❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛
❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦
▼❡t♦❞♦❧♦❣✐❛ ♣❛r❛ ❙✉❜❛♠♦str❛❣❡♠ ❡♠ ●r❛♥❞❡s
❇❛♥❝♦s ❞❡ ❉❛❞♦s ❆♠♦str❛✐s ❈♦♠♣❧❡①♦s ♣❛r❛
❘❡❛❧✐③❛çã♦ ❞❡ ❚❡st❡s ❞❡ ❍✐♣ót❡s❡s
♣♦r●✐❧❜❡rt♦ ❘❡③❡♥❞❡ ❞❡ ❆❧♠❡✐❞❛ ❏✉♥✐♦r
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆❧❛♥ ❘✐❝❛r❞♦ ❞❛ ❙✐❧✈❛
●✐❧❜❡rt♦ ❘❡③❡♥❞❡ ❞❡ ❆❧♠❡✐❞❛ ❏ú♥✐♦r
▼❡t♦❞♦❧♦❣✐❛ ♣❛r❛ ❙✉❜❛♠♦str❛❣❡♠ ❡♠ ●r❛♥❞❡s
❇❛♥❝♦s ❞❡ ❉❛❞♦s ❆♠♦str❛✐s ❈♦♠♣❧❡①♦s ♣❛r❛
❘❡❛❧✐③❛çã♦ ❞❡ ❚❡st❡s ❞❡ ❍✐♣ót❡s❡s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ à ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠
❊st❛tíst✐❝❛✳
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
✏❚❤❡r❡ ✇❡r❡ ✺ ❊①❛❜②t❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❝r❡❛t❡❞ ❜❡t✇❡❡♥ t❤❡ ❞❛✇♥ ♦❢ ❝✐✈✐❧✐③❛t✐♦♥ t❤r♦✉❣❤ ✷✵✵✸✱ ❜✉t t❤❛t ♠✉❝❤ ✐♥❢♦r♠❛t✐♦♥ ✐s ♥♦✇ ❝r❡❛t❡❞ ❡✈❡r② ✷ ❞❛②s✳✑ ●♦♦❣❧❡✬s ❈❊❖✱ ❊r✐❝ ❙❝❤♠✐❞t✱ ✷✵✶✵✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ ♠❡ ❛♣♦✐♦✉ s❡♠♣r❡ q✉❡ ♥❡❝❡ssár✐♦✳
❆♦s ♠❡✉s ❛♠✐❣♦s✱ q✉❡ s♦✉❜❡r❛♠ ❡♥t❡♥❞❡r ❛ ❛✉sê♥❝✐❛ ♥❡st❛ ❢❛s❡ ❞❡ ❡st✉❞♦s✳ ❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s✱ q✉❡ ❡♥s✐♥❛r❛♠ ❛ t❡♦r✐❛ ❡ ❛♣❧✐❝❛çã♦ ❞❛ ❡st❛tíst✐❝❛ ❛t✐ç❛♥❞♦ ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦ q✉❡st✐♦♥❛♠❡♥t♦ ❞❡ ❛♥t✐❣❛s ❡ ✐♠❛❣✐♥❛çã♦ ❞❡ ♥♦✈❛s té❝♥✐❝❛s✳
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦s ✐✐
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✹
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ✼
❘❡s✉♠♦ ✽
❆❜str❛❝t ✾
■♥tr♦❞✉çã♦ ✶✵
✶ ❚❡st❡s ❞❡ ❍✐♣ót❡s❡s ✶✸
✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷ ❚❡st❡s ❞❡ ❍✐♣ót❡s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✶ ❚❡st❡ ♣❛r❛ ♠é❞✐❛ ❡♠ ✉♠❛ ♣♦♣✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✶✳✶ ❚❡st❡ ♣❛r❛ ♠é❞✐❛ ❝♦♠ ✈❛r✐â♥❝✐❛ ❝♦♥❤❡❝✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✶✳✷ ❚❡st❡ ♣❛r❛ ♠é❞✐❛ ❝♦♠ ✈❛r✐â♥❝✐❛ ❞❡s❝♦♥❤❡❝✐❞❛ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✷ ❈♦♠♣❛r❛çã♦ ❞❡ ✈❛r✐â♥❝✐❛s ❡♥tr❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷✳✸ ❈♦♠♣❛r❛çã♦ ❞❡ ♠é❞✐❛s ❡♥tr❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✸✳✶ ❚❡st❡t ♣❛r❛ ❛♠♦str❛s ♣❛r❡❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✸✳✷ ❚❡st❡ t ♣❛r❛ ❛♠♦str❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛r✐â♥❝✐❛
❝♦♥❤❡❝✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✸✳✸ ❚❡st❡t ♣❛r❛ ❛♠♦str❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛r✐â♥❝✐❛s
❞❡s❝♦♥❤❡❝✐❞❛s ✐❣✉❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✶✳✷✳✸✳✹ ❚❡st❡t ♣❛r❛ ❛♠♦str❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛r✐â♥❝✐❛s ❞❡s❝♦♥❤❡❝✐❞❛s ❞✐❢❡r❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✸ ❈♦♥s✐stê♥❝✐❛ ❞❡ ❡st✐♠❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷ ❆♠♦str❛❣❡♠ ❈♦♠♣❧❡①❛ ✷✷
✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ P❡s♦ ❛♠♦str❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸ ❚❡st❡ ♣❛r❛ ♠é❞✐❛ ❝♦♠ ✈❛r✐â♥❝✐❛ ❞❡s❝♦♥❤❡❝✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ❊st✐♠❛çã♦ ❞❡ ✈❛r✐â♥❝✐❛ ❡♠ ♣❡sq✉✐s❛s ❛♠♦str❛✐s ❝♦♠♣❧❡①❛s ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✺ ❆ P◆❆❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸ ▼❛t❡r✐❛❧ ❡ ♠ét♦❞♦s ✸✷
✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ▼❛t❡r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷✳✶ ❚❛♠❛♥❤♦ ❛♠♦str❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✸ ▼ét♦❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✸✳✶ Pr✐♠❡✐r❛ ❛♠♦str❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✸✳✷ ❙✉❜❛♠♦str❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✸✳✸ ❚é❝♥✐❝❛ ❞❡ ❙✉❜❛♠♦str❛❣❡♠ ♠✐st❛ ❡♠ ●r❛♥❞❡s ❆♠♦str❛s ✳ ✳ ✳ ✹✷ ✸✳✸✳✹ ❚❡st❡ ❞❡ ❊❢❡✐t♦ ❞♦ ❚❛♠❛♥❤♦ ❞❛ ❆♠♦str❛ ♥❛ ❙✐❣♥✐✜❝â♥❝✐❛ ❞❡
❚❡st❡s ❞❡ ❍✐♣ót❡s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹ ❆♥á❧✐s❡ ❞♦s ❘❡s✉❧t❛❞♦s ✹✺
✹✳✶ ❆♠♦str❛ ❆❧❡❛tór✐❛ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✶✳✶ Pr✐♠❡✐r❛ ❛♠♦str❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✶✳✷ ❙✉❜❛♠♦str❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✷ ❆♠♦str❛ ❊str❛t✐✜❝❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✷✳✶ Pr✐♠❡✐r❛ ❛♠♦str❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✷✳✷ ❙✉❜❛♠♦str❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✸ ❆♠♦str❛ ❈♦♠♣❧❡①❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✸✳✶ Pr✐♠❡✐r❛ ❛♠♦str❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✸✳✷ ❙✉❜❛♠♦str❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✹✳✹ P❡s♦s ❯t✐❧✐③❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✺ ❚é❝♥✐❝❛ ❞❡ s✉❜❛♠♦str❛❣❡♠ ♠✐st❛ ♥❛ P◆❆❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✺ ❈♦♥❝❧✉sõ❡s ✻✷
✺✳✶ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✺✳✷ ▲✐♠✐t❛çõ❡s ❞♦ tr❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✺✳✸ ❘❡❝♦♠❡♥❞❛çõ❡s ♣❛r❛ tr❛❜❛❧❤♦s ❢✉t✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✺
