.001)	(0.62, 1.000)

Table 5 Range of k₂ at the fixed value of k₁ for different populations.

Estimators	Population 1	Population 2	Population 3	Population 4
	($k_1=0.98)$	($k_1=0.97)$	($k_1=1.00)$	($k_1=0.80)$
tm	(–1.21,–2.08)	(–0.17,–2.09)	(–0.45,–1.99)	(–2.76, 1.15)
tr	(–0.93,–2.35)	(–0.75,–1.51)	(–1.00,–1.44)	(–2.09, 0.47)
treg	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
ts	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
tw	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
tsd	(–0.98,–2.30)	(–0.46,–1.79)	(–1.001,–1.43)	(–2.09, 0.47)
tre	(–1.35,–1.94)	(–0.62,–1.64)	(–0.941,–1.49)	(–2.29, 0.68)
tsk	(–1.26,–2.02)	(–0.22,–2.04)	(–1.04,–1.40)	(–2.09, 0.47)
$t_{u{p}_1}$	(–1.04,–2.24)	(–0.22,–2.03)	(–1.04,–1.39)	(–2.20, 0.62)
$t_{u{p}_2}$	(–1.35,–1.94)	(–0.34,–1.91)	(–1.05,–1.41)	(–2.70, 1.11)
$t_{s_1}$	(–1.25,–2.04)	(–0.75,–1.51)	(–1.08,–1.09)	(–2.40, 0.79)
tsn	(–1.35,–1.94)	(–0.62,–1.64)	(–0.95,–1.49)	(–2.29, 0.68)
tyk	(–1.35,–1.93)	(–0.62,–1.64)	(–0.95,–1.49)	(–2.22, 0.60)
tgk	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
tia	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
$t_{p_1}$	(–0.93,–2.36)	(–0.28,–1.98)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_2}$	(–0.93,–2.36)	(–0.40,–1.86)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_3}$	(–0.94,–2.36)	(–0.20,–2.06)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_4}$	(–0.93,–2.35)	(–0.23,–2.02)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_5}$	(–0.93,–2.36)	(–0.76,–1.49)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_6}$	(–0.93,–2.36)	(–0.78,–1.48)	(–1.01,–1.43)	(–2.09, 0.47)
$t_{p_7}$	(–0.93,–2.36)	(–0.78,–1.48)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_8}$	(–0.93,–2.36)	(–0.78,–1.48)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_9}$	(–0.93,–2.35)	(–0.65,–1.60)	(–1.01,–1.44)	(–2.09, 0.47)

Table 6 Bias, MSE and PRE for several estimators.

