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▲✐♥❡❛r r❡❣r❡ss✐♦♥

❙t❛t✐st✐❝❛❧ ♠♦❞❡❧❧✐♥❣

• ✐❧❧❡s ●✉✐❧❧♦t

❣✐❣✉❅❞t✉✳❞❦

❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶ ✴ ✸✸

❊①❛♠♣❧❡

❊①❛♠♣❧❡

❈♦♥❝❡♥tr❛t✐♦♥ ♦❢ ❉❉❚ ✭❛ t♦①✐❝ ❝❤❡♠✐❝❛❧✮ ✐♥ ✶✺ ♣✐❦❡ ✜s❤ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✜s❤ ❛❣❡✳✳✳✳ ❙❡❡ ✜❧❡ ♣✐❦❡❴❞❛t❛✳t①t ✐♥ ❞❛t❛ ❢♦❧❞❡r✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷ ✴ ✸✸

❊①❛♠♣❧❡

❱❛r✐♦✉s q✉❡st✐♦♥s

❉♦❡s ❝♦♥❝❡♥tr❛t✐♦♥ ✐♥❝r❡❛s❡ ✇✐t❤ ❛❣❡❄ ❍♦✇ ♠✉❝❤❄ P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ✭❛❦❛ ✐♥❢❡r❡♥❝❡✮ ❍♦✇ ♠✉❝❤ ❝♦♥✜❞❡♥❝❡ s❤♦✉❧❞ ✇❡ ♣❧❛❝❡ ✐♥ t❤❡ ❛♥s✇❡r❄ ❚❡st✐♥❣ s✐❣♥✐✜❝❛♥❝❡✱ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ❲❤❛t ✐s t❤❡ ❛✈❡r❛❣❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ❛ ✸✳✺ ♦r ✽ ②❡❛r ♦❧❞ ✜s❤❄ Pr❡❞✐❝t✐♦♥

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✸ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥

❊♠♣✐r✐❝❛❧ ❝♦✈❛r✐❛♥❝❡ ❛♥❞ ❝♦rr❡❧❛t✐♦♥ ■

❉❡✜♥✐t✐♦♥✿ ❊♠♣✐r✐❝❛❧ ❝♦✈❛r✐❛♥❝❡ Cov(x, y) = 1 n

• i

(xi − ¯ x)(yi − ¯ y) ❚❡♥❞s t♦ ❜❡ ❧❛r❣❡ ✇❤❡♥ xi ❛♥❞ yi ❛r❡ ❧❛r❣❡ s✐♠✉❧t❛♥❡♦✉s❧②✳ ❍❡♥❝❡ q✉❛♥t✐✜❡s ❤♦✇ ♠✉❝❤ x ❛♥❞ y ✏❝♦✲✈❛r② t♦❣❡t❤❡r✑✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ✐s s❝❛❧❡✲❞❡♣❡♥❞❡♥t✿ Cov(ax, y) = aCov(x, y)

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✹ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥

❊♠♣✐r✐❝❛❧ ❝♦✈❛r✐❛♥❝❡ ❛♥❞ ❝♦rr❡❧❛t✐♦♥ ■■

❉❡✜♥✐t✐♦♥✿ ❊♠♣✐r✐❝❛❧ ❝♦rr❡❧❛t✐♦♥ ✭❛❦❛ P❡❛rs♦♥✬s ❝♦rr❡❧❛t✐♦♥ ❝♦❡❢✳✮ Cor(x, y) =

• i(xi − ¯

x)(yi − ¯ y)

• i(xi − ¯

x)2

i(yi − ¯

y)2 ❈♦r✳ ❂ ❈♦✈✳ r❡s❝❛❧❡❞ ❜② st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥s

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✺ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥

❈♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t✿ ✐♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ♣✐t❢❛❧❧s

❈♦rr❡❧❛t✐♦♥ ❢♦r ✈❛r✐♦✉s ❞❛t❛ ♣❛tt❡r♥s ✭r❡♣r✐♥❡t❞ ❢r♦♠ ✇✐❦✐♣❡❞✐❛✮

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✻ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥

❚❤❡ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t ρ ❞♦❡s ♥♦t t❡❧❧ t❤❡ ✇❤♦❧❡ st♦r②

❆❧❧ ❢♦✉r s❡ts ❤❛✈❡ ✐❞❡♥t✐❝❛❧ ρ ≈ 0.816✱ ❜✉t ✈❛r② ❝♦♥s✐❞❡r❛❜❧② ✇❤❡♥ ❣r❛♣❤❡❞✳ ❉❛t❛ ♣❧♦tt❡❞ ❛❜♦✈❡ ❛r❡ s②♥t❤❡t✐❝ ❞❛t❛ ♠❛❞❡ ✉♣ ❜② ❋✳ ❆♥s❝♦♠❜❡ ✐♥ ✶✾✼✸ t♦ ✐❧❧✉str❛t❡ t❤❡ ♣✐t❢❛❧❧s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ρ✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✼ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥

❙tr✉❝t✉r❡ ❡✛❡❝t ✭❛❦❛ ❙✐♠♣s♦♥✬s ♣❛r❛❞♦①✮

◗✉✐❝❦ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ρ ✈❛❧✉❡s ❝❛♥ ❧❡❛❞ t♦ ❡rr♦♥❡❛♦✉s ❝♦♥❝❧✉s✐♦♥s✳ ❙❡❡ ❛❧s♦ ❙✐♠♣s♦♥✬s ❡✛❡❝t ♦♥ ✇✐❦✐♣❡❞✐❛ ❢♦r ❢✉rt❤❡r ❞❡t❛✐❧ ❛♥❞ ❡①❛♠♣❧❡s✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✽ ✴ ✸✸

❚❤❡ s✐♠♣❧❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

❚❤❡ s✐♠♣❧❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆♦t❛t✐♦♥✿ x1, ..., xn ❛❣❡s ♦❢ t❤❡ ✈❛r✐♦✉s ✐♥❞✐✈✐❞✐❞✉❛❧s✱ y1, ..., yn ❉❉❚ ❝♦♥❝❡♥tr❛t✐♦♥s ❚❤❡ s✐♠♣❧❡ ❧✐♥❡❛r ♠♦❞❡❧ Yi = axi + b + εi t❤❡ xis ❛r❡ ❞❡t❡r♠✐♥✐st✐❝ ✈❛r✐❛❜❧❡s

✭❛ s♦♠❡❤♦✇ ❛r❜✐tr❛r② ♠♦❞❡❧❧✐♥❣ ❝❤♦✐❝❡✮

a ❛♥❞ b ❛r❡ ✉♥❦♥♦✇♥ ❞❡t❡r♠✐♥✐st✐❝ ❝♦❡✛✜❝✐❡♥ts t❤❡ εi✬s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❡❛❧✐s❛t✐♦♥s ♦❢ ❛ N(0, σ2) ✈❛r✐❛❜❧❡

✭♠❛❞❡ ❢♦r ❝♦♥✈✐❡♥✐❡♥❝❡✱ ❝♦♥s✐st❡♥❝② ✇✐t❤ ❞❛t❛ ❤❛s t♦ ❜❡ ❝❤❡❝❦❡❞✱ s❡❡ ❧❛t❡r✮

❚❤❡r❡ ❛r❡ t❤r❡❡ ♣❛r❛♠❡t❡rs ✐♥ t❤✐s ♣r♦❜❧❡♠✿ (a, b, σ) = θ

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✾ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