❆ ❆♣ê♥❞✐❝❡ ✻✼
❆✳✶ ❆♠♦str❛ ❆❧❡❛tór✐❛ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ❆✳✷ ❆♠♦str❛ ❊str❛t✐✜❝❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ❆✳✸ ❆♠♦str❛ ❈♦♠♣❧❡①❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ ▼é❞✐❛ ❡ ✈❛r✐â♥❝✐❛ ❡st✐♠❛❞❛ ❞❛ ♠é❞✐❛ ♣♦r t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ✲ ❇❡r✲ ♥♦✉❧❧✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷ ▼é❞✐❛ ❡ ✈❛r✐â♥❝✐❛ ❡st✐♠❛❞❛ ❞❛ ♠é❞✐❛ ♣♦r t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ✲
◆♦r♠❛❧✭0 ❀ 25✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸ ▼é❞✐❛ ❡ ✈❛r✐â♥❝✐❛ ❡st✐♠❛❞❛ ❞❛ ♠é❞✐❛ ♣♦r t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ✲
◆♦r♠❛❧✭0 ❀ 625✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✸✳✶ ❍✐st♦❣r❛♠❛ ❞♦ r❡♥❞✐♠❡♥t♦ ♠❡♥s❛❧ ❞❡ t♦❞❛s ❛s ❢♦♥t❡s ♣❛r❛ ♣❡ss♦❛s ❞❡ ✶✵ ❛♥♦s ♦✉ ♠❛✐s ❞❡ ✐❞❛❞❡ ✲ P◆❆❉ ✷✵✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✹✳✶ ❉✐str✐❜✉✐çõ❡s ♥♦r♠❛✐s s✐♠✉❧❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✷ ❆♠♣❧✐t✉❞❡ ❞♦ ✐♥t❡r✈❛❧♦ ❞❡ ❝♦♥✜❛♥ç❛ ❛♦ ❛✉♠❡♥t❛r ♦ t❛♠❛♥❤♦ ♠í♥✐♠♦
❞❛ ❛♠♦str❛ ✲ ◆♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✸ ❆♠♣❧✐t✉❞❡ ❞♦ ✐♥t❡r✈❛❧♦ ❞❡ ❝♦♥✜❛♥ç❛ ❛♦ ❛✉♠❡♥t❛r ♦ t❛♠❛♥❤♦ ♠í♥✐♠♦
❞❛ ❛♠♦str❛ ✲ ▲♦❣✲♥♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✹ ❆♠♣❧✐t✉❞❡ ▼é❞✐❛ ✲ ■♥t❡r✈❛❧♦ ❞❡ ❈♦♥✜❛♥ç❛ ✲ ❆♠♦str❛❣❡♠ ❊str❛t✐✜✲
❝❛❞❛ ✲ ◆♦r♠❛❧ ❡ ▲♦❣✲♥♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✺ ❱❛❧♦r ❞❛ ❡st❛tíst✐❝❛tr❡❢❡r❡♥t❡s ❛ ❞✐❢❡r❡♥t❡sµD ✲ ❛♠♦str❛❣❡♠ ♣r✐♥❝✐♣❛❧
❡ ♠✐st❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✶✳✶ ❚✐♣♦s ❞❡ ❡rr♦ ❡♠ t❡st❡s ❞❡ ❤✐♣ót❡s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✶ ❉❛❞♦s ♣♦♣✉❧❛❝✐♦♥❛✐s ✜❝tí❝✐♦s ♣❛r❛ ❡①❡♠♣❧✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ P❡s♦s ❛tr✐❜✉í❞♦s ❛♦ ❡①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ❊st✐♠❛çã♦ ❞❡ ♠é❞✐❛s ❡♠ ❛♠♦str❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ❯❧t✐♠❛t❡ ❝❧✉st❡r ❞❡ t❛♠❛♥❤♦ na= 4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✶ ❆❧❣♦rít♠♦ ❞❡ ❣❡r❛çã♦ ❞❛s ✈❛r✐á✈❡✐s ❞♦ ❡st✉❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷ ❉❛❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ s✐♠✉❧❛çã♦ ❞❛ r❡♥❞❛ ✲ P◆❆❉ ✷✵✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✸ ❱❛r✐á✈❡✐s P◆❆❉ ✷✵✶✹ ✲ ▼❡❞✐❞❛s ❘❡s✉♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✹ ●r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ♣❛r❛ ♦ t❡st❡ t ♣♦r t✐♣♦ ❞❡ ❛♠♦str❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹✳✶ ❚❛♠❛♥❤♦ ❛♠♦str❛❧ ✲ ❆♠♦str❛❣❡♠ ❛❧❡❛tór✐❛ s✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✷ P❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s ❞❛ ❡st✐♠❛t✐✈❛ ❞❛ ♠é❞✐❛ ♥❛s s✐♠✉❧❛çõ❡s ♣❛r❛
❆❆❙ ❝♦♠ ❞❡s✈✐♦ ♣❛❞rã♦ ❡ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ✈❛r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✸ P❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s ❞❛ ❡st✐♠❛t✐✈❛ ❞❛ ♠é❞✐❛ ♥❛s s✐♠✉❧❛çõ❡s ♣❛r❛
❆❆❙ ❝♦♠ ❞❡s✈✐♦ ♣❛❞rã♦ ❡ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ✈❛r✐á✈❡✐s ✲ ❇❛s❡s ✏♣r✐♥✲ ❝✐♣❛❧✑✱ ✏s✉❜❛♠♦str❛❣❡♠✑ ❡ ✏❝❛s♦ ♠✐st♦✑ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✹ ❚❛♠❛♥❤♦ ❛♠♦str❛❧ ✲ ❆♠♦str❛❣❡♠ ❡str❛t✐✜❝❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✺ P❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s ❞♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝♦♥✜❛♥ç❛ ❣❡r❛❞♦s ♣❡❧❛s ❛♠♦s✲
tr❛s ❡str❛t✐✜❝❛❞❛s ♣♦r t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ❡ ♥í✈❡❧ ❞❡ ❝♦♥✜❛♥ç❛ ❞♦ t❡st❡ ✲ ◆♦r♠❛❧ ❡ ▲♦❣✲♥♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✹✳✻ P❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s ❞❛ ❡st✐♠❛t✐✈❛ ❞❛ ♠é❞✐❛ ♥❛s s✐♠✉❧❛çõ❡s ♣❛r❛ ❛♠♦str❛❣❡♠ ❡str❛t✐✜❝❛❞❛ ❝♦♠ ❞❡s✈✐♦ ♣❛❞rã♦ ❡ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ✈❛r✐á✈❡✐s ✲ ❇❛s❡s ✏♣r✐♥❝✐♣❛❧✑✱ ✏s✉❜❛♠♦str❛❣❡♠✑ ❡ ✏❝❛s♦ ♠✐st♦✑ ✲ ❙✉❜❛✲ ♠♦str❛❣❡♠ ❝♦♠ ♣❧❛♥♦ ❛♠♦str❛❧ ❡str❛t✐✜❝❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✼ P❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s ❞❛ ❡st✐♠❛t✐✈❛ ❞❛ ♠é❞✐❛ ♥❛s s✐♠✉❧❛çõ❡s ♣❛r❛
❛♠♦str❛❣❡♠ ❡str❛t✐✜❝❛❞❛ ❝♦♠ ❞❡s✈✐♦ ♣❛❞rã♦ ❡ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ✈❛r✐á✈❡✐s ✲ ❇❛s❡s ✏♣r✐♥❝✐♣❛❧✑✱ ✏s✉❜❛♠♦str❛❣❡♠✑ ❡ ✏❝❛s♦ ♠✐st♦✑ ✲ ❙✉❜❛✲ ♠♦str❛❣❡♠ ❝♦♠ ♣❧❛♥♦ ❛♠♦str❛❧ ❛❧❡❛tór✐♦ s✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✽ P❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s ❞♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝♦♥✜❛♥ç❛ ❣❡r❛❞♦s ♣❡❧❛s ❛♠♦s✲
tr❛s ❝♦♠♣❧❡①❛s ❡ ❛♠♣❧✐t✉❞❡ ❞♦s ■❈ ♣♦r t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✾ ❊st✐♠❛çã♦ ❡q✉✐✈♦❝❛❞❛ ❞❡ ♣❛râ♠❡tr♦s ❡♠ ❛♠♦str❛❣❡♠ ❝♦♠♣❧❡①❛ ✲ ❆♠✲