Estimators	Population 1			Population 2			Population 3			Population 4
	Bias	MSE	PRE	Bias	MSE	PRE	Bias	MSE	PRE	Bias	MSE	PRE
t m	0.00	3.1	100.0	0.00	0.0004	100.0	0.0000	0.55	100.00	0.00	142775.0	100.00
t r	0.17	7.0	45.2	0.00	0.0001	397.6	0.0012	0.25	212.79	–1.87	70220.4	203.32
t reg	0.00	1.6	189.8	0.00	0.0001	434.9	0.0000	0.23	235.79	0.00	70172.4	203.46
ts	0.03	1.6	189.8	0.00	0.0001	434.9	0.0004	0.23	235.79	–0.96	70172.4	203.46
tw	–0.00	1.6	189.8	0.00	0.0001	434.9	0.0000	0.23	235.79	0.00	70172.4	203.46
tsd	0.15	6.1	51.7	–0.00	0.0002	185.6	0.0012	0.25	212.96	–1.88	70220.6	203.32
tre	0.05	1.9	161.3	–0.00	0.0001	275.1	–0.0000	0.27	200.68	–9.84	89267.9	159.93
tsk	0.04	2.6	119.1	–0.00	0.0004	110.0	0.0010	0.25	218.91	–1.95	70224.6	203.31
$t_{u{p}_1}$	0.12	5.2	60.4	–0.00	0.0004	110.4	0.3943	32.55	1.69	–18.43	83781.9	170.41
$t_{u{p}_2}$	–0.00	2.0	158.3	–0.00	0.0003	141.0	–0.0008	0.34	158.12	–2.37	138101.3	103.38
$t_{s_1}$	0.05	2.8	112.85	–0.00	0.0001	396.8	0.0007	0.24	225.10	–17.26	100042.3	142.71
tsn	0.08	1.9	161.3	0.00	0.0001	275.1	0.0006	0.27	200.68	–0.93	89267.9	159.93
tyk	–0.04	1.9	162.7	–0.00	0.0001	276.8	–0.0030	0.27	200.69	–84.42	81632.0	174.90
tgk	0.03	1.6	189.8	0.00	0.0001	434.9	0.0004	0.23	235.79	–0.96	70172.4	203.46
tia	0.06	1.6	189.8	0.00	0.0001	434.9	0.0009	0.23	235.79	–1.92	70172.4	203.46
$t_{p_1}$	0.17	6.9	45.3	–0.00	0.0003	123.4	0.0012	0.25	212.80	–1.87	70220.4	203.32
$t_{p_2}$	0.17	6.9	45.3	–0.00	0.0002	158.7	0.0012	0.25	212.79	–1.87	70220.4	203.32
$t_{p_3}$	0.17	6.8	46.0	–0.00	0.0004	105.3	0.0012	0.25	212.75	–1.87	70220.4	203.32
$t_{p_4}$	0.17	6.9	45.6	–0.00	0.0004	113.3	0.0012	0.25	212.79	–1.87	70220.4	203.32
$t_{p_5}$	0.17	6.9	45.3	–0.00	0.0001	412.5	0.0012	0.25	212.81	–1.87	70220.4	203.32
$t_{p_6}$	0.17	6.9	45.3	0.00	0.0001	434.7	0.0012	0.25	212.79	–1.88	70220.4	203.32
$t_{p_7}$	0.17	6.9	45.3	0.00	0.0001	433.8	0.0012	0.25	212.79	–1.87	70220.4	203.32
$t_{p_8}$	0.17	6.9	45.3	–0.00	0.0001	434.7	0.0012	0.25	212.79	–1.88	70220.4	203.32
$t_{p_9}$	0.17	6.9	45.5	–0.00	0.0001	301.5	0.0012	0.25	212.99	–1.87	70220.4	203.32
T(1)	0.81	0.9	343.9	0.00	0.0000	936.8	0.0176	0.23	236.21	166.68	16408.4	870.12
T(2)	0.71	1.0	292.2	0.00	0.0001	437.9	0.0176	0.23	236.21	166.66	16418.3	869.60
T(3)	0.79	0.9	331.6	0.00	0.0000	466.6	0.0176	0.23	236.21	166.68	16408.8	870.11
T(4)	0.14	1.6	193.7	0.00	0.0001	436.4	0.0075	0.23	235.89	166.45	16507.6	864.90
T(5)	0.69	1.1	282.8	–0.00	0.0001	436.1	0.0173	0.23	236.20	166.67	16412.5	869.91
T(6)	0.28	1.5	202.4	0.00	0.0001	436.2	0.0163	0.23	236.16	166.28	16578.5	861.20
T(7)	0.60	1.2	255.6	–0.00	0.0001	436.1	0.0140	0.23	236.07	166.67	16410.8	870.00
T(8)	0.78	0.9	328.6	0.00	0.0001	442.0	0.0175	0.23	236.21	166.68	16408.6	870.12

Table 7 Bias, MSE, and PRE of several estimators for simulated normal population.