▲❡❛st sq✉❛r❡ ❡rr♦r ❡st✐♠❛t✐♦♥

❋♦r ❛r❜✐tr❛r② ✈❛❧✉❡s a ❛♥❞ b✱ ei = yi − axi − b ♠❡❛s✉r❡s t❤❡ ❡rr♦r ♠❛❞❡ ❜② t❤❡ ❧✐♥❡❛r ♠♦❞❡❧ ♦♥ ♦❜s✳ i ❆ ❣♦♦❞ ♠♦❞❡❧ s❤♦✉❧❞ ②✐❡❧❞ ❧♦✇ ❡rr♦rs ♦♥ ❛❧❧ ♦❜s❡r✈❛t✐♦♥s ❉❡✜♥✐t✐♦♥✿ t❤❡ ▲❡❛st ❙q✉❛r❡ ❡st✐♠❛t♦r ❚❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡rs (a, b) ❛r❡ ❡st✐♠❛t❡❞ ❛s (a, b)LS t❤❛t ❥♦✐♥t❧② ♠✐♥✐♠✐③❡

i(yi − axi − b)2✳

■♥ ♠❛t❤ st②❧❡✿

• (a, b)LS = Argmina,b
• i

(yi − axi − b)2

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✵ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ❧❡❝t✉r❡ ♦♥ ❙t❛t✐st✐❝❛❧ ❊st✐♠❛t✐♦♥ ✭❧❡❝t✉r❡ ✶✮

❚❤❡ ✈❡❝t♦r (a, b)LS ✐s ❛♥ ❡st✐♠❛t❡ ♦❢ (a, b) ❚❤❡ ♣r♦❝❡❞✉r❡ ❉❛t❛ − → (a, b)LS✱ ✐✳❡✳ t❤❡ ❣❡♥❡r✐❝ ♣r♦❝❡ss ❛ss♦❝✐❛t✐♥❣ ❛♥ ❡st✐♠❛t❡ t♦ ❛ ❞❛t❛s❡t ✐s ❛♥ ❡st✐♠❛t♦r ❚❤✐s ♣r♦❝❡❞✉r❡ ♦r ❢✉♥❝t✐♦♥ ✐s ❞❡t❡r♠✐♥✐st✐❝ ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡ s❛♠❡ ❞❛t❛s❡t ✇✐❧❧ ②✐❡❧❞ t❤❡ s❛♠❡ ❡st✐♠❛t❡ ■♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ t❤✐s ❝♦✉rs❡✱ ❉❛t❛ = (xi, yi)i=1,...,n ✇✐t❤ Yi = axi + b + εi ❛♥❞ ✇❤❡r❡ εi ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❉❛t❛ ❛r❡ r❛♥❞♦♠ t❤❡r❡❢♦r❡ (a, b)LS s❤♦✉❧❞ ❜❡ s❡❡♥ ❛s r❛♥❞♦♠✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✶ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❘❡♠❛r❦s ❛♥❞ q✉❡st✐♦♥s ♦♥ t❤❡ ▲❡❛st ❙q✉❛r❡ ♣r✐♥❝✐♣❧❡

❲❤❛t ✐❢ ✇❡ ❛tt❡♠♣t t♦ ♠✐♥✐♠✐③❡ |yi − axi − b| ❄ ❲❤❛t ✐❢ ✇❡ s✇❛♣ x ❛♥❞ y❄ ❲❤❛t ✐❢ t❤❡ ❞❛t❛ ♣♦✐♥ts ❛r❡ ❛♣♣r♦①✐♠❛t❡❧② ❧♦❝❛t❡❞ ♦♥ ❛ ❝✐r❝❧❡❄ ❲❤② t❤❡ sq✉❛r❡❞ ❡rr♦r❄

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✷ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❊①♣❧✐❝✐t ❡①♣r❡ss✐♦♥ ♦❢ (a, b)LS

• (a, b)LS = Argmina,b
• i

(yi − axi − b)2 SSE(a, b) =

i(yi −axi −b)2 ✐s ❛ s❡❝♦♥❞ ♦r❞❡r ♣♦❧②♥♦♠✐❛❧ ✐♥ a ❛♥❞ b

❙♦❧✈✐♥❣ ∂ ∂aSSE(a, b) = 0 ❛♥❞ ∂ ∂bSSE(a, b) = 0 ②✐❡❧❞s✿ ❊①♣r❡ss✐♦♥ ♦❢ ▼❙❊ ❡st✐♠❛t❡s✿ ˆ aLS =

• i(xi − ¯

x)(yi − ¯ y)