♣❧✐t✉❞❡ ❞♦ ■❈ ❡ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦ ❞❛ ♠é❞✐❛ ❞❡♥tr♦ ❞♦ ■❈ ♣♦r t❛✲ ♠❛♥❤♦ ❞❡ ❛♠♦str❛ ✲ ❆♠♦str❛❣❡♠ ❛❧❡❛tór✐❛ s✐♠♣❧❡s ❡ ❡str❛t✐✜❝❛❞❛ ✳ ✳ ✺✺ ✹✳✶✵ ❆♠♣❧✐t✉❞❡ ❞♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝♦♥✜❛♥ç❛ ❞❛ ♠é❞✐❛ ♥❛s s✐♠✉❧❛çõ❡s ♣❛r❛
❛♠♦str❛❣❡♠ ✐♥✐❝✐❛❧ ❝♦♠♣❧❡①❛ ❡ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ té❝♥✐❝❛s ❞❡ s✉❜❛✲ ♠♦str❛❣❡♠ ✲ ❇❛s❡s ✏♣r✐♥❝✐♣❛❧✑✱ ✏s✉❜❛♠♦str❛❣❡♠✑ ❡ ✏❝❛s♦ ♠✐st♦✑ ✳ ✳ ✳ ✳ ✺✼ ✹✳✶✶ P❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s ❞❛ ❡st✐♠❛t✐✈❛ ❞❛ ♠é❞✐❛ ♥❛s s✐♠✉❧❛çõ❡s ♣❛r❛
❛♠♦str❛❣❡♠ ✐♥✐❝✐❛❧ ❝♦♠♣❧❡①❛ ❡ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ té❝♥✐❝❛s ❞❡ s✉❜❛✲ ♠♦str❛❣❡♠ ✲ ❇❛s❡s ✏♣r✐♥❝✐♣❛❧✑✱ ✏s✉❜❛♠♦str❛❣❡♠✑ ❡ ✏❝❛s♦ ♠✐st♦✑ ✳ ✳ ✳ ✳ ✺✼ ✹✳✶✷ ▼é❞✐❛ ❞❛ s♦♠❛ ❞♦s ♣❡s♦s ♦❜t✐❞♦s ♥❛s s✐♠✉❧❛çõ❡s ♣♦r t✐♣♦ ❞❡ ❛♠♦s✲
tr❛❣❡♠✱ s✉❜❛♠♦str❛❣❡♠ ❡ ✉t✐❧✐③❛çã♦ ❞❡ ♣❡s♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✶✸ ❚❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ✲ P◆❆❉ ✷✵✶✷✱ ✷✵✶✹ ❡ s✉❜❛♠♦str❛❣❡♠ ♠✐st❛ ✳ ✳ ✺✾ ✹✳✶✹ ❊st✐♠❛t✐✈❛s ❞❡ r❡♥❞✐♠❡♥t♦ ♠é❞✐♦ ✲ P◆❆❉ ✷✵✶✷ ❡ ✷✵✶✹ ✲ ❆♠♦str❛❣❡♠
✏♣r✐♥❝✐♣❛❧✑ ❡ s✉❜❛♠♦str❛❣❡♠ ♠✐st❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✶✺ ❚❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ✲ P◆❆❉ ✷✵✶✷✱ ✷✵✶✹ ❡ s✉❜❛♠♦str❛❣❡♠ ♠✐st❛ ✳ ✳ ✻✵
❆✳✶ ❙✐♠✉❧❛çõ❡s ♣❛r❛ ❆❆❙ ❝♦♠ ❞❡s✈✐♦ ♣❛❞rã♦ ❡ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ✈❛r✐á✈❡✐s ✻✽ ❆✳✷ ❆♠♣❧✐t✉❞❡ ♠é❞✐❛ ❞♦ ■❈ ❣❡r❛❞♦ ♥❛s s✐♠✉❧❛çõ❡s ♣❛r❛ ❆♠♦str❛❣❡♠ ❆❧❡✲
❛tór✐❛ ❙✐♠♣❧❡s ✭❆❆❙✮ ✲ ❈❛s♦s ✧♣r✐♥❝✐♣❛❧✧✱ ✧s✉❜❛♠♦str❛❣❡♠✧❡ ✧♠✐st♦✧ ✻✾ ❆✳✸ ❙✐♠✉❧❛çõ❡s ♣❛r❛ ❛♠♦str❛❣❡♠ ❡str❛t✐✜❝❛❞❛ s❡❣✉♥❞♦ ♦ ♥í✈❡❧ ❞❡ ❝♦♥✜✲
❛♥ç❛ ❡ ♦ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ✈❛r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
❆✳✹ ❆♠♣❧✐t✉❞❡ ♠é❞✐❛ ❞♦ ■❈ ❣❡r❛❞♦ ♥❛s s✐♠✉❧❛çõ❡s ♣❛r❛ ❛♠♦str❛❣❡♠ ❡s✲ tr❛t✐✜❝❛❞❛ ✲ ❈❛s♦s ✧♣r✐♥❝✐♣❛❧✧✱ ✧s✉❜❛♠♦str❛❣❡♠✧❡ ✧♠✐st♦✲ ❙✉❜❛♠♦s✲ tr❛❣❡♠ ❡str❛t✐✜❝❛❞❛ ❡ ❛❧❡❛tór✐❛ s✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ❆✳✺ ❚❛♠❛♥❤♦ ❛♠♦str❛❧ ♥♦ s❡❣✉♥❞♦ ❡stá❣✐♦ ✲ ❆♠♦str❛❣❡♠ ❝♦♠♣❧❡①❛ ✳ ✳ ✳ ✳ ✼✶ ❆✳✻ ❉❛❞♦s ♣❛r❛ ❛♠♦str❛❣❡♠ ❝♦♠♣❧❡①❛ ✲ ❡str❛t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ❆✳✼ ❉❛❞♦s ♣❛r❛ ❛♠♦str❛❣❡♠ ❝♦♠♣❧❡①❛ ✲ ❝♦♥❣❧♦♠❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
❘❡s✉♠♦
❆ ❛♠♦str❛❣❡♠ é ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛✉①✐❧✐❛r ❛ s❡❧❡çã♦ ❞❡ ❛♠♦str❛s ❡ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ❝♦♠ ❜❛s❡ ♥❡ss❛s ❛♠♦str❛s✳ ❯s✉❛❧♠❡♥t❡ é ❞✐s❝✉t✐❞♦ ♦ t❛♠❛✲ ♥❤♦ ♠í♥✐♠♦ q✉❡ ❞❡✈❡ s❡ t♦♠❛r ❡♠ ✉♠❛ ❛♠♦str❛✳ ◆♦ ❡♥t❛♥t♦✱ ❛♦ ✉t✐❧✐③❛r ❛♠♦str❛s ❣r❛♥❞❡s✱ ♣♦❞❡♠ s✉r❣✐r ♣r♦❜❧❡♠❛s ♥❛ r❡❛❧✐③❛çã♦ ❞❡ t❡st❡s ❞❡ ❤✐♣ót❡s❡s ♣♦✐s✱ s❡❣✉♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❝♦♥s✐stê♥❝✐❛ ❞♦s ❡st✐♠❛❞♦r❡s✱ ❛♦ ❛✉♠❡♥t❛r ♦ t❛♠❛♥❤♦ ❛♠♦str❛❧ ❛ ✈❛r✐â♥❝✐❛ ❞♦ ❡st✐♠❛❞♦r ❞✐♠✐♥✉✐✱ ♣♦❞❡♥❞♦ ✐♥✢✉❡♥❝✐❛r ♥♦ ✈❛❧♦r ❞❛ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ ❞❡ ❤✐♣ót❡s❡✳ ❖ ♣r♦❜❧❡♠❛ s❡ ❛❣r❛✈❛ ❡♠ ❛♠♦str❛s ❝♦♠♣❧❡①❛s✳
◆❡st❡ tr❛❜❛❧❤♦ é ♣r♦♣♦st♦ ✉♠❛ té❝♥✐❝❛ ❞❡ s✉❜❛♠♦str❛❣❡♠ ♣❛r❛ s❡r ❛♣❧✐❝❛❞❛ ♥❡s✲ s❛s ❣r❛♥❞❡s ❛♠♦str❛s✱ ❛ss✐♠ ❝♦♠♦ ♦ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❢❛③❡r ✉♠❛ s✉❜❛♠♦str❛❣❡♠ ❞❡ ♠❛♥❡✐r❛ ❝♦rr❡t❛✳ ❯♠ t❡st❡ ♣❛r❛ ♦ ❡❢❡✐t♦ ❞♦ t❛♠❛♥❤♦ ❛♠♦str❛❧ ♥❛ s✐❣♥✐✜❝â♥❝✐❛ ❞❡ t❡st❡ ❞❡ ❤✐♣ót❡s❡s t❛♠❜é♠ é ❛♣r❡s❡♥t❛❞♦✳ ❋♦r❛♠ s✐♠✉❧❛❞♦s ❞❛❞♦s ❡♠ q✉❡ ♦s r❡s✉❧✲ t❛❞♦s ♠♦str❛r❛♠ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ss❛ ✈❡r✐✜❝❛çã♦✳ ❚❛♠❜é♠ ❢♦✐ ❢❡✐t❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ✉t✐❧✐③❛♥❞♦ ♦s ❞❛❞♦s ❞❛ P❡sq✉✐s❛ ◆❛❝✐♦♥❛❧ ♣♦r ❆♠♦str❛ ❞❡ ❉♦♠✐❝í❧✐♦s ✲ P◆❆❉✱ ❡ ♦s r❡s✉❧t❛❞♦s ♠♦str❛r❛♠ ✉♠❛ ♠✉❞❛♥ç❛ ♥❛ ✐♥❢❡rê♥❝✐❛ q✉❛♥❞♦ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❢♦✐ r❡❞✉③✐❞♦✳
P❛❧❛✈r❛s✲❈❤❛✈❡✿ ❆♠♦str❛❣❡♠❀ ●r❛♥❞❡s ❇❛♥❝♦s ❞❡ ❉❛❞♦s❀ ❚❡st❡ ❞❡ ❍✐♣ót❡s❡s❀ ❙✉❜❛♠♦str❛❣❡♠ ✳
❆❜str❛❝t
❙❛♠♣❧✐♥❣ ✐s ❛ st❛t✐st✐❝❛❧ ♠❡t❤♦❞♦❧♦❣② ✉s❡❞ t♦ ❛✐❞ t❤❡ s❛♠♣❧❡ s❡❧❡❝t✐♦♥ ❛♥❞ t❤❡ ♣❛r❛♠❡t❡rs ❡st✐♠❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤✐s s❛♠♣❧❡✳ ■t ✐s ❝♦♠♠♦♥ t♦ ❞✐s❝✉ss t❤❡ ♠✐♥✐♠✉♠ s✐③❡ t♦ ❜❡ t❛❦❡♥ ✐♥ ❛ s❛♠♣❧❡✱ ❤♦✇❡✈❡r✱ ✇❤❡♥ ✉s✐♥❣ ❧❛r❣❡ s❛♠♣❧❡s✱ ♣r♦❜❧❡♠s ♠❛② ❛r✐s❡ ✐♥ ♣❡r❢♦r♠✐♥❣ ❤②♣♦t❤❡s✐s t❡sts ❜❡❝❛✉s❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝♦♥s✐st❡♥❝② ♣r♦♣❡rt② ♦❢ t❤❡ ❡st✐♠❛t♦rs✱ ❜② ✐♥❝r❡❛s✐♥❣ t❤❡ s❛♠♣❧❡ s✐③❡ t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❡st✐♠❛t♦r ❞❡❝r❡✲ ❛s❡s✳ ❚❤✐s ♠❛② ✐♥✢✉❡♥❝❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❤②♣♦t❤❡s✐s t❡st st❛t✐st✐❝✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ❡①❛❝❡r❜❛t❡❞ ✐♥ ❝♦♠♣❧❡① s❛♠♣❧❡s✳
■♥ t❤✐s ✇♦r❦ ✇❡ ♣r♦♣♦s❡ ❛ s✉❜s❛♠♣❧✐♥❣ t❡❝❤♥✐q✉❡ t♦ ❜❡ ❛♣♣❧✐❡❞ ✐♥ ❧❛r❣❡ s❛♠♣❧❡s✱ ❛s ✇❡❧❧ ❛s ❛♥ ❛❧❣♦r✐t❤♠ t♦ ❝♦♥❞✉❝t r❡s❛♠♣❧✐♥❣✳ ❆ t❡st ❢♦r t❤❡ ❡✛❡❝t ♦❢ s❛♠♣❧❡ s✐③❡ ♦♥ t❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ t❤❡ ❤②♣♦t❤❡s✐s t❡st ✐s ❛❧s♦ ♣r❡s❡♥t❡❞✳ ❙✐♠✉❧❛t❡❞ ❞❛t❛ ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ❛♥❞ t❤❡ r❡s✉❧ts s❤♦✇❡❞ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s ♠❡t❤♦❞✳ ■♥ ❛❞❞✐t✐♦♥✱ ✐♥ ❛♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ ❇r❛③✐❧✐❛♥ ◆❛t✐♦♥❛❧ ❍♦✉s❡❤♦❧❞ ❙❛♠♣❧❡ ❙✉r✈❡② ✭P◆❆❉✮ s❤♦✇❡❞ t❤❛t t❤❡ ✐♥❢❡r❡♥❝❡ ✇❛s ❝❤❛♥❣❡❞ ✇❤❡♥ t❤❡ s❛♠♣❧❡ s✐③❡ ✇❛s r❡❞✉❝❡❞✳