${\rho }_{\mathit{\text{xy}}}$	0.1			0.5			0.9
Estimators	Bias	MSE	PRE	Bias	MSE	PRE	Bias	MSE	PRE
t m	0.00000	0.0014	100.00	0.00000	0.0014	100.00	0.00000	0.0014	100.00
t r	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.17
t reg	0.00000	0.0014	100.00	0.00000	0.0011	123.14	0.00000	0.0002	489.86
ts	0.00005	0.0014	100.00	0.00001	0.0011	123.14	–0.00008	0.0002	489.86
tw	0.00000	0.0014	100.00	0.00000	0.0011	123.14	–0.00007	0.0002	489.86
tsd	0.00002	0.0018	75.31	0.00003	0.0012	122.20	–0.00001	0.0004	339.42
tre	0.00001	0.0015	91.55	–0.00000	0.0012	119.20	–0.00009	0.0007	186.90
tsk	0.00002	0.0018	76.35	0.00002	0.0012	122.54	–0.00001	0.0004	329.48
$t_{u{p}_1}$	0.00000	0.0015	92.93	–0.00003	0.0012	119.56	–0.00001	0.0008	172.05
$t_{u{p}_2}$	0.00000	0.0014	96.45	–0.00003	0.0012	113.59	–0.00001	0.0009	148.38
$t_{s_1}$	0.00000	0.0014	96.34	–0.00003	0.0012	114.69	–0.00001	0.0009	146.15
tsn	0.00001	0.0015	91.55	0.00002	0.0011	119.20	–0.00006	0.0007	186.90
tyk	–0.00007	0.0015	99.68	–0.00005	0.0011	125.53	–0.00003	0.0007	191.15
tgk	0.00000	0.0014	100.00	0.00001	0.0011	123.14	–0.00008	0.0002	489.86
tia	0.00000	0.0014	100.00	0.00003	0.0011	123.14	–0.00001	0.0002	489.86
$t_{p_1}$	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.03
$t_{p_2}$	0.00002	0.0019	72.89	0.00004	0.0012	121.47	–0.00001	0.0003	368.87
$t_{p_3}$	0.00002	0.0019	72.89	0.00004	0.0012	121.36	–0.00001	0.0003	374.87
$t_{p_4}$	0.00002	0.0019	72.89	0.00004	0.0012	121.27	–0.00001	0.0003	381.78
$t_{p_5}$	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.10
$t_{p_6}$	0.00002	0.0019	72.90	0.00004	0.0012	121.47	–0.00001	0.0003	369.01
$t_{p_7}$	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.03
$t_{p_8}$	0.00002	0.0018	77.78	0.00004	0.0012	121.47	–0.00001	0.0003	368.99
$t_{p_9}$	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.14
T(1)	0.00007	0.0010	128.89	0.00006	0.0009	150.97	0.00007	0.0001	852.63
T(2)	0.00006	0.0011	124.33	0.00005	0.0009	147.53	0.00006	0.0001	721.11
T(3)	0.00007	0.0011	127.24	0.00005	0.0009	149.75	0.00007	0.0001	798.36
T(4)	0.00006	0.0011	125.56	0.00005	0.0009	147.86	0.00006	0.0001	759.96
T(5)	0.00007	0.0010	128.88	0.00005	0.0009	150.08	0.00007	0.0001	787.86
T(6)	0.00006	0.0011	122.65	0.00005	0.0009	145.69	0.00005	0.0002	689.73
T(7)	0.00007	0.0010	128.88	0.00005	0.0009	150.48	0.00007	0.0001	820.51
T(8)	0.00007	0.0010	128.88	0.00005	0.0009	150.67	0.00007	0.0001	828.79

Table 8 Bias, MSE, and PRE of several estimators for simulated gamma population.