• i(xi − ¯

x)2 ❛♥❞ ˆ bLS = ¯ y − ˆ aLS ¯ x

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✸ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❈♦♠♣✉t❛t✐♦♥❛❧ ❞❡t❛✐❧ ■

❩❡r♦✲✐♥❣ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ✐♥ a ❛♥❞ b ✇❡ ❣❡t ∂ ∂aSSE(a, b) = −2

• i

xi(yi − axi − b) = 0 ✭✶✮ ∂ ∂bSSE(a, b) = −2

• i

(yi − axi − b) = 0 ✭✷✮ ❍❡♥❝❡ a

• i

x2

i + b

• xi

=

• i

xiyi ✭✸✮ a

• i

xi + nb =

• i

yi ✭✹✮ ❲❡ ❤❛✈❡ ❛ ❧✐♥❡❛r s②st❡♠ ✐♥ a ❛♥❞ b✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✹ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❈♦♠♣✉t❛t✐♦♥❛❧ ❞❡t❛✐❧ ■■

❆ s✉❜s✐t✉t✐♦♥ ②✐❡❧❞s✿ ˆ aLS = n

i xiyi − xi

yi n x2

i − ( i xi)2

✭✺✮ ❛♥❞ ˆ bLS = 1/n

i xiyi − ¯

x¯ y 1/n x2

i − ¯

x2 ✭✻✮

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✺ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❊st✐♠❛t✐♥❣ t❤❡ ✈❛r✐❛♥❝❡ σ2 ♦❢ t❤❡ r❡s✐❞✉❛❧s ■

❚❤❡ ▲❙❊ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ❛♥ ❡st✐♠❛t❡ ♦❢ σ2✳ ❆ r❡❛s♦♥❛❜❧❡ ✐❞❡❛ ❝♦✉❧❞ ❜❡ t♦ ❞❡✜♥❡ ˆ εi = yi − ˆ aLSx + ˆ bLS ❛♥❞ ❡st✐♠❛t❡ σ2 ❛s 1 n

• i

ˆ ε2

i ✳

• ❚❤✐s ✇♦✉❧❞ ❧❡❛❞ t♦ ❛ ❜✐❛s❡❞ ❡st✐♠❛t♦r
• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✻ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❊st✐♠❛t✐♥❣ t❤❡ ✈❛r✐❛♥❝❡ σ2 ♦❢ t❤❡ r❡s✐❞✉❛❧s ■■

❯♥❜✐❛s❡❞ ❡st✐♠❛t❡ ♦❢ σ2✿ ❲❡ ❞❡✜♥❡ ˆ σ2 = 1 n − 2

• i

ˆ ε2

i

✇✐t❤ ˆ εi = yi − ˆ aLSxi + ˆ bLS ■t ✐s ❛♥ ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦r ♦❢ σ2✱ ✐✳❡✳ E[ˆ σ2] = σ2

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✼ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❈♦♥♥❡❝t✐♦♥ t♦ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ■

❘❡♠❡♠❜❡r✿ ✐♥ t❤❡ ▲✐♥❡❛r ▼♦❞❡❧✱ t❤❡ xi✬s ❛r❡ ❝♦♥s✐❞❡r❡❞ ❞❡t❡r♠✐♥✐st✐❝✳ ❆♥❞ ♦✉r ♠♦❞❡❧ s❛②s✿ Yi = axi + b + εi ❛♥❞ (εi)i=1,...,n

✐✳✐✳❞

∼ N(0, σ2)✳

✭✐✳✐✳❞ st❛♥❞s ❢♦r ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✮

❚❤❡ t✇♦ ❧✐♥❡s ❛❜♦✈❡ ❝❛♥ ❜❡ r❡✲✇r✐tt❡♥ ❡q✉✐✈❛❧❡♥t❧② ❛s (Yi)i=1,...,n

✐♥❞❡♣✳

∼ N(axi + b, σ2)

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✽ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❈♦♥♥❡❝t✐♦♥ t♦ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ■■