❑❡②✇♦r❞s✿ ❙❛♠♣❧✐♥❣❀ ❇✐❣ ❉❛t❛❀ ❍②♣♦t❤❡s✐s ❚❡st❀ ❙✉❜s❛♠♣❧✐♥❣ ✳
■♥tr♦❞✉çã♦
❙❡❣✉♥❞♦ ❈❛s❡❧❧❛ ❡ ❇❡r❣❡r ✭✷✵✵✷✮✱ ❛ ✐♥❢❡rê♥❝✐❛ ❡st❛tíst✐❝❛ ❜✉s❝❛ ❣❡♥❡r❛❧✐③❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❛♠♦str❛ ❝♦♠ ♦❜❥❡t✐✈♦ ❞❡ ❡st✐♠❛r ♣❛râ♠❡tr♦s ♣♦♣✉❧❛❝✐♦♥❛✐s✳ ❖s ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ té❝♥✐❝❛s ❞❡ ❛♠♦str❛❣❡♠✱ q✉❡ ✈❛r✐❛♠ ❞❡s❞❡ ❛ ❢♦r♠❛ ❞❡ ❝♦❧❡t❛ à s❡❧❡çã♦ ❞❛s ♦❜s❡r✈❛çõ❡s✱ tr❛t❛♠ ❞❛ ❡s♣❡❝✐✜❝✐❞❛❞❡ ❞❡ ❝❛❞❛ ♣❧❛♥♦ ❛♠♦str❛❧✳ ❍❡❡r✐♥❣❛ ❡t ❛❧✳ ✭✷✵✶✵✮ r❡ss❛❧t❛♠ q✉❡ q✉❛♥❞♦ ❤á ♠✐st✉r❛ ❞❡ té❝♥✐❝❛s ❞❡ ❛♠♦str❛❣❡♠ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ✐♥❡r❡♥t❡ ❛♦ ♣❧❛♥♦ r❡q✉❡r tr❛t❛♠❡♥t♦ ❡s♣❡❝✐❛❧ ♥❛ ❡st✐✲ ♠❛çã♦ ♣♦♥t✉❛❧ ❡ ♥❛ ✈❛r✐â♥❝✐❛ ❞❛s ❡st✐♠❛t✐✈❛s✳
❈♦♠ ❛ ❡✈♦❧✉çã♦ ❞❛ t❡❝♥♦❧♦❣✐❛ ❞❡ ✐♥❢♦r♠❛çã♦ ❡ ❛ ❢❛❝✐❧✐❞❛❞❡ ❞❡ ♦❜t❡♥çã♦ ❞❡ ❞❛❞♦s✱ ♦ ♣❡sq✉✐s❛❞♦r s❡ ❞❡♣❛r❛✱ ❝❛❞❛ ✈❡③ ♠❛✐s✱ ❝♦♠ ❣r❛♥❞❡s q✉❛♥t✐❞❛❞❡s ❞❡ ✐♥❢♦r♠❛çõ❡s✳ ❖ ❢❡♥ô♠❡♥♦ ❇✐❣ ❉❛t❛ é ✉♠ ❡①❡♠♣❧♦ ❞❡ss❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥❢♦r♠❛çõ❡s à ❞✐s♣♦s✐çã♦ ❞♦ ♣❡sq✉✐s❛❞♦r✳
◆❡ss❡ ❝♦♥t❡①t♦✱ ❛♦ ✉t✐❧✐③❛r ❡st✐♠❛t✐✈❛s ✐♥t❡r✈❛❧❛r❡s ✐♥❢❡r❡♥❝✐❛✐s ❡♠ ❣r❛♥❞❡s ❛♠♦s✲ tr❛s✱ ♣♦❞❡ ❛❝♦♥t❡❝❡r ❞❡ q✉❡ ❛❧❣✉♠❛s ❛♥á❧✐s❡s ❡st❛tíst✐❝❛s ✐♥❞✐q✉❡♠ ✉♠❛ ❞✐❢❡r❡♥ç❛ s✐❣♥✐✜❝❛t✐✈❛ ❡♥tr❡ ♦ q✉❡ ❡stá s❡♥❞♦ t❡st❛❞♦✱ ♠❛s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦ ❢❡♥ô♠❡♥♦ ❛♣r❡s❡♥t❛ t❛❧ ❞✐❢❡r❡♥ç❛✳ ❈♦♠♦ ❡①❡♠♣❧♦ ♣♦❞❡♠♦s ❝✐t❛r ♦ ❝❛s♦ ❞♦ t❡st❡ ♣❛r❛ ♠é❞✐❛ ❡♠ ✉♠❛ ♣♦♣✉❧❛çã♦ ♥♦r♠❛❧ ❡♠ q✉❡ ❛ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ é ❝❛❧❝✉❧❛❞❛ ❝♦♠♦✿
z = x¯−µ0
σ/√n ✭✶✮
♦♥❞❡ x¯ é ❛ ♠é❞✐❛ ❛♠♦str❛❧✱ µ0 é ❛ ♠é❞✐❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ❛ s❡r t❡st❛❞❛✱ σ é ♦ ❞❡s✈✐♦✲
♣❛❞rã♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ n é ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛✳
❈❛s♦ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ n ❝r❡sç❛ ♦ ✈❛❧♦r ❞❛ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ z t❛♠❜é♠ t❡♥❞❡ ❛ ❝r❡s❝❡r✱ ❝♦♥❢♦r♠❡ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ❊q✉❛çã♦ ✭✷✮✳ ❈♦♠♦ ♦ ✈❛❧♦r ❞❛ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ é ❣r❛♥❞❡ ❡ ♦ ✈❛❧♦r ❝rít✐❝♦ ♥ã♦ ♠✉❞❛✱ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ♣♦❞❡ ❧❡✈❛r à
r❡❥❡✐çã♦ ❞❛ ❤✐♣ót❡s❡ ♥✉❧❛✱ ♣✉r❛♠❡♥t❡ ♣♦r q✉❡stõ❡s ♠❛t❡♠át✐❝❛s✳
lim
n→∞
¯ x−µ0
σ
√
n=∞ ✭✷✮
◆♦ ❝❛s♦ ❡♠ q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ é ♣❡q✉❡♥♦✱ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ✭✸✮✱ é tr❛t❛❞♦✱ ❝♦♠✉♠❡♥t❡✱ ❛♦ ❝♦♥s✐❞❡r❛r ✉♠ t❛♠❛♥❤♦ ♠í♥✐♠♦ ❞❡ ❛♠♦str❛✱ ❝❛❧❝✉❧❛❞♦ ♣❛r❛ ♦ ♣❧❛♥♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s ❝♦♠♦ n = z
2 γ/2σ2
E2 ✱ ♦♥❞❡ γ ❡ ♦ ♥í✈❡❧ ❞❡ ❝♦♥✜❛♥ç❛ ❡E é ❡rr♦ ❛♠♦str❛❧ ♠á①✐♠♦ ❛❞♠✐t✐❞♦✳
lim
n→0
¯ x−µ0
σ
√
n = 0 ✭✸✮
P❛r❛ ❡①❡♠♣❧✐✜❝❛r ♦ ❡①♣♦st♦✱ ❢♦✐ ❣❡r❛❞❛ ✈✐❛ s✐♠✉❧❛çã♦ ✉♠❛ ✈❛r✐á✈❡❧ ❞❡ ❞✐str✐❜✉✐çã♦ ❇❡r♥♦✉❧❧✐ ❝♦♠ ♠é❞✐❛ ✵✱✻ ❡ ❞✉❛s ◆♦r♠❛✐s ❝♦♠ ♠é❞✐❛ ✷✵ ❡ ✈❛r✐â♥❝✐❛s ✷✺ ❡ ✻✷✺✱ t♦❞❛s ❝♦♠ t❛♠❛♥❤♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❞❡ ✶✵✵ ♠✐❧✳ ❋♦r❛♠ r❡t✐r❛❞❛s ❛♠♦str❛s ❛❧❡❛tór✐❛s s✐♠♣❧❡s ❝♦♠ t❛♠❛♥❤♦s ✈❛r✐❛♥❞♦ ❞❡ ✶✵✵ ❛ ✹✵ ♠✐❧✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♠é❞✐❛ ❡ ❞♦ ❡rr♦ ♣❛❞rã♦ ❞❛ ♠é❞✐❛ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ♥❛s ❋✐❣✉r❛s ✶✱ ✷ ❡ ✸✳ ❆ ❢ór♠✉❧❛ ❞♦ t❛♠❛♥❤♦ ❛♠♦str❛❧ ♣❛r❛ ❡ss❡ ❝❛s♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛ ❙❡çã♦ ✸✳✷✳✶✳
❋✐❣✉r❛ ✶✿ ▼é❞✐❛ ❡ ✈❛r✐â♥❝✐❛ ❡st✐♠❛❞❛ ❞❛ ♠é❞✐❛ ♣♦r t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ✲ ❇❡r♥♦✉❧❧✐
➱ ❡✈✐❞❡♥t❡ q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ✐♥✢✉❡♥❝✐❛ ❞✐r❡t❛♠❡♥t❡ ♦ ✈❛❧♦r ❞♦ ❡rr♦ ♣❛❞rã♦ ❞❛ ♠é❞✐❛✱ ✐♥✢✉ê♥❝✐❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❝♦♥s✐stê♥❝✐❛ ❞♦s ❡st✐✲ ♠❛❞♦r❡s✱ ❞✐s❝✉t✐❞❛ ♥❛ ❙❡çã♦ ✶✳✸✳ P❡r❝❡❜❡♠♦s t❛♠❜é♠ q✉❡ ❛ ✈❛r✐â♥❝✐❛ ❝♦♠❡ç❛ ❛ s❡ ❡st❛❜✐❧✐③❛r ❛ ♣❛rt✐r ❞❡ ❛♠♦str❛s ♠❛✐♦r❡s ❞♦ q✉❡ ✶✵ ♠✐❧✳
❊st❡ tr❛❜❛❧❤♦ ✈✐s❛ ❞✐s❝✉t✐r ❛ ✐♥✢✉ê♥❝✐❛ ❞♦ t❛♠❛♥❤♦ ❛♠♦str❛❧ ❡ s❡✉s ✐♠♣❛❝t♦s ❡♠ t❡st❡s ❞❡ ❤✐♣ót❡s❡s ❛♣❧✐❝❛❞♦s ❛ ♣❧❛♥♦s ❛♠♦str❛✐s ❝♦♠♣❧❡①♦s✳ ❯♠❛ s♦❧✉çã♦ ♣❛r❛ r❡❞✉③✐r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❞❛❞♦s ❛♠♦str❛✐s✱ ♣♦r q✉❡stõ❡s ❞❡ t❡♠♣♦ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦✱ ❢♦✐ ❞❛❞❛ ♣♦r ❩❤✉ ❡t ❛❧✳ ✭✷✵✶✺✮✱ ❡♠ q✉❡ ❢♦✐ r❡t✐r❛❞❛ ✉♠❛ ❛♠♦str❛ ❛❧❡❛tór✐❛ s✐♠♣❧❡s ❞♦s
❋✐❣✉r❛ ✷✿ ▼é❞✐❛ ❡ ✈❛r✐â♥❝✐❛ ❡st✐♠❛❞❛ ❞❛ ♠é❞✐❛ ♣♦r t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ✲ ◆♦r♠❛❧✭0❀ 25✮
❋✐❣✉r❛ ✸✿ ▼é❞✐❛ ❡ ✈❛r✐â♥❝✐❛ ❡st✐♠❛❞❛ ❞❛ ♠é❞✐❛ ♣♦r t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ✲ ◆♦r♠❛❧✭0❀ 625✮
❞❛❞♦s✳ ◆♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❛❞✈✐♥❞♦s ❞❡ ♣❡sq✉✐s❛s ❛♠♦str❛✐s ❝♦♠♣❧❡①❛s✱ ❞❡✈❡✲s❡ t♦♠❛r ✉♠ ❝✉✐❞❛❞♦ ❡s♣❡❝✐❛❧ q✉❛♥t♦ à r❡t✐r❛❞❛ ❞❡ s✉❜❛♠♦str❛s✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❡str✉t✉r❛ ✐♥✐❝✐❛❧ ❞♦ ♣❧❛♥♦ ❞❡ ❛♠♦str❛❣❡♠ ❞❡✈❡ s❡r ♣r❡s❡r✈❛❞❛✳