${\rho }_{\mathit{\text{xy}}}$	0.1			0.5			0.9
Estimators	Bias	MSE	PRE	Bias	MSE	PRE	Bias	MSE	PRE
t m	0.00000	0.0008	100.00	0.00000	0.0008	100.00	0.00000	0.0008	100.00
t r	0.00001	0.0009	87.63	–0.00005	0.0005	142.93	–0.00001	0.0002	325.40
t reg	0.00000	0.0007	103.81	0.00000	0.0004	142.94	0.00000	0.0001	536.95
ts	0.00002	0.0007	103.81	–0.00002	0.0004	142.94	–0.00001	0.0001	536.95
tw	0.00000	0.0007	103.81	0.00002	0.0004	142.94	–0.00001	0.0001	536.95
tsd	0.00001	0.0009	88.52	–0.00003	0.0005	142.91	–0.00009	0.0002	317.93
tre	0.00005	0.0008	102.40	–0.00002	0.0006	129.00	–0.00001	0.0004	173.34
tsk	0.00008	0.0008	96.36	–0.00003	0.0006	140.12	–0.00008	0.0003	239.92
$t_{u{p}_1}$	0.00006	0.0014	56.95	0.00004	0.0008	98.34	–0.00001	0.0002	491.24
$t_{u{p}_2}$	–0.00001	0.0008	103.78	–0.00004	0.0006	117.86	–0.00009	0.0006	132.61
$t_{s_1}$	0.00009	0.0008	94.91	–0.00002	0.0005	140.92	–0.00001	0.0003	257.90
tsn	0.00008	0.0008	102.40	–0.00002	0.0006	129.00	–0.00006	0.0004	173.34
tyk	–0.00006	0.0008	102.50	–0.00005	0.0006	129.08	–0.00003	0.0004	173.39
tgk	0.00002	0.0007	103.81	–0.00002	0.0004	142.94	–0.00001	0.0001	536.95
tia	0.00005	0.0007	103.81	–0.00005	0.0004	142.94	–0.00002	0.0001	536.95
$t_{p_1}$	0.00001	0.0009	87.65	–0.00008	0.0005	142.93	–0.00001	0.0002	324.68
$t_{p_2}$	0.00001	0.0009	87.64	–0.00006	0.0005	142.93	–0.00001	0.0002	325.24
$t_{p_3}$	0.00001	0.0009	87.73	–0.00001	0.0005	142.93	–0.00001	0.0002	321.18
$t_{p_4}$	0.00001	0.0009	87.66	–0.00008	0.0005	142.93	–0.00001	0.0002	324.45
$t_{p_5}$	0.00001	0.0009	87.63	–0.00006	0.0005	142.93	–0.00001	0.0002	325.24
$t_{p_6}$	0.00001	0.0009	87.64	–0.00006	0.0005	142.93	–0.00001	0.0002	325.36
$t_{p_7}$	0.00001	0.0009	87.63	–0.00005	0.0005	142.93	–0.00001	0.0002	325.36
$t_{p_8}$	0.00001	0.0009	87.65	–0.00006	0.0005	142.93	–0.00001	0.0002	325.36
$t_{p_9}$	0.00001	0.0009	87.64	–0.00007	0.0005	142.93	–0.00001	0.0002	324.70
T(1)	0.00007	0.0007	104.25	0.00005	0.0004	143.50	0.00005	0.0001	539.96
T(2)	0.00007	0.0007	104.18	0.00005	0.0004	143.34	0.00004	0.0001	538.89
T(3)	0.00007	0.0007	104.20	0.00005	0.0004	143.35	0.00004	0.0001	539.01
T(4)	0.00001	0.0007	103.92	0.00006	0.0004	143.04	0.00004	0.0001	537.62
T(5)	0.00007	0.0007	104.19	0.00005	0.0004	143.31	0.00004	0.0001	538.46
T(6)	0.00004	0.0007	104.02	0.00003	0.0004	143.17	0.00002	0.0001	537.55
T(7)	0.00006	0.0007	104.14	0.00003	0.0004	143.20	0.00001	0.0001	537.49
T(8)	0.00007	0.0007	104.20	0.00005	0.0004	143.34	0.00004	0.0001	538.87

Sarndal CE, Swensson B, Wretman J. 2003. Model assisted survey sampling. New York (NY): Springer-Verlag.

 Google Scholar

Singh S. 2003b. Advanced sampling theory with applications: how Michael “selected” Amy. Vol. 2. Dordrecht: Springer.

 Google Scholar

Alomair MA, Shahzad U. 2023. Compromised-imputation and EWMA-based memory-type mean estimators using quantile regression. Symmetry. 15:1888.

 Google Scholar

Kadilar C, Cingi H. 2003. Ratio estimators in stratified random sampling. Biometrical J. 45:218–225.

 Web of Science ®Google Scholar