❚❤❡ ❞❡♥s✐t② ♦❢ ♣r♦❜❛❜✐❧✐t② ♦❢ Yi ✐s ●❛✉ss✐❛♥ ✇✐t❤ ♠❡❛♥ µi = axi + b ❛♥❞ ✈❛r✐❛♥❝❡ σ2✿ fYi(y) = 1 σ √ 2π exp

• −1

2 y − axi − b σ 2 ❚❤❡ ❧✐❦❡❧✐❤♦♦❞ ✐♥ t❤✐s ♣r♦❜❧❡♠ ✐s L(y1, ..., yn; a, b, σ) =

n

• i=1

fYi(yi) =

n

• i=1

1 σ √ 2π exp

• −1

2 yi − axi − b σ 2

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✾ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❈♦♥♥❡❝t✐♦♥ t♦ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ■■■

❚❤❡ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞ ✭✉♣ t♦ ❛♥ ❛❞❞✐t✐✈❡ ❝♦♥st❛♥t✮ ✐s✿ l(a, b, σ) = ln L(a, b, σ) = −n ln σ − 1/2

• i

yi − axi − b σ 2 ❚❤❡ ▼▲❊ ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❜② ③❡r♦✲✐♥❣

∂ ∂al(a, b, σ) ✱ ∂ ∂bl(a, b, σ) ❛♥❞ ∂ ∂σl(a, b, σ)✿

∂ ∂al(a, b, σ) = 1 σ2

• i

xi(yi − axi − b) = 0 ✭✼✮ ∂ ∂bl(a, b, σ) = 1 σ2

• i

(yi − axi − b) = 0 ✭✽✮ ∂ ∂σl(a, b, σ) = −n σ + 1 σ3

• i

(yi − axi − b)2 = 0 ✭✾✮

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✵ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥

❈♦♥♥❡❝t✐♦♥ t♦ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ■❱

■♥ ❊q✳ ✭✼✲✽✮✱ ✇❡ r❡❝♦❣♥✐③❡ t❤❡ ❡①♣r❡ss✐♦♥s ✐♥ t❤❡ ▲❙ ❡st✐♠❛t♦r✳ ❍❡♥❝❡ ˆ aML = ˆ aLS ❛♥❞ ˆ bML = ˆ bLS P❧✉❣❣✐♥❣ ˆ aML ❛♥❞ ˆ bML ✐♥ ❊q✳ ✭✾✮ ②✐❡❧❞s✿ ˆ σ2

ML = 1

n

• i

(yi − ˆ aMLxi − ˆ bML)2 ❲❤✐❝❤ ✐s ❜✐❛s❡❞✳✳✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✶ ✴ ✸✸

• ❡♦♠❡tr✐❝ ✐♥t❡r♣r❡t❛t✐♦♥
• ❡♦♠❡tr✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❧✐♥❡❛r r❡❣r❡ss✐♦♥

x = (x1, ..., xn) ❛♥❞ y = (y1, ..., yn) ❛r❡ ✈❡❝t♦rs ✐♥ Rn t❤❡ ✈❛❧✉❡s axi + b ❝❛♥ ❜❡ s❡❡♥ ❛s t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ ✈❡❝t♦r ax + b1 ✐♥ Rn

• i(yi − axi − b)2 ✐s t❤❡ sq✉❛r❡ ♦❢ t❤❡ ♥♦r♠ ♦❢ y − ax − b1

ˆ ax + ˆ b1 ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ y ♦♥ Span(1, x) Cor(x, y) ✐s t❤❡ ❝♦s✐♥❡ ♦❢ t❤❡ ❛♥❣❧❡ ❢♦r♠❡❞ ❜② x ❛♥❞ y ✐♥ Rn