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ tr❛t❛♠❡♥t♦ ❞❡ t❡s✲ t❡s ❞❡ ❤✐♣ót❡s❡s ❡♠ ❣r❛♥❞❡s ❜❛♥❝♦s ❞❡ ❞❛❞♦s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ♣❡sq✉✐s❛ ❛♠♦str❛✐s ❝♦♠♣❧❡①❛s✳ P❛r❛ ✐ss♦✱ s❡rá ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ❢♦r♠❛ ❞❡ s✉❜❛♠♦str❛❣❡♠ q✉❡ ♠❛♥té♠ ❛ ❡str✉t✉r❛ ❝♦♠♣❧❡①❛ ❞❛ ❛♠♦str❛❣❡♠✱ ♣r♦❞✉③✐♥❞♦ ❛ss✐♠ ✉♠ t❡st❡ ♠❛✐s r♦❜✉st♦✳
❖ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ♦ ❈❛♣ít✉❧♦ ✶ tr❛t❛ ❞❡ t❡st❡s ❞❡ ❤✐♣ót❡s❡s✳ ❖ ❈❛♣ít✉❧♦ ✷ ✐♥tr♦❞✉③ ♦ ❝♦♥❝❡✐t♦ ❞❡ t❡st❡s ❞❡ ❤✐♣ót❡s❡s ❡♠ ♣❧❛♥♦s ❛♠♦s✲ tr❛✐s ❝♦♠♣❧❡①♦s✳ ❖ ❈❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛ ❛ ♠❡t♦❞♦❧♦❣✐❛ ❞♦ tr❛❜❛❧❤♦ ❡ ♥♦ ❈❛♣ít✉❧♦ ✹ ❝♦♥st❛ ❛ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s✳ P♦r ✜♠ ♦ ❈❛♣ít✉❧♦ ✺ ❛♣r❡s❡♥t❛ ❛s ❝♦♥❝❧✉sõ❡s✱ ❧✐♠✐t❛çõ❡s ❞♦ tr❛❜❛❧❤♦ ❡ r❡❝♦♠❡♥❞❛çõ❡s ♣❛r❛ tr❛❜❛❧❤♦s ❢✉t✉r♦s✳
❈❛♣ít✉❧♦ ✶
❚❡st❡s ❞❡ ❍✐♣ót❡s❡s
✶✳✶ ■♥tr♦❞✉çã♦
❖s t❡st❡s ❞❡ ❤✐♣ót❡s❡s sã♦ ♠ét♦❞♦s ✐♥❢❡r❡♥❝✐❛✐s q✉❡ ❛✉①✐❧✐❛♠ ❛ q✉❛♥t✐✜❝❛çã♦ ❞❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦✳ ❖ ♦❜❥❡t✐✈♦ ❞♦ t❡st❡ ❞❡ ❤✐♣ót❡s❡s é t♦♠❛r ❞❡❝✐sõ❡s✱ ❜❛s❡❛❞♦ ❡♠ ✉♠❛ ❛♠♦str❛ ❞❛ ♣♦♣✉❧❛çã♦✳
❲❛❧♣♦❧❡ ❡t ❛❧✳ ✭✶✾✾✸✮ ❝♦♠❡♥t❛♠ q✉❡ ♥✉♥❝❛ s❛❜❡♠♦s ❝♦♠ ❛❜s♦❧✉t❛ ❝❡rt❡③❛ s❡ ✉♠❛ ❤✐♣ót❡s❡ ❡st❛tíst✐❝❛ é ✈❡r❞❛❞❡✐r❛ ♦✉ ❢❛❧s❛✳ ❆ ♥ã♦ s❡r q✉❡ ❡①❛♠✐♥❡♠♦s ❛ ♣♦♣✉❧❛çã♦ ✐♥t❡✐r❛✱ ♦ q✉❡ s❡r✐❛ ✐♠♣r❛t✐❝á✈❡❧ ♥❛ ♠❛✐♦r✐❛ ❞❛s s✐t✉❛çõ❡s✳ P♦❞❡♠♦s r❡t✐r❛r ✉♠❛ ❛♠♦str❛ ❛❧❡❛tór✐❛✱ ❣r❛♥❞❡ ♦ s✉✜❝✐❡♥t❡✱ ❞❛ ♣♦♣✉❧❛çã♦ ❡ ✉s❛r♠♦s ♦s ❞❛❞♦s ❝♦♥t✐❞♦s ♥❡❧❛ ♣❛r❛ ❢♦r♥❡❝❡r ❡✈✐❞ê♥❝✐❛s q✉❡ ❛♣♦✐❡ ♦✉ r❡❥❡✐t❡ ❛ ❤✐♣ót❡s❡✳ ❙❡❣✉♥❞♦ ❇✉ss❛❜ ❡ ▼♦r❡tt✐♥ ✭✷✵✶✵✮✱ ♦ ♦❜❥❡t✐✈♦ ❞♦ t❡st❡ ❡st❛tíst✐❝♦ ❞❡ ❤✐♣ót❡s❡s é ❢♦r♥❡❝❡r ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ q✉❡ ♥♦s ♣❡r♠✐t❛ ✈❡r✐✜❝❛r s❡ ♦s ❞❛❞♦s ❛♠♦str❛✐s tr❛③❡♠ ❡✈✐❞ê♥❝✐❛s q✉❡ ❛♣♦✐❡♠ ♦✉ ♥ã♦ ✉♠❛ ❤✐♣ót❡s❡ ❢♦r♠✉❧❛❞❛✳
❊st❡ ❝❛♣ít✉❧♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ♠♦str❛r t❡st❡s ❞❡ ❤✐♣ót❡s❡s ❝❧áss✐❝♦s ❛♣❧✐❝❛❞♦s ❡♠ ❛♠♦str❛s ❛❧❡❛tór✐❛s s✐♠♣❧❡s ❝♦♠ r❡♣♦s✐çã♦ ❡ ✈❡r✐✜❝❛r✱ ♣❛r❛ ❝❛❞❛ ✉♠✱ s✉❛ r❡❧❛çã♦ ❝♦♠ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛✳
✶✳✷ ❚❡st❡s ❞❡ ❍✐♣ót❡s❡s
❊♠ ✉♠ t❡st❡ ❞❡ ❤✐♣ót❡s❡s ❡①✐st❡♠ ❞✉❛s ❤✐♣ót❡s❡s ❝♦♠♣❧❡♠❡♥t❛r❡s✿ ❛ ❤✐♣ót❡s❡ ♥✉❧❛✱H0✱ ❡ ❛ ❛❧t❡r♥❛t✐✈❛✱HA✳ ➱ ❞❡✜♥✐❞♦ ❛q✉✐θ ❝♦♠♦ ♣❛râ♠❡tr♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❛ s❡r
t❡st❛❞♦✳ ❙ã♦ ❡①❡♠♣❧♦s ❞❡ ❤✐♣ót❡s❡s ❝♦♠✉♠❡♥t❡ t❡st❛❞❛sθ0 = 0❡θ0 6= 0 ♦✉θ0 ≥0❡
θ0 <0✳ ❆♣ós ❞❡✜♥✐r ❛s ❤✐♣ót❡s❡s ❡ ✈❡r✐✜❝❛r ♦ ♣✲✈❛❧♦r ❞♦ t❡st❡✱ ♦ ♣❡sq✉✐s❛❞♦r ♣♦❞❡
t♦♠❛r s✉❛ ❞❡❝✐sã♦ ❡♠ r❡❥❡✐t❛r ♦✉ ♥ã♦ ❛ ❤✐♣ót❡s❡ ♥✉❧❛ ✭❈❛s❡❧❧❛ ❡ ❇❡r❣❡r✱ ✷✵✵✷✮✳ ◆❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ t❡st❡ ❞❡ ❤✐♣ót❡s❡s é ❝❛❧❝✉❧❛❞❛ ❛ r❡❣✐ã♦ ❝rít✐❝❛ ❞♦ t❡st❡✱ ❞❡✲ ✜♥✐❞❛ ♣❡❧♦ ✈❛❧♦r ❛♠♦str❛❧ ♠í♥✐♠♦✱ ♦✉ ♠á①✐♠♦✱ q✉❡ ♦ ♣❡sq✉✐s❛❞♦r ♣♦❞❡ ♦❜s❡r✈❛r ♣❛r❛ r❡❥❡✐t❛r ❛ ❤✐♣ót❡s❡ ♥✉❧❛✱ ✜①❛♥❞♦ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛✳ ❊ss❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ é ❞❡✜♥✐❞❛ ❡♠ ❞♦✐s t✐♣♦s ❞❡ ❡rr♦s✳ ❖ ❡rr♦ ❞♦ t✐♣♦I ❡ ♦ ❡rr♦ ❞♦ t✐♣♦II ✭❈❛s❡❧❧❛ ❡ ❇❡r❣❡r✱ ✷✵✵✷✮✳
❖ ❡rr♦ ❞♦ t✐♣♦ I é ❞❡✜♥✐❞♦ q✉❛♥❞♦ ♦ ♣❛râ♠❡tr♦ ♣♦♣✉❧❛❝✐♦♥❛❧θ ❡stá ❝♦♥t✐❞♦ ♥♦ ❡s♣❛ç♦ ♣❛r❛♠étr✐❝♦ ❞❛ ❤✐♣ót❡s❡ ♥✉❧❛✱ ♣♦ré♠ ♦ t❡st❡ ❞❡ ❤✐♣ót❡s❡✱ ✐♥❝♦rr❡t❛♠❡♥t❡✱ ❞❡❝✐❞❡ r❡❥❡✐t❛r ❛ ❤✐♣ót❡s❡ ♥✉❧❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ❡rr♦ ❞♦ t✐♣♦ II ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ♦ ♣❛râ♠❡tr♦ ♣♦♣✉❧❛❝✐♦♥❛❧ θ ❡stá ❝♦♥t✐❞♦ ♥❛ ❤✐♣ót❡s❡ ❛❧t❡r♥❛t✐✈❛ ♠❛s ♦ t❡st❡ ✐♥❞✐❝❛ ❛ ♥ã♦ r❡❥❡✐çã♦ ❞❛ ❤✐♣ót❡s❡ ♥✉❧❛ ✭▼♦♦❞ ❡t ❛❧✳✱ ✶✾✼✹✮✳ ❊ss❛s ❞❡✜♥✐çõ❡s sã♦ ♠❡❧❤♦r ✈✐s✉❛❧✐③❛❞❛s ♥❛ ❚❛❜❡❧❛ ✶✳✶✳ ❖ ❡rr♦ ❞♦ t✐♣♦ I t❛♠❜é♠ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛α ❡ ♦ ❡rr♦ ❞♦ t✐♣♦ II ❝♦♠♦ β✳
❯♠ ❡①❡♠♣❧♦ ❝❧áss✐❝♦ ❞❡ ❡rr♦ ❞♦ t✐♣♦ I ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞♦ ♥❛ ❛♣❧✐❝❛çã♦ ❞❛ ❧❡✲ ❣✐s❧❛çã♦ ❜r❛s✐❧❡✐r❛✱ ❡♠ q✉❡ ✉♠ ❥✉✐③ ❡✈✐t❛ ❛♦ ♠á①✐♠♦ ❛ ❝♦♥❞❡♥❛çã♦ ❞❡ ✉♠ ✐♥♦❝❡♥t❡✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❈♦♥st✐t✉✐çã♦ ❋❡❞❡r❛❧ ❞❡ ✶✾✽✽ ❡st❛❜❡❧❡❝❡ ❝♦♠♦ ♣r❡♠✐ss❛ ❜ás✐❝❛ ✭❤✐♣ó✲ t❡s❡ ♥✉❧❛✮ ❛ ♣r❡s✉♥çã♦ ❞❡ ✐♥♦❝ê♥❝✐❛ ❞❡s❝r✐t❛ ❡♠ s❡✉ ❛rt✐❣♦ ✺✱ ✐♥❝✐s♦ ▲❱■■✱ ✏♥✐♥❣✉é♠ s❡rá ❝♦♥s✐❞❡r❛❞♦ ❝✉❧♣❛❞♦ ❛té ♦ trâ♥s✐t♦ ❡♠ ❥✉❧❣❛❞♦ ❞❡ s❡♥t❡♥ç❛ ♣❡♥❛❧ ❝♦♥❞❡♥❛tór✐❛✑ ✭❇❘❆❙■▲✱ ✶✾✽✽✮✳ ◆❡ss❡ ♠❡s♠♦ ❝♦♥t❡①t♦✱ ♦ ❡rr♦ ❞♦ t✐♣♦ II s❡r✐❛ ❛ ♥ã♦ ❝♦♥❞❡♥❛çã♦ ❞❡ ✉♠ ❝✐❞❛❞ã♦ ❝✉❧♣❛❞♦✳
❇✉ss❛❜ ❡ ▼♦r❡tt✐♥ ✭✷✵✶✵✮ ❞❡s❝r❡✈❡♠ ❛ ❢✉♥çã♦ ♣♦❞❡r ❞♦ t❡st❡ ✭π✮ ❝♦♠♦ ❛ ♣r♦❜❛✲ ❜✐❧✐❞❛❞❡ ❞❡ s❡ r❡❥❡✐t❛r ❛ ❤✐♣ót❡s❡ ♥✉❧❛✱ ❞❛❞♦ ✉♠ ✈❛❧♦r q✉❛❧q✉❡r ❞❡µ✱ ❡s♣❡❝✐✜❝❛❞♦ ♦✉ ♥ã♦ ♣❡❧❛ ❤✐♣ót❡s❡ ❛❧t❡r♥❛t✐✈❛✱ ♦✉π = 1−β✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s é ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡rr♦ ❞♦ t✐♣♦ II✳
❙❡❣✉♥❞♦ ❲❛❧♣♦❧❡ ❡t ❛❧✳ ✭✶✾✾✸✮✱ ♦ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ❡s❝♦❧❤✐❞♦ ♣❛r❛ s❡ t❡r ✉♠ ❜♦♠ ♣♦❞❡r ❞❡ t❡st❡ ♣❛r❛ ✉♠ ❞❛❞♦ α é ♦❜t✐❞♦✱ ❡♠ t❡st❡s ✉♥✐❧❛t❡r❛✐s✱ ♣♦r✿
n = (zα+zβ)
2σ2
E2 ✭✶✳✶✮
❚❛❜❡❧❛ ✶✳✶✿ ❚✐♣♦s ❞❡ ❡rr♦ ❡♠ t❡st❡s ❞❡ ❤✐♣ót❡s❡s
