Table 1 Averages of estimated expected values of the superpopulation mean \(\overline {\widehat {E(\mu )}}\) and their estimated analytical variances \(\overline {\widehat {V}}\Big [\widehat {E(\mu )}\Big ]\), corresponding MSE \(MSE\Big [\widehat {E(\mu )}\Big ]\), empirical variances \(V_{emp}\Big [\widehat {E(\mu )}\Big ]\), and estimated relative bias \(\widehat {RelBIAS}\): Case A – standard model-based inference, Case B – hybrid inference, Case C −− model-based inference with hierarchical modelling: C.1 – two-stage lest squares regression, C.2 – hierarchical model-based estimation, C.3 – simplified hierarchical model-based estimation

m=50 g r i d c e l l s, M=500 g r i d c e l l s

A

102.96

159.05

156.97

157.43

155.72

2.14

0.80

B

104.74

47.74

49.05

53.12

52.90

-9.77

-7.28

C

C.1

104.65

43.48

44.56

48.37

48.23

-9.84

-7.60

C.2

43.73

44.99

-9.33

-6.72

C.3

13.46

13.46

-72.08

-72.08

m=100 g r i d c e l l s, M=1000 g r i d c e l l s

A

103.68

76.78

76.27

75.67

75.33

1.93

1.25

B

104.54

23.82

24.37

25.15

25.07

-5.00

-2.80

C

C.1

104.48

21.69

22.16

22.69

22.64

-4.19

-2.15

C.2

21.75

22.27

-3.92

-1.61

C.3

6.54

6.54

-71.12

-71.12

m=500 g r i d c e l l s, M=5000 g r i d c e l l s

A

104.13

15.01

14.99

14.98

14.96

0.32

0.20

B

104.30

4.75

4.78

4.80

4.80

-1.12

-0.46

C

C.1

104.29

4.32

4.35

4.31

4.31

0.28

0.92

C.2

4.33

4.36

0.33

1.05

C.3

1.27

1.27

-70.43

-70.43

m=1000 g r i d c e l l s, M=10000 g r i d c e l l s

A

104.21

7.48

7.48

7.44

7.44

0.60

0.53

B

104.30

2.38

2.39

2.41

2.41

-1.36

-0.95

C

C.1

104.29

2.17

2.17

2.17

2.17

-0.12

0.29

C.2

2.17

2.18

-0.09

0.36

C.3

0.64

0.64

-70.69

-70.69