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✷ ✴ ✸✸

• ♦♦❞♥❡ss ♦❢ ✜t
• ♦♦❞♥❡ss ♦❢ ✜t

❚❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♠♦❞❡❧ ❝❛♥ ❜❡ ❛ss❡ss❡❞ ❜② t❤❡ ❝♦❡✣❝❡♥t ♦❢ ❞❡t❡r♠✐♥❛t✐♦♥✿ ❈♦❡✣❝✐❡♥t ♦❢ ❞❡t❡r♠✐♥❛t✐♦♥✿ R2 ˆ =1 − SSerr SStot ✇✐t❤ SStot =

i(yi − ¯

y)2 ❛♥❞ SSerr =

i(ˆ

yi − yi)2 0 ≤ R2 ≤ 1 ✏●♦♦❞✑ ♠♦❞❡❧ ⇔ ❧♦✇ SSerr ⇔ ❤✐❣❤ R2 ■❢ t❤❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ✐♥❝❧✉❞❡s ❛♥ ✐♥t❡r❝❡♣t✱ t❤❡♥ R2 = ρ2

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✸ ✴ ✸✸

❚❡st✐♥❣ ♣❛r❛♠❡t❡rs

❚❡st✐♥❣ H0 : a = a0 ■

❲❡ ❛r❡ ♦❢t❡♥ ✐♥t❡r❡st❡❞ ✐♥ ❛ss❡ss✐♥❣ ✐❢ a ✐s s✐❣♥✐✜❝❛♥t❧② ❞✐✛❡r❡♥t ❢r♦♠ ❛ ♣❛rt✐❝✉❧❛r ✈❛❧✉❡ a0✳ ❖❢t❡♥ H0 : a = 0 ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ x ❛♥❞ y✳ ❚❤❡ q✉❡st✐♦♥ ✐s✿ s❤♦✉❧❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ˆ a ❛♥❞ a0 ❜❡ ❝♦♥s✐❞❡r❡❞ ❧❛r❣❡ ❡♥♦✉❣❤ t♦ r❡❥❡❝t H0❄

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✹ ✴ ✸✸

❚❡st✐♥❣ ♣❛r❛♠❡t❡rs

❚❡st✐♥❣ H0 : a = a0 ■■

❚❤❡ ❙t✉❞❡♥t t❡st ❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥s ❣✐✈❡♥

❤❡r❡ t❤❡♥

T = ˆ a − a0

• sd(ˆ

a) ∼ Stn−2 H0 ✐s r❡❥❡❝t❡❞ ❛t ❧❡✈❡❧ α ✐❢ |t| ✐s ❧❛r❣❡r t❤❛♥ t❤❡ q✉❛♥t✐❧❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 1 − α/2 ♦❢ ❛ Stn−2 ❞✐str✐❜✉t✐♦♥

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✺ ✴ ✸✸

❈❤❡❝❦✐♥❣ ♠♦❞❡❧ ❛ss✉♠♣t✐♦♥s

❈❤❡❝❦✐♥❣ ♠♦❞❡❧ ❛ss✉♠♣t✐♦♥s

❚❤❡ εi✬s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ✐✳✐✳❞✳ ■❢ (a, b) ❡st✐♠❛t❡s (a, b) ❝♦rr❡❝t❧②✱ t❤❡ ˆ εi✬s s❤♦✉❧❞ ❜❡ ❝❧♦s❡ t♦ ✐✳✐✳❞✳ ❆ ♣❧♦t ♦❢ (ˆ εi)i=1,...,n ❛❣❛✐♥st (xi)i=1,...,n s❤♦✉❧❞ ♥♦t ❞✐s♣❧❛② ❛♥② ♣❛tt❡r♥✳ ❱✐s✉❛❧ ❝❤❡❝❦ ♦❢ r❡s✐❞✉❛❧s ❱✐s✉❛❧ ❝❤❡❝❦ ♦❢ st❛♥❞❛r❞✐s❡❞ r❡s✐❞✉❛❧s ♣❧♦t✭❧♠✭✳✳✳✮✮ ✐♥ ❘

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✻ ✴ ✸✸

Pr❡❞✐❝t✐♦♥ ♦✉t✲♦❢ s❛♠♣❧❡ ✈❛❧✉❡ ✉♥❞❡r ❛ ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ♠♦❞❡❧