❉❡❝✐sã♦ H ❙✐t✉❛çã♦
0 é ✈❡r❞❛❞❡✐r❛ H0 é ❢❛❧s❛
◆ã♦ r❡❥❡✐t❛r H0 ❉❡❝✐sã♦ ❝♦rr❡t❛ ❊rr♦ ❞♦ t✐♣♦ II
❘❡❥❡✐t❛r H0 ❊rr♦ ❞♦ t✐♣♦ I ❉❡❝✐sã♦ ❝♦rr❡t❛
♦♥❞❡ zα é ♦ q✉❛♥t✐❧ ❞❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♣❛r❛ ✉♠ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛ α ❡ zβ é
♦ q✉❛♥t✐❧ ❞❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♣❛r❛ ✉♠ ❡rr♦ ❞♦ t✐♣♦ ■■ β ❡ E é ♦ ❡rr♦ ♠á①✐♠♦ ❞❡ ❡st✐♠❛t✐✈❛✳
❊♠ t❡st❡ ❜✐❧❛t❡r❛✐s é ❞❛❞❛ ♣♦r✿
n= (zα/2+zβ/2)
2σ2
E2 ✭✶✳✷✮
❈❛s❡❧❧❛ ❡ ❇❡r❣❡r ✭✷✵✵✷✮ ❝♦♠❡♥t❛♠ q✉❡ ♦ t❡st❡ ❞❡ ❤✐♣ót❡s❡s ♣♦❞❡ s❡r ❡①♣r❡ss♦ ♣❡❧❛ ❡st❛tíst✐❝❛ ❞♦ t❡st❡W(X1, ..., Xn) =W(X)q✉❡ é ❢✉♥çã♦ ❞❛ ❛♠♦str❛ ❝♦❧❡t❛❞❛✳
❈♦♠♦ ❞❡s❝r✐t♦ ❡♠ ▼❛❣❛❧❤ã❡s ❡ ❞❡ ▲✐♠❛ ✭✷✵✵✽✮✱ ❛♣ós ❛ ♠❡♥s✉r❛çã♦ ❞♦ t❡st❡✱ ❛s ❞❡❝✐sõ❡s ♣♦❞❡♠ s❡r t♦♠❛❞❛s ❛♦ ✐♥t❡r♣r❡t❛r ♦ ♥í✈❡❧ ❞❡s❝r✐t✐✈♦ ❞♦ t❡st❡✱ ♦✉ p✲✈❛❧♦r✳ ❖ p✲✈❛❧♦r✱ p(X) é ✉♠❛ ❡st❛tíst✐❝❛ ❞❡ t❡st❡ q✉❡ s❛t✐s❢❛③ 0 ≤ p(X) ≤ 1 ❡ q✉❛♥❞♦ ❛♣r❡s❡♥t❛ ✈❛❧♦r ♣❡q✉❡♥♦ ✐♥❞✐❝❛ q✉❡ ❤á ♣♦✉❝❛s ❡✈✐❞ê♥❝✐❛s ❞❡ q✉❡ H0 s❡❥❛ ✈❡r❞❛❞❡✐r❛
✭❈❛s❡❧❧❛ ❡ ❇❡r❣❡r✱ ✷✵✵✷✮✳
❈❛❜❡ ❛♦ ♣❡sq✉✐s❛❞♦r ❞❡✜♥✐r q✉ã♦ ♣❡q✉❡♥♦ ❞❡✈❡ s❡r ♦p✲✈❛❧♦r ❞♦ t❡st❡ ♣❛r❛ r❡❥❡✐t❛r ❛ ❤✐♣ót❡s❡ ♥✉❧❛✳ ❉❡♣❡♥❞❡♥❞♦ ❞♦ ❢❡♥ô♠❡♥♦ ❡st✉❞❛❞♦ ❡st❡ ✈❛❧♦r ❞❡ ❝♦rt❡ ♣♦❞❡ ✈❛r✐❛r✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝✉st♦ ❞❡ t❡r ✉♠ ❢❛❧s♦ ♣♦s✐t✐✈♦ ♦✉ ✉♠ ❢❛❧s♦ ♥❡❣❛t✐✈♦✳
❈♦♠♦ ❡①❡♠♣❧♦✱ ❆r✐③♦❧❛ ❡ ❚❡✐①❡✐r❛ ✭✷✵✶✺✮ ✉t✐❧✐③❛r❛♠ ✉♠ t❡st❡ ❞❡ ❤✐♣ót❡s❡ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ♦ ✐♠♣❛❝t♦ ❞♦ ③✉♠❜✐❞♦ ✭♣❡r❝❡♣çã♦ ❞♦ s♦♠ s❡♠ ❛ ♣r❡s❡♥ç❛ ❞❡ ❡stí♠✉❧♦ s♦♥♦r♦ ❡①t❡r♥♦✮ é ❞✐❢❡r❡♥t❡ ❡♥tr❡ ✐❞♦s♦s ♣r❛t✐❝❛♥t❡s ♦✉ ♥ã♦ ❞❡ ❛t✐✈✐❞❛❞❡s ❢ís✐❝❛s✳ ❋♦✐ ✈❡r✐✜❝❛❞♦ q✉❡ ♦ ✐♠♣❛❝t♦ ❞♦ ③✉♠❜✐❞♦ é ♠❡♥♦r ❡♥tr❡ ♦s ✐❞♦s♦s ♣r❛t✐❝❛♥t❡s ❞❡ ❛t✐✈✐❞❛❞❡ ❢ís✐❝❛✱ ❝♦♠ p✲✈❛❧♦r ✐❣✉❛❧ ❛ ✵✱✵✵✹✱ s❡♥❞♦ ❡ss❡ ♣✲✈❛❧♦r ❝♦♥s✐❞❡r❛❞♦ ❛❝❡✐tá✈❡❧ ♣❛r❛ ❛ ♣❡sq✉✐s❛✳
✶✳✷✳✶ ❚❡st❡ ♣❛r❛ ♠é❞✐❛ ❡♠ ✉♠❛ ♣♦♣✉❧❛çã♦
❖s t❡st❡s ♣❛r❛ ♠é❞✐❛ sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ t❡st❛r s❡ ❛ ♠é❞✐❛ ♣♦♣✉❧❛❝✐♦♥❛❧ é ✐❣✉❛❧✱ ♠❛✐♦r ♦✉ ♠❡♥♦r ❛♦ ✈❛❧♦r ❞❡✜♥✐❞♦ ♥❛ ❤✐♣ót❡s❡ ♥✉❧❛✳ ❖ t❡st❡ ♣❛r❛ ♠é❞✐❛ ♣❛r❛♠étr✐❝♦ ♣♦❞❡ ❞✐❢❡r✐r q✉❛♥❞♦ ❝♦♥❤❡❝❡♠♦s ♦✉ ♥ã♦ ❛ ✈❛r✐â♥❝✐❛ ♣♦♣✉❧❛❝✐♦♥❛❧✳ P♦❞❡✲s❡ ❝✐t❛r ❝♦♠♦ ❡①❡♠♣❧♦ ❞❡ ❤✐♣ót❡s❡s t❡st❛❞❛sH0 :µX =µ0 ❡HA:µX 6=µ0✳
✶✳✷✳✶✳✶ ❚❡st❡ ♣❛r❛ ♠é❞✐❛ ❝♦♠ ✈❛r✐â♥❝✐❛ ❝♦♥❤❡❝✐❞❛
❙❡❣✉♥❞♦ ▼❛❣❛❧❤ã❡s ❡ ❞❡ ▲✐♠❛ ✭✷✵✵✽✮✱ ♦ t❡st❡ z ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ t❡st❛r s❡ ❛ ♠é❞✐❛ ♣♦♣✉❧❛❝✐♦♥❛❧ µ é ✐❣✉❛❧ ❛ ♠é❞✐❛ ❛♠♦str❛❧ x¯✳ ◆❡st❡ ❝❛s♦✱ é ❝♦♥s✐❞❡r❛❞♦ q✉❡ ♦ ♠♦❞❡❧♦ ♥♦r♠❛❧ é ❛❞❡q✉❛❞♦ ❛♦s ❞❛❞♦s✳ ❆ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ z é ❝❛❧❝✉❧❛❞❛ ❝♦♠♦✿
z = x¯−µ0
σ/√n ∼N(0,1) ✭✶✳✸✮
♦♥❞❡ x¯ é ❛ ♠é❞✐❛ ❛♠♦str❛❧✱ µ0 é ♦ ✈❛❧♦r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛ ♠é❞✐❛
♥❛ ❤✐♣ót❡s❡ ♥✉❧❛✱ σ é ♦ ❞❡s✈✐♦ ♣❛❞rã♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❝♦♥❤❡❝✐❞♦ ❡ n é ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛✳ ❆ ✈❛r✐â♥❝✐❛ ♣♦♣✉❧❛❝✐♦♥❛❧ σ2 é ❝❛❧❝✉❧❛❞❛ ♣♦r ✭✶✳✹✮✱ ❡♠ q✉❡ N é ♦ t❛♠❛♥❤♦
❞❛ ♣♦♣✉❧❛çã♦✳
σ2 =
N
X
i=1
(Xi−µ)2
N ✭✶✳✹✮
❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❡♠ ❣r❛♥❞❡s ❛♠♦str❛s é ✈❡r✐✜❝❛❞♦ ❡♠ ✭✶✳✺✮
lim
n→∞
¯ x−µ0
σ
√
n=∞ ✭✶✳✺✮
❱❡r✐✜❝❛✲s❡ q✉❡ à ♠❡❞✐❞❛ q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❝r❡s❝❡✱ ❛ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ t❛♠❜é♠ ❝r❡s❝❡✱ ❝♦♥❝❧✉✐♥❞♦✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ q✉❡ ❛ ♠é❞✐❛ ❛♠♦str❛❧ é ❞✐❢❡r❡♥t❡ ❞❛ ♠é❞✐❛ ♣♦♣✉❧❛❝✐♦♥❛❧ q✉❡ ❡stá s❡♥❞♦ t❡st❛❞❛✳
✶✳✷✳✶✳✷ ❚❡st❡ ♣❛r❛ ♠é❞✐❛ ❝♦♠ ✈❛r✐â♥❝✐❛ ❞❡s❝♦♥❤❡❝✐❞❛
❙❡❣✉♥❞♦ ❇✉ss❛❜ ❡ ▼♦r❡tt✐♥ ✭✷✵✶✵✮✱ ♦ t❡st❡ t ❞❡ ❙t✉❞❡♥t é ❛❞❡q✉❛❞♦ ♣❛r❛ t❡st❛r ✈❛r✐á✈❡✐s q✉❡ s❡❣✉❡♠ ✉♠❛ s❡❣✉❡♠ ❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ❝♦♠ ♠é❞✐❛µ❡ ✈❛r✐â♥❝✐❛σ2
❞❡s❝♦♥❤❡❝✐❞❛✳
❆ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ é ❞❛❞❛ ♣♦r✿
t= x¯−µ0
s/√n ∼tn−1 ✭✶✳✻✮
♦♥❞❡ x¯é ❛ ♠é❞✐❛ ❛♠♦str❛❧✱ µ0 é ♦ ✈❛❧♦r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛ ♠é❞✐❛ ♥❛
❤✐♣ót❡s❡ ♥✉❧❛✱ s é ♦ ❞❡s✈✐♦ ♣❛❞rã♦ ❞❛ ❛♠♦str❛✱ ❝❛❧❝✉❧❛❞♦ ❡♠ ✭✶✳✼✮✱ ❡n é ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛✳ ❆ ❡st❛tíst✐❝❛ ❞♦ t❡st❡✱ s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ t ❞❡ ❙t✉❞❡♥t ❝♦♠ n−1 ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳
s2 =
n
X
i=1
(Xi−x)¯ 2
n−1 ✭✶✳✼✮
❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❡♠ ❣r❛♥❞❡s ❛♠♦str❛s é ✈❡r✐✜❝❛❞♦ ❡♠ ✭✶✳✽✮
lim
n→∞
¯ x−µ0
s
√
n=∞ ✭✶✳✽✮
❱❡r✐✜❝❛✲s❡ q✉❡ à ♠❡❞✐❞❛ q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❝r❡s❝❡✱ ❛ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ t❛♠❜é♠ ❝r❡s❝❡✱ ❝♦♥❝❧✉✐♥❞♦✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ q✉❡ ❛ ♠é❞✐❛ ❛♠♦str❛❧ é ❞✐❢❡r❡♥t❡ ❞❛ ♠é❞✐❛ ♣♦♣✉❧❛❝✐♦♥❛❧ q✉❡ ❡stá s❡♥❞♦ t❡st❛❞❛✳
✶✳✷✳✷ ❈♦♠♣❛r❛çã♦ ❞❡ ✈❛r✐â♥❝✐❛s ❡♥tr❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s