Pr❡❞✐❝t✐♦♥

❲❤❛t ✈❛❧✉❡ ynew s❤♦✉❧❞ ❜❡ ❡①♣❡❝t❡❞ ❢♦r ❛♥ ❡①tr❛ ✐♥❞✐✈✐❞✉❛❧ ✇✐t❤ ♦❜s❡r✈❡❞ ❡①♣❧❛♥❛t♦r② ✈❛r✐❛❜❧❡ xnew❄ ❉❡✜♥✐t✐♦♥✿ ♣r❡❞✐❝t✐♦♥ ynew = ˆ axnew + ˆ b ❙tr❛✐❣❤t❢♦r✇❛r❞ ✐♥ ❘✱ s❡❡ ❛❧s♦ ✉s❡ ♦❢ t❤❡ ❣❡♥❡r✐❝ ❢✉♥❝t✐♦♥ ♣r❡❞✐❝t✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✼ ✴ ✸✸

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ✐♥ ♣r❛❝t✐❝❡ ✇✐t❤ ❘

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ✐♥ ♣r❛❝t✐❝❡ ✇✐t❤ ❘

❆ss✉♠✐♥❣ ❞❛t❛ ♦❜❥❡❝ts ❛r❡ ♥❛♠❡❞ ① ❛♥❞ ② ✐♥ ②♦✉r ❘ s❡ss✐♦♥ ★ ❢✐t ❛ ❧✐♥❡❛r ♠♦❞❡❧ ❛♥❞ st♦r❡ t❤❡ ✭❧♦♥❣✮ ♦✉t♣✉t ❧✐st r❡s✳❧♠ ❂ ❧♠✭❢♦r♠✉❧❛ ❂ ② ⑦ ①✮ ★ ❡①tr❛❝t ❡st✐♠❛t❡❞ ❝♦❡❢ r❡s✳❝♦❡❢ ❂ ❝♦❡❢❢✐❝✐❡♥ts✭r❡s✳❧♠✮

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✽ ✴ ✸✸

❇❛❝❦ t♦ ♣✐❦❡s

❘ ❝♦❞❡ t♦ ✜t ❛ ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ♦♥ t❤❡ ♣✐❦❡ ❞❛t❛

♣✐❦❡❂r❡❛❞✳t❛❜❧❡✭✧❤tt♣✿✴✴✇✇✇✷✳✐♠♠✳❞t✉✳❞❦✴❝♦✉rs❡s✴✵✷✹✶✽✴❧❡❝t✉r❡✸❴s✐♠♣❧❡❴r❡❣r❡ss✐♦♥✴❞❛t❛✴♣✐❦❡❴❞❛t❛✳t①t✧✱ ❤❡❛❞❡r❂❚❘❯❊✮ ★★ ❢✐tt✐♥❣ r❡❣r❡ss✐♦♥ ❧✐♥❡ ❧♠✳r❡s ❂ ❧♠✭❢♦r♠✉❧❛❂❉❉❚⑦✶✰❆❣❡✱❞❛t❛❂♣✐❦❡✮