❙❡❣✉♥❞♦ ▼❛❣❛❧❤ã❡s ❡ ❞❡ ▲✐♠❛ ✭✷✵✵✽✮✱ ♦s t❡st❡s ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ ✈❛r✐â♥❝✐❛s sã♦ ✐♠♣♦rt❛♥t❡s ❝♦♠♦ ♣r♦❝❡❞✐♠❡♥t♦s ♣r❡❧✐♠✐♥❛r❡s ♣❛r❛ t❡st❡s ❞❡ ❝♦♠♣❛r❛çõ❡s ❞❡ ♠é❞✐❛s ❡♥tr❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s ♣♦✐s ❝♦♥❤❡❝❡r s❡ ❛s ♣♦♣✉❧❛çõ❡s t❡st❛❞❛s ♣♦ss✉❡♠ ✈❛r✐â♥❝✐❛s ✐❣✉❛✐s é ❞❡t❡r♠✐♥❛♥t❡ ♥❛ ❡s❝♦❧❤❛ ❞♦ t❡st❡ ✉t✐❧✐③❛❞♦✱ ❝♦♠♦ ✈❡r✐✜❝❛❞♦ ♥❛ ♣ró①✐♠❛ s✉❜s❡çã♦✳
❙❡❥❛ ❳ ✉♠❛ ♥♦r♠❛❧ ❝♦♠ ♠é❞✐❛ µX ❡ ✈❛r✐â♥❝✐❛ σ2X ❡ ❨ ✉♠❛ ♥♦r♠❛❧ ❝♦♠ ♠é❞✐❛
µY ❡ ✈❛r✐â♥❝✐❛ σY2✳ ❉❡s❡❥❛♠♦s t❡st❛r ❛s ❤✐♣ót❡s❡s H0 :σX2 =σ2Y ❡ HA:σ2X 6=σY2✳
❆ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ é ❞❛❞❛ ♣♦r✿
F = s
2 X
s2 Y
✭✶✳✾✮
♦♥❞❡s2
X é ❛ ✈❛r✐â♥❝✐❛ ❛♠♦str❛❧ ❞❡ ❳ ❡s2Y ❛ ✈❛r✐â♥❝✐❛ ❛♠♦str❛❧ ❞❡ ❨✳ ❆ ❡st❛tíst✐❝❛ F
❞♦ t❡st❡ s❡❣✉❡ ♦ ♠♦❞❡❧♦ ❞❡ ❋✐s❤❡r✲❙♥❡❞❡❝♦r ❝♦♠nX−1❡nY −1❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳
nX ❡ nY sã♦ ♦s t❛♠❛♥❤♦s ❛♠♦str❛✐s ❞❡ ❳ ❡ ❨✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
✶✳✷✳✸ ❈♦♠♣❛r❛çã♦ ❞❡ ♠é❞✐❛s ❡♥tr❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s
❙❡❣✉♥❞♦ ▼❛❣❛❧❤ã❡s ❡ ❞❡ ▲✐♠❛ ✭✷✵✵✽✮✱ ♦ t❡st❡ t ❞❡ ❙t✉❞❡♥t t❛♠❜é♠ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❝♦♠♣❛r❛r ♠é❞✐❛ ❡♥tr❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s✳ ❆ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ♦✉ ♥ã♦ ❞❛s ❛♠♦str❛s é ✉♠ ❢❛t♦r ✐♠♣♦rt❛♥t❡ ❛ s❡r ❝♦♥s✐❞❡r❛❞♦ q✉❛♥❞♦ ❞❛ ❝♦♠♣❛r❛çã♦ ❞❡ ♠é❞✐❛ ❡♥tr❡ ❞✉❛s ❛♠♦str❛s✱ ❛ss✐♠ ❝♦♠♦ ♦ ✈❛❧♦r ❞❛s ✈❛r✐â♥❝✐❛s✳ ➱ ✉t✐❧✐③❛❞♦ ❛q✉✐ ♦ t❡st❡ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ ✈❛r✐â♥❝✐❛s ❡♥tr❡ ❛s ❞✉❛s ♣♦♣✉❧❛çõ❡s✳
❆ss✐♠✱ s❡rã♦ ❝✐t❛❞♦s ♦s ❝❛s♦s ❡♠ q✉❡ ❛s ❛♠♦str❛s sã♦ ♣❛r❡❛❞❛s✱ ❡♠ q✉❡ ❛s ❛♠♦s✲ tr❛s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ❝♦♠ ✈❛r✐â♥❝✐❛ ❝♦♥❤❡❝✐❞❛✱ ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛r✐â♥❝✐❛s ❞❡s❝♦♥❤❡❝✐❞❛s ♠❛s ✐❣✉❛✐s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛r✐â♥❝✐❛s ❞❡s❝♦♥❤❡❝✐❞❛s ❡ ❞✐❢❡r❡♥✲ t❡s✳
✶✳✷✳✸✳✶ ❚❡st❡ t ♣❛r❛ ❛♠♦str❛s ♣❛r❡❛❞❛s
❙❡❣✉♥❞♦ ▼❛❣❛❧❤ã❡s ❡ ❞❡ ▲✐♠❛ ✭✷✵✵✽✮✱ ♦ t❡st❡ t ♣❛r❛ ❛♠♦str❛s ♣❛r❡❛❞❛s é ❛❞❡✲ q✉❛❞♦ q✉❛♥❞♦ ❤á ✉♠❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ ❛♠♦str❛s t❡st❛❞❛s✳ ❊ss❡ ❝❛s♦ é ✉t✐❧✐③❛❞♦ ♣❛r❛ ❝♦♠♣❛r❛r ❞✉❛s ♠é❞✐❛s ♣♦♣✉❧❛❝✐♦♥❛✐s s❡♥❞♦ q✉❡ ♣❛r❛ ❝❛❞❛ ✉♥✐✲ ❞❛❞❡ ❛♠♦str❛❧ é r❡❛❧✐③❛❞❛ ❞✉❛s ♠❡❞✐çõ❡s ❞❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ✐♥t❡r❡ss❡✳ ●❡r❛❧♠❡♥t❡ ❡ss❛s ♠❡❞✐❞❛s sã♦ t♦♠❛❞❛s ❛♥t❡s ❡ ❛♣ós ✉♠❛ ❞❛❞❛ ✐♥t❡r✈❡♥çã♦✳ ❆s ♠❡❞✐❞❛s t♦♠❛❞❛s ❛♥t❡s ❡ ❞❡♣♦✐s s❡rã♦ r❡♣r❡s❡♥t❛❞❛s ♣♦r Xi ❡ Yi r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❙❡❥❛ Di = Yi−Xi✱ D¯ = Pn
i=1Di
n ❡ s
2
D = n−11
Pn
i=1(Di−D)¯ 2✳ ❆s ❤✐♣ót❡s❡s ❞♦
t❡st❡ sã♦H0 :µD = 0 ❡ HA:µD 6= 0✳
❆ ❡st❛tíst✐❝❛ ❞♦ t❡st❡ é é ❞❛❞❛ ♣♦r✿
¯ D−µD
p s2
D/n
∼tn−1 ✭✶✳✶✵✮
❡ s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ t ❞❡ ❙t✉❞❡♥t ❝♦♠ n−1 ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳
✶✳✷✳✸✳✷ ❚❡st❡ t ♣❛r❛ ❛♠♦str❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛r✐â♥❝✐❛ ❝♦♥❤❡❝✐❞❛
❙❡❣✉♥❞♦ ▼❛❣❛❧❤ã❡s ❡ ❞❡ ▲✐♠❛ ✭✷✵✵✽✮✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ❝❛s♦ ❡♠ q✉❡ q✉❡r❡♠♦s ❝♦♠♣❛r❛r ❛s ♠é❞✐❛s ❞❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠ q✉❡ ❝♦♥❤❡❝❡♠♦s ❛s ✈❛r✐✲
â♥❝✐❛s✳ ❆ ✐♥❢♦r♠❛çã♦ ❞♦s ✈❛❧♦r❡s ❞❛s ✈❛r✐â♥❝✐❛s ♣♦♣✉❧❛❝✐♦♥❛✐s ♣♦❞❡ s❡r ❡st✐♠❛❞❛ ✈✐❛ ❡st✉❞♦s ❛♥t❡r✐♦r❡s ♦✉ ❡①♣❡r✐♠❡♥t♦s s✐♠✐❧❛r❡s✳ ❈♦♥s✐❞❡r❡ X ❡ Y ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s r❡♣r❡s❡♥t❛♥❞♦ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ✐♥t❡r❡ss❡ ❡♠ ❝❛❞❛ ♣♦♣✉❧❛çã♦ ❝♦♠ ♠é❞✐❛sµX ❡µY✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❙❡❥❛ D¯ = ¯X − Y¯✱ ❡ V ar( ¯D) = σ2
X/n1 +σY2/n2✳ ❆s ❤✐♣ót❡s❡s ❞♦ t❡st❡ sã♦
H0 :µD = 0 ❡HA:µD 6= 0✳ ❖ t❡st❡ é ❞❛❞♦ ♣♦r✿
¯ D−µD
p σ2
X/n1+σY2/n2
∼N(0,1) ✭✶✳✶✶✮
❡ s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ◆♦r♠❛❧ ❝♦♠ ♠é❞✐❛ 0❡ ✈❛r✐â♥❝✐❛ 1✳
✶✳✷✳✸✳✸ ❚❡st❡ t ♣❛r❛ ❛♠♦str❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛r✐â♥❝✐❛s ❞❡s❝♦♥❤❡❝✐✲ ❞❛s ✐❣✉❛✐s
▼❛❣❛❧❤ã❡s ❡ ❞❡ ▲✐♠❛ ✭✷✵✵✽✮ ❝✐t❛♠ ♦ ❝❛s♦ ❡♠ q✉❡ q✉❡r❡♠♦s ❝♦♠♣❛r❛r ❛s ♠é❞✐❛s ❞❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠ q✉❡ ♥ã♦ ❝♦♥❤❡❝❡♠♦s s✉❛s ✈❛r✐â♥❝✐❛s✳ ◆❡st❡ ❝❛s♦✱ ♣♦❞❡ s❡r r❡❛❧✐③❛❞♦ ✉♠ t❡st❡ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ ✈❛r✐â♥❝✐❛s ❡♥tr❡ ❛s ❞✉❛s ♣♦♣✉❧❛çõ❡s ❛ ✜♠ ❞❡ ✈❡r✐✜❝❛r s❡ sã♦ ✐❣✉❛✐s ♦✉ ❞✐❢❡r❡♥t❡s✳ ❈❛s♦ ♦ t❡st❡ r❡s✉❧t❡ ❡♠ q✉❡ ❛s ❞✉❛s ♣♦♣✉❧❛çõ❡s t❡st❛❞❛s ♣♦ss✉❡♠ ✈❛r✐â♥❝✐❛s ✐❣✉❛✐s✱ ❡st❡ ❝❛s♦ é ❛❞❡q✉❛❞♦✳
❙❡❥❛ D¯ = ¯X−Y¯ ❡ s❡❥❛ ❛ ✈❛r✐â♥❝✐❛✿
s2c = (n1 −1)s
2
X + (n2−1)s2Y
(n1−1) + (n2−1) ✭✶✳✶✷✮
❈♦♥s✐❞❡r❡ ❛s ❤✐♣ót❡s❡s ❞♦ t❡st❡ H0 :µD = 0 ❡ HA:µD 6= 0✳ ❆ss✐♠✱ ❛ ❡st❛tíst✐❝❛
❞♦ t❡st❡ é ❞❛❞❛ ♣♦r✿
¯ D−µD
p s2
c(1/n1 + 1/n2)
∼tn1+n2−2 ✭✶✳✶✸✮
❡ s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ t ❞❡ ❙t✉❞❡♥t ❝♦♠ n1+n2−2 ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳
✶✳✷✳✸✳✹ ❚❡st❡ t ♣❛r❛ ❛♠♦str❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛r✐â♥❝✐❛s ❞❡s❝♦♥❤❡❝✐✲ ❞❛s ❞✐❢❡r❡♥t❡s
❈♦♠♣❧❡♠❡♥t❛r♠❡♥t❡ à ú❧t✐♠❛ s✉❜s❡çã♦✱ ♦ ❝❛s♦ ❡♠ q✉❡ q✉❡r❡♠♦s ❝♦♠♣❛r❛r ❛s ♠é❞✐❛s ❞❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠ q✉❡ ♥ã♦ ❝♦♥❤❡❝❡♠♦s ❛s ✈❛r✐â♥❝✐❛s ❡
q✉❡✱ ✈✐❛ t❡st❡s ❛❞❡q✉❛❞♦s ♦✉ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦✱ ❛s ✈❛r✐â♥❝✐❛s ❞❛s ♣♦♣✉❧❛çõ❡s sã♦ ❞✐❢❡r❡♥t❡s✱ ❡st❡ t❡st❡ é ❛❞❡q✉❛❞♦ ✭▼❛❣❛❧❤ã❡s ❡ ❞❡ ▲✐♠❛✱ ✷✵✵✽✮✳
❈♦♥s✐❞❡r❡ D¯ = ¯X−Y¯ ❡ ❛ ✈❛r✐â♥❝✐❛ ❞❡ D ❝♦♠♦✿
σD2¯ = s 2 X n1 +s 2 Y
n2 ✭✶✳✶✹✮
❈♦♥s✐❞❡r❡ ❛s ❤✐♣ót❡s❡s ❞♦ t❡st❡ H0 :µD = 0 ❡ HA:µD 6= 0✳ ❆ss✐♠✱ ❛ ❡st❛tíst✐❝❛
❞♦ t❡st❡ é ❞❛❞❛ ♣♦r✿
¯ D−µD
q σ2
¯ D
∼tν ✭✶✳✶✺✮
q✉❡ s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ t ❞❡ ❙t✉❞❡♥t ❝♦♠ ν ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳ ❖s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞♦ t❡st❡ sã♦ ❞❛❞♦s ♣♦r✿
ν = σ
4 ¯ D (s2
X/n1)2
n1−1 +
(s2 Y/n2)2
n2−1
✭✶✳✶✻✮
✶✳✸ ❈♦♥s✐stê♥❝✐❛ ❞❡ ❡st✐♠❛❞♦r❡s
❙❡❣✉♥❞♦ ❈❛s❡❧❧❛ ❡ ❇❡r❣❡r ✭✷✵✵✷✮✱ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❛ss✐♥tót✐❝❛s ❞♦s ❡st✐♠❛❞♦r❡s à ♠❡❞✐❞❛ q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ s❡ t♦r♥❛ ❣r❛♥❞❡ ♦✉ t❡♥❞❡♥❞♦ ❛♦ ✐♥✜♥✐t♦✳ ❆ ❡st❛tíst✐❝❛ ✐♥❢❡r❡♥❝✐❛❧ ❛ss✐♥tót✐❝❛ ✉s❛ ❞❡ té❝♥✐❝❛s ❞❡ ♠❛♥❡✐r❛ ❛ s✐♠♣❧✐✜❝❛r ❛❧❣✉♥s ❝á❧❝✉❧♦s q✉❡✱ s❡r✐❛♠ ❞✐❢í❝❡✐s ❡♠ ♣♦♣✉❧❛çõ❡s ✜♥✐t❛s✳ ❉❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❡st✐♠❛❞♦r❡s✱ ❛ q✉❡ ❛♣r❡s❡♥t❛ r❡❧❛çã♦ ❞✐r❡t❛ ❛ ❡st❡ tr❛❜❛❧❤♦ é ❛ ❝♦♥s✐stê♥❝✐❛✱ q✉❡ é ✉♠❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡ ✉♠ ❡st✐♠❛❞♦r✳
❆ ❞❡✜♥✐çã♦ ❞❡ ✉♠ ❡st✐♠❛❞♦r ❝♦♥s✐st❡♥t❡ é ❛♣r❡s❡♥t❛❞❛ ❡♠ ❈❛s❡❧❧❛ ❡ ❇❡r❣❡r ✭✷✵✵✷✮✳ ❯♠ ❡st✐♠❛❞♦r é ❞✐t♦ ❝♦♥s✐st❡♥t❡ s❡ ♦ ❡st✐♠❛❞♦r ❝♦♥✈❡r❣❡ ❡♠ ♣r♦❜❛❜✐❧✐✲ ❞❛❞❡ ♣❛r❛ ♦ ✈❛❧♦r ♣♦♣✉❧❛❝✐♦♥❛❧ à ♠❡❞✐❞❛ q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❝r❡s❝❡✳ ❙❡❥❛♠ X1, X2, ..., Xn✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞❛s s❡❣✉✐♥❞♦ ❛ ❞✐str✐✲
❜✉✐çã♦f(x|θ)✳ P♦❞❡♠♦s ❝♦♥str✉✐r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡st✐♠❛❞♦r❡sWn(X1, ..., Xn)q✉❡
♣♦❞❡ s❡r ❞❡✜♥✐❞❛ ❝♦♠♦ ✉♠ ❡st✐♠❛❞♦r ❝♦♥s✐st❡♥t❡ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ θ✱ ♣❡rt❡♥❝❡♥t❡ ❛♦ ❡s♣❛ç♦ ♣❛r❛♠étr✐❝♦Θ✱ s❡ ❛s ❡q✉❛çõ❡s ✭✶✳✶✼✮ ❡ ✭✶✳✶✽✮ ❢♦r❡♠ ✈á❧✐❞❛s✳
lim
n→∞V arθWn= 0 ✭✶✳✶✼✮
lim
n→∞V iesθWn = 0 ✭✶✳✶✽✮
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ é ❞❡s❡❥á✈❡❧ q✉❡ ♦ ❡st✐♠❛❞♦r s❡❥❛ ❝♦♥s✐st❡♥t❡ ♣♦✐s ❛♦ ❛✉♠❡♥t❛r ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛✱ ❡❧❡ s❡ ❛♣r♦①✐♠❛rá ❞♦ ✈❛❧♦r r❡❛❧ ❞♦ ♣❛râ♠❡tr♦ ❡st✐♠❛❞♦✳ P❛r❛ q✉❡ ♦ ❡st✐♠❛❞♦r s❡❥❛ ❝♦♥s✐st❡♥t❡✱ ♦ ❡st✐♠❛❞♦r ✉t✐❧✐③❛❞♦ ❞❡✈❡ ❞✐♠✐♥✉✐r s✉❛ ✈❛r✐â♥❝✐❛ à ♠❡❞✐❞❛ q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❝r❡s❝❡ ❡ ❛♣r❡s❡♥t❛r ✉♠ ✈✐és t❡♥❞❡♥❞♦ à ③❡r♦✳
❆ ✐♥✈❡st✐❣❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ ♣❛rt❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❝♦♥s✐stê♥❝✐❛ ❞♦s ❡st✐♠❛✲ ❞♦r❡s✱ ✐♥✈❡st✐❣❛♥❞♦ ♦ ✐♠♣❛❝t♦ ❞♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ♥❛ ✈❛r✐â♥❝✐❛ ❞❡ss❡s ❡ ♥♦ r❡s✉❧t❛❞♦s ❞❡ t❡st❡s ✐♥❢❡r❡♥❝✐❛✐s ✐♥✢✉❡♥❝✐❛❞♦s ♣❡❧❛ ♣❡q✉❡♥❛ ✈❛r✐â♥❝✐❛✱ ❡st✐♠❛❞❛ ❝❛❞❛ ✈❡③ ♠❡♥♦r à ♠❡❞✐❞❛ q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❝r❡s❝❡✳
❈❛♣ít✉❧♦ ✷
❆♠♦str❛❣❡♠ ❈♦♠♣❧❡①❛
✷✳✶ ■♥tr♦❞✉çã♦
❙❡❣✉♥❞♦ ❑✐s❤ ❡ ❋r❛♥❦❡❧ ✭✶✾✼✹✮✱ ♦s ♣r✐♥❝✐♣❛✐s ♠ét♦❞♦s ❡st❛tíst✐❝♦s ✐♥❢❡r❡♥❝✐❛✐s ❢♦✲ r❛♠ ❝r✐❛❞♦s ❝♦♠ ❛ s✉♣♦s✐çã♦ ❞❡ ❆♠♦str❛❣❡♠ ❆❧❡❛tór✐❛ ❙✐♠♣❧❡s ✭❆❆❙✮✳ ❆ss✉♠✐r q✉❡ ❛s ♦❜s❡r✈❛çõ❡s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s é ✉♠❛ ❡s❝♦❧❤❛ q✉❡ ❢❛❝✐❧✐t❛ ♦s ❝á❧❝✉❧♦s ❡ s✐♠♣❧✐✜❝❛ ❛s ✐♥t❡r♣r❡t❛çõ❡s✳ P♦ré♠✱ ♥❡♠ s❡♠♣r❡ ♦ ❡st✉❞♦ ❛♣r❡s❡♥t❛ ❡ss❛s ❝♦♥❞✐çõ❡s ✐❞❡❛✐s✳
❙❡❣✉♥❞♦ ❈❤❛♠❜❡rs ❡ ❙❦✐♥♥❡r ✭✷✵✵✸✮✱ ✉♠❛ ❞❛s ♣r✐♠❡✐r❛s ♣❡r❣✉♥t❛s q✉❡ ♦ ♣❡s✲ q✉✐s❛❞♦r ❞❡✈❡ ❢❛③❡r q✉❛♥❞♦ ❡stá tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ❞❛❞♦s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ✉♠ ♣❧❛♥♦ ❛♠♦str❛❧ ❝♦♠♣❧❡①♦ é ❝♦♠♦ ❡ ♦♥❞❡ ♦ t✐♣♦ ❞❡ ❛♠♦str❛❣❡♠ ✐♥✢✉❡♥❝✐❛ ♥❛ ❛♥á❧✐s❡✳ ❆ ❛♠♦str❛❣❡♠ é ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♣❧❡①❛ q✉❛♥❞♦ ❛ ❛♠♦str❛ é ❝♦❧❡t❛❞❛ ✉t✐❧✐③❛♥❞♦ ❝♦♥❣❧♦✲ ♠❡r❛❞♦s✱ ❡str❛t✐✜❝❛çã♦ ❡✴♦✉ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡s✐❣✉❛❧ ❞❡ s❡❧❡çã♦✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ ❞✉❛s ♦✉ ♠❛✐s té❝♥✐❝❛s ❞❡ ❛♠♦str❛❣❡♠ sã♦ ❝♦♠❜✐♥❛❞❛s✱ ❝❛♣t✉r❛♥❞♦ ❛ss✐♠ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ❢❡♥ô♠❡♥♦✳
❍❡❡r✐♥❣❛ ❡t ❛❧✳ ✭✷✵✶✵✮ ❝♦♠♣❧❡♠❡♥t❛♠ q✉❡ ♦ ♣❡sq✉✐s❛❞♦r ♣♦❞❡ ❛♣r♦✈❡✐t❛r ♦ ❝♦✲ ♥❤❡❝✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❛♠♦str❛❧ ♣❛r❛ ❣❡r❛r r❡s✉❧t❛❞♦s ♠❛✐s ♣r❡❝✐s♦s✳ ▲✐✈r♦s✲❚❡①t♦ ❡ ♣r♦❣r❛♠❛s ❡st❛tíst✐❝♦s ✉t✐❧✐③❛❞♦s ♣♦r ♣❡sq✉✐s❛❞♦r❡s✱ ❞❡ ♠❛♥❡✐r❛ ❣❡r❛❧✱ ♥ã♦ ❞✐s❝✉t❡♠ ❛ ❢♦r♠❛ ❞❡ tr❛❜❛❧❤❛r ❝♦♠ ♦s ❞❛❞♦s q✉❡ ❢♦r❛♠ ❝♦❧❡t❛❞♦s ❡♠ ✉♠ ♣❧❛♥♦ ❛♠♦str❛❧ q✉❡ ❡♥✈♦❧✈❛ ❛♠♦str❛❣❡♠ ♣♦r ❝♦♥❣❧♦♠❡r❛❞♦s✱ ❡str❛t✐✜❝❛çã♦ ❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡s✐❣✉❛❧ ❞❡ s❡❧❡çã♦✳
❆ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s ❡st❛tíst✐❝♦s s❡♠ ❧❡✈❛r ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛s ❝❛r❛❝t❡ríst✐✲ ❝❛s ❞♦ ♣❧❛♥♦ ❛♠♦str❛❧ ❣❡r❛ t❛♥t♦ ❡st✐♠❛t✐✈❛s ♣♦♥t✉❛✐s ❡rr❛❞❛s q✉❛♥t♦ ❡st✐♠❛t✐✈❛s