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷✾ ✴ ✸✸

❇❛❝❦ t♦ ♣✐❦❡s

P✐❦❡ ❞❛t❛ ❛♥❛❧②s✐s ♦✉t♣✉t ✐♥ ❘

★★ ❞✐s♣❧❛② t❤❡ ❘ ♦❜❥❡❝t ❧♠✳r❡s s✉♠♠❛r②✭❧♠✳r❡s✮ ❈❛❧❧✿ ❧♠✭❢♦r♠✉❧❛ ❂ ❉❉❚ ⑦ ✶ ✰ ❆❣❡✱ ❞❛t❛ ❂ ♣✐❦❡✮ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✵✳✷✹✶✸✸ ✲✵✳✶✵✺✵✵ ✵✳✵✶✶✸✸ ✵✳✵✽✸✵✵ ✵✳✸✵✼✸✸ ❈♦❡❢❢✐❝✐❡♥ts✿ ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r t ✈❛❧✉❡ Pr✭❃⑤t⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✵✳✷✸✺✸✸ ✵✳✶✶✷✻✾ ✲✷✳✵✽✽ ✵✳✵✺✼ ✳ ❆❣❡ ✵✳✶✼✶✸✸ ✵✳✵✷✻✺✻ ✻✳✹✺✵ ✷✳✶✻❡✲✵✺ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ❵✯✯✯✬ ✵✳✵✵✶ ❵✯✯✬ ✵✳✵✶ ❵✯✬ ✵✳✵✺ ❵✳✬ ✵✳✶ ❵ ✬ ✶ ❘❡s✐❞✉❛❧ st❛♥❞❛r❞ ❡rr♦r✿ ✵✳✶✹✺✺ ♦♥ ✶✸ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ▼✉❧t✐♣❧❡ ❘✲sq✉❛r❡❞✿ ✵✳✼✻✶✾✱ ❆❞❥✉st❡❞ ❘✲sq✉❛r❡❞✿ ✵✳✼✹✸✻ ❋✲st❛t✐st✐❝✿ ✹✶✳✻✶ ♦♥ ✶ ❛♥❞ ✶✸ ❉❋✱ ♣✲✈❛❧✉❡✿ ✷✳✶✻✺❡✲✵✺

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✸✵ ✴ ✸✸

❇❛❝❦ t♦ ♣✐❦❡s

❘ ❝♦❞❡ ▲✐♥❦ t♦ t❤❡ ❘ s❝r✐♣t ✉s❡❞ ✐♥ t❤✐s ❧❡❝t✉r❡ ✭❛♥❞ ♠♦r❡✮

✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✸✶ ✴ ✸✸

❊①❡r❝✐s❡

❊①❡r❝✐s❡

❇✐♥❣❤❛♠ ❛♥❞ ❋r②✱ ❡①❡r❝✐s❡ ✶✳✸ ♣✳ ✷✾✳ ❉❛t❛ ✜❧❡ ❤❡r❡

❍✐♥ts✿ ✲ ❞♦✇♥❧♦❛❞ ✇✐t❤ ❞♦✇♥❧♦❛❞✳❢✐❧❡✭✉r❧❂✧✧✱ ❞❡st❢✐❧❡❂✧✳✴r✉♥♥✐♥❣✳t①t✧✮ ✲ r❡❛❞ ✐♥ ❘ ✇✐t❤ r❡❛❞✳t❛❜❧❡✭✧✳✴r✉♥♥✐♥❣✳t①t✧✱❤❡❛❞❡r❂❚❘❯❊✮ ✲ ♠♦❞❡❧ ✜t ♦♥ ❧♦❣ ❞❛t❛ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❧♠✭❧♦❣✭②✮ ⑦ ❧♦❣✭①✮✮ ✲ ✉s❡ ❘ ❢✉♥❝t✐♦♥ ❝♦♥❢✐♥t ❢♦r ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧

❙❤❡❛t❤❡r✱ ❡①❡r❝✐s❡ ✶ ♣✳ ✸✽ ❉❛t❛ ❛r❡ ❛✈❛✐❧❛❜❧❡ ♦♥ t❤❡ ❜♦♦❦ ✇❡❜ s✐t❡ ❛s ✜❧❡ ♣❧❛②❜✐❧❧✳❝s✈

❍✐♥ts✿ ❋♦r t❡st✐♥❣ β0 = 10000✱ ✉s❡ ❢✉♥❝t✐♦♥ t❡st✳❝♦❡❢ ❛✈❛✐❧❛❜❧❡ ❢r♦♠ ❤❡r❡✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✸✷ ✴ ✸✸

❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s

❙✉❣❣❡st❡❞ r❡❛❞✐♥❣ ❈❤❛♣t❡r ❘❡❣r❡ss✐♦♥ ❛♥❞ ❝♦rr❡❧❛t✐♦♥✱ ■♥tr♦❞✉❝t♦r② st❛t✐st✐❝s ✇✐t❤ ❘ ✱ P✳ ❉❛❧❣❛❛r❞✱ ❙❡r✐❡s ❙t❛t✐st✐❝s ❛♥❞ ❈♦♠♣✉t✐♥❣✱ ❙♣r✐♥❣❡r✱ ✷✵✵✽✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✸✸ ✴ ✸✸