勉強会資料 / “Asymptotic Statistics” Section 3.1

A.W. van der Vaart著 “Asymptotic Statistics”の勉強会資料を公開します。

担当範囲：
第3章 Delta Method
3.1節 Basic Result

Transcript

1. ୈ 6 ճ ษڧձ A.W. van der Vaart ”Asymptotic Statistics” Chapter 3 ぶ゚の๏ Minato 2022 ೥ 12 ݄ 10 ೔
2. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ 〤」〶〠 2 / 39 ࠓճ〣಺༰ Chapter 3ɿDelta Method [1] Section 3.1 Basic Result ๯಄ ʙ Example 3.2 (pp.25, 26) ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ɿଟม਺ؔ਺〣ඍ෼ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿぶ゚の๏ぇ༻⿶〔ඪຊ෼ࢄ〣઴ۙ࿦〣ղੳ A.W. van der Vaart ”Asymptotic Statistics”
3. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ه๏〣४උ Table: ओ〟ه๏ N ࣗવ਺શମ〣ू߹ R ࣮਺શମ〣ू߹ Rk ू߹ {x = (x1 , . . . , xk )⊤ : xi ∈ R (i = 1, . . . , k)} E[·] ظ଴஋ V[·] ෼ࢄ Xn ⇝ X Xn ⿿ X 〠෼෍ऩଋ『぀ Xn P − → X Xn ⿿ X 〠֬཰ऩଋ『぀ N(µ, σ2) ฏۉ µɺ෼ࢄ σ2 〣ਖ਼ن෼෍ Nk (µ, Σ) ฏۉ µɺ෼ࢄڞ෼ࢄߦྻ Σ 〣 (k ࣍ݩ) ଟมྔਖ਼ن෼෍ 3 / 39
4. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ໨࣍ 1 ぶ゚の๏〣֓ཁ 2 શඍ෼〣⿼《〾⿶ 3 Theorem 3.1ɿぶ゚の๏ 4 Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ 3 / 39
5. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ໨࣍ 1 ぶ゚の๏〣֓ཁ 2 શඍ෼〣⿼《〾⿶ 3 Theorem 3.1ɿぶ゚の๏ 4 Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ 4 / 39
6. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ぶ゚の๏〣֓ཁ ぶ゚の๏ ϕ(Tn ) 〣ܗࣜ〜ද《ぁ぀֬཰ม਺ぇɺふぐ゘がల։ぇ༻⿶〛 Tn − θ 〣ଟ߲ࣜ ϕ(θ) + ϕ′(θ)(Tn − θ) + . . . 〜ۙࣅ『぀〈〝 Tn − θ ⿾〾 ϕ(Tn ) − ϕ(θ) ぇਪଌ『぀ࡍ〠༗༻ ௨ৗ〣෼ࢄ (normal variances) 〹෼ࢄ҆ఆԽม׵ (variance stabilizing transformations) 〠ؔ『぀じぐೋ৐ݕఆ〣ඇؤ݈ੑ (nonrobustness) 〟〞⿿ Ԡ༻ྫ (࣍ճ〣ൣғ) ຊઅ〜〤 1 ࣍〣߲〳〜〣ふぐ゘がల։〠〽぀ۙࣅぇߟ⿺぀ 5 / 39
7. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ৚݅ઃఆ〝ふがろ む゘ゐがの θ 〣ਪఆྔ Tn ⿿ط஌〜⿴〿ɺ⿴぀ط஌〣ؔ਺ ϕ ぇ༻⿶〛ද《ぁ぀ ྔ ϕ(θ) ぇਪఆ「〔⿶ɻ ふがろ 1 ϕ(θ) 〣ࣗવ〟ਪఆྔ〝「〛 ϕ(Tn ) ⿿ߟ⿺〾ぁ぀ɻ Tn 〠ؔ『぀઴ۙڍಈぇ༻⿶〛ɺϕ(Tn ) 〠ؔ『぀઴ۙڍಈぇղੳ〜　぀⿾ʁ ɹ Ans. ࿈ଓࣸ૾ఆཧ 1 ⿿ద༻〜　぀ঢ়گ〟〾௚〖〠ղੳ〜　぀ɻ Tn P − → θɺ⿾〙ɺϕ ⿿఺ θ 〠⿼⿶〛࿈ଓ〜⿴぀〝『぀ɻ 〈〣〝　ɺ࿈ଓࣸ૾ఆཧ〽〿ɺϕ(Tn ) P − → ϕ(θ) ⿿੒〿ཱ〙ɻ 1࿈ଓࣸ૾ఆཧ (Theorem 2.3 (ii))ɿg : Rk → Rm ぇɺP (X ∈ C) = 1 ぇຬ〔『ू߹ C ্〜࿈ଓ 〟ؔ਺〝『぀ɻ〈〣〝　ɺXn P − → X 〟〾〥 g(Xn) P − → g(X) ⿿੒〿ཱ〙ɻ 6 / 39
8. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ৚݅ઃఆ〝ふがろ ふがろ 2 √ n(Tn − θ) ⿿⿴぀֬཰෼෍〠෼෍ऩଋ『぀〝　ɺ √ n(ϕ(Tn ) − ϕ(θ)) 〷⿴぀෼ ෍〠෼෍ऩଋ『぀〕あ⿸⿾ʁ Ans. ϕ ⿿ඍ෼Մೳ〟〾 Yes. ݫີ〟දه〜〤〟⿶⿿ɺҎԼ〣ۙࣅࣜ⿿੒〿ཱ〙ɿ √ n(ϕ(Tn ) − ϕ(θ)) ≈ ϕ′(θ) √ n(Tn − θ). ্ࣜ⿾〾 √ n(Tn − θ) ⇝ T 〟〾〥 √ n(ϕ(Tn ) − ϕ(θ)) ⇝ ϕ′(θ)T 〝༧૝〜　぀ɻಛ〠 2 √ n(Tn − θ) ⇝ N(0, σ2) 〟〾〥 √ n(ϕ(Tn ) − ϕ(θ)) ⇝ N(0, ϕ′(θ)2σ2). ্ه〣ओு (ぇҰൠԽ「〔〷〣) 〤 Theorem3.1 〜ূ໌『぀ɻ 2ਖ਼ن෼෍〣ઢܗม׵〷〳〔ਖ਼ن෼෍ɿX ∼ N (µ, σ2) =⇒ aX ∼ N (µ, a2σ2) 7 / 39
9. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ໨࣍ 1 ぶ゚の๏〣֓ཁ 2 શඍ෼〣⿼《〾⿶ 3 Theorem 3.1ɿぶ゚の๏ 4 Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ 8 / 39
10. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ଟม਺よぜぷ゚஋ؔ਺〣ඍ෼ લらがで〠ొ৔「〔֬཰ม਺ Tn 〤 1 ࣍ݩ〜⿴぀〈〝ぇ҉〠Ծఆ「〛⿶〔ɻ 〈〈⿾〾〤ɺTn ⿿ଟ࣍ݩ〜⿴぀৔߹ぇߟ⿺぀ɻ 〒〣〔〶〠〤ɺଟม਺よぜぷ゚஋ؔ਺〠ؔ『぀ඍ෼〠〙⿶〛⿼《〾⿶『぀ɻ ४උɿ゘アはげ〣 o ه๏ [2] x = a 〣ۙ๣〜ఆٛ《ぁ〛⿶぀ؔ਺ f, g 〠〙⿶〛ɺ f(x) = o(g(x)) (x → a) def ⇐⇒ lim x→a f(x) g(x) = 0 f(x) = o(g(x)) (x → a) 〤ɺ ʮx = a 〣ۙ。〜〤 f(x) 〤 g(x) 〽〿〤぀⿾ 〠খ《⿶ʯ〈〝ぇද「〛⿶぀ɻ f(x) = 2x2 + o(x2) (x → 0) 〝〤ɺf(x) = 2x2 + h(x), lim x→0 h(x) x2 = 0 〜 ⿴぀〈〝ぇҙຯ『぀ɻ 9 / 39
11. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ଟม਺よぜぷ゚஋ؔ਺〣ඍ෼ ଟม਺よぜぷ゚஋ؔ਺〣ඍ෼ ϕ : Rk → Rmɿ఺ θ ∈ Rk 〣ۙ๣〠⿼⿶〛ఆٛ《ぁ〛⿶぀ؔ਺ ؔ਺ ϕ ⿿఺ θ 〜ඍ෼Մೳ (differentiable) 〜⿴぀〝〤ɺ⿴぀ઢܗࣸ૾ (ߦྻ) ϕ′ θ : Rk → Rm ⿿ଘࡏ「〛ɺ ϕ(θ + h) − ϕ(θ) = ϕ′ θ (h) + o( h ), h → 0 ⿿੒〿ཱ〙〈〝ぇ⿶⿸ɻ ্ࣜ〤 m ࣍ݩよぜぷ゚〠ؔ『぀ࣜ〜⿴぀〈〝〠஫ҙɻ · 〤ゕがぜ゙ひへぽ゚わɻ ઢܗࣸ૾ h → ϕ′ θ (h) 〤શඍ෼ (total derivative) 〝〷ݺ〥ぁ぀〈〝〷⿴぀ɻ ઢܕࣸ૾ h → ϕ′ θ (h) ⿿࿈ଓ〜⿴぀〝　ɺϕ 〤࿈ଓඍ෼Մೳ (continuously differentiable) 〝ݺ〥ぁ぀ɻ زԿֶత〠〤ɺぎやくアม׵ h → ϕ(h) + ϕ′ θ (h) 〤఺ θ 〠⿼々぀ؔ਺ ϕ 〣 ઀ฏ໘〠〟぀ɻ 10 / 39
12. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ଟม਺よぜぷ゚஋ؔ਺〣ඍ෼ શඍ෼〣ੑ࣭ ؔ਺ ϕ : Rk → Rm ⿿఺ θ ∈ Rk 〜શඍ෼Մೳ〜⿴぀〝　ɺϕ 〤ภඍ෼Մೳ〜⿴ 〿ɺಋؔ਺ (derivative map) h → ϕ′ θ (h) 〤࣍〣ߦྻ ϕ′ θ =     ∂ϕ1 ∂x1 (θ) · · · ∂ϕ1 ∂xk (θ) . . . . . . ∂ϕm ∂x1 (θ) · · · ∂ϕm ∂xk (θ)     ぇ༻⿶〔ߦྻੵ〠〽〿ද《ぁ぀ɻ(i.e. ϕ′ θ (h) = ϕ′ θ h.) શඍ෼Մೳ〜⿴぀〈〝〣े෼৚݅ ؔ਺ ϕ : Rk → Rm ⿿఺ θ ∈ Rk 〜શඍ෼Մೳ〜⿴぀〈〝〣े෼৚݅〤ɺθ 〣ۙ ๣〣શ〛〣఺ x 〠⿼⿶〛ภඍ෼ ∂ϕj (x)/∂xi ⿿ଘࡏ「ɺ⿾〙఺ θ 〠⿼⿶〛࿈ଓ 〜⿴぀〈〝〜⿴぀ɻ ୯〠ภඍ෼⿿ଘࡏ『぀〕々〜〤ɺશඍ෼Մೳ〝〤ݶ〾〟⿶ɻ 11 / 39
13. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ࢀߟɿ1 ม਺࣮਺஋ؔ਺〣ඍ෼⿾〾ଟม਺࣮਺஋ؔ਺〣ඍ෼〭 ؔ਺ ϕ : R → R 〣఺ θ ∈ R 〠⿼々぀ඍ෼܎਺ ϕ′(θ) 〤 ϕ′(θ) = lim h→0 ϕ(θ + h) − ϕ(θ) h 〠〽〿ఆ〶〾ぁ぀ɻ3 ্ࣜ〤 ϕ(θ + h) − ϕ(θ) = ϕ′(θ)h + o(h) (h → 0) 〝ॻ　׵⿺〾ぁ぀ɻ『぀〝ɺϕ ⿿ଟม਺࣮਺஋ؔ਺〜⿴぀৔߹〠〷ඍ෼Մೳੑ 〣֓೦ぇఆ〶〾ぁ぀ɻ ଟม਺࣮਺஋ؔ਺〣ඍ෼ ϕ : Rk → R, θ ∈ Rk 〝『぀ɻ⿴぀ n ࣍ݩྻよぜぷ゚ c ⿿ଘࡏ「ɺ ϕ(θ + h) − ϕ(θ) = hc + o( h ) (h → 0) 〝〟぀〝　ɺϕ 〤 θ 〜 (શ) ඍ෼Մೳ〜⿴぀〝⿶⿶ɺc = ϕ′(θ) 〝ද『ɻ 3ࢀߟらがで〤 Web ্〜ެ։《ぁ〛⿶぀౻Ԭರઌੜ〣ߨٛࢿྉ http://www2.itc.kansai-u.ac.jp/~afujioka/2012-2016/2016/g3/161006g3.pdf ぇࢀর「〛࡞ ੒「〳「〔ɻ 12 / 39
14. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ࢀߟɿଟม਺࣮਺஋ؔ਺〣ඍ෼ 〈〈〜〤ɺRk 〣ݩぇߦよぜぷ゚〝「〛⿶぀ɻҰํɺc 〤ྻよぜぷ゚〜⿴぀ 〔〶ɺよぜぷ゚ಉ࢜〣ੵ hc 〤಺ੵぇද『ɻ ϕ ⿿೚ҙ〣 θ ∈ Rk 〜ඍ෼Մೳ〟〝　ɺn ࣍ݩྻよぜぷ゚〠஋ぇ〝぀よぜぷ ゚஋ؔ਺ ϕ′ : θ → ϕ′(θ) ぇ ϕ 〣ಋؔ਺〝⿶⿸ɻ શඍ෼Մೳ〟ଟม਺࣮਺஋ؔ਺〣ಋؔ਺ ؔ਺ ϕ : Rk → R 〣ಋؔ਺ ϕ′ ⿿ଘࡏ『぀〟〾〥ɺϕ 〤 Rk 〣֤࠲ඪ〠〙⿶〛ภ ඍ෼Մೳ〜ɺ ϕ′ =     ∂ϕ ∂x1 . . . ∂ϕ ∂xk     . ϕ′ 〤ؔ਺ ϕ 〣ޯ഑ (gradient) 〝〷ݺ〥ぁ぀ɻ 13 / 39
15. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ࢀߟɿଟม਺࣮਺஋ؔ਺〣ඍ෼⿾〾ଟม਺よぜぷ゚஋ؔ਺〣ඍ෼〭 ଟม਺よぜぷ゚஋ؔ਺〣ඍ෼〠〙⿶〛〷ಉ༷〠ߟ⿺〾ぁ぀ɻ ଟม਺よぜぷ゚஋ؔ਺〣ඍ෼ ϕ : Rk → Rm, θ ∈ Rk 〝『぀ɻ⿴぀ m × k ߦྻ M ⿿ଘࡏ「ɺ ϕ(θ + h) − ϕ(θ) = Mh + o( h ) (h → 0) 〝〟぀〝　ɺϕ 〤 θ 〜 (શ) ඍ෼Մೳ〜⿴぀〝⿶⿸ɻ〈〣〝　ɺM = ϕ′(θ) 〝ද 「ɺ〈ぁぇ ϕ 〣 θ 〠⿼々぀ඍ෼܎਺〝⿶⿸ɻ ϕ ⿿೚ҙ〣 θ ∈ Rk 〜ඍ෼Մೳ〟〝　ɺn ࣍ݩྻよぜぷ゚〠஋ぇ〝぀よぜぷ ゚஋ؔ਺ ϕ′ : θ → ϕ′(θ) ぇ ϕ 〣ಋؔ਺〝⿶⿸ɻ 14 / 39
16. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ࢀߟɿଟม਺よぜぷ゚஋ؔ਺〣ඍ෼〣ੑ࣭ ଟม਺よぜぷ゚஋ؔ਺〣ඍ෼〠〙⿶〛〷ಉ༷〣ੑ࣭⿿੒〿ཱ〙ɻ ଟม਺よぜぷ゚஋ؔ਺〣ඍ෼〣ੑ࣭ ϕ : Rk → Rm 〣 ϕ 〣ಋؔ਺ ϕ′ ⿿ଘࡏ『぀〟〾〥ɺϕ 〤 Rk 〣֤੒෼〠〙⿶〛ภ ඍ෼Մೳ〜ɺ ϕ′ =     ∂ϕ1 ∂x1 · · · ∂ϕ1 ∂xk . . . . . . ∂ϕm ∂x1 · · · ∂ϕm ∂xk     . ߦྻ ϕ′ 〤んぢもߦྻ〝〷ݺ〥ぁ぀ɻ 15 / 39
17. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ ໨࣍ 1 ぶ゚の๏〣֓ཁ 2 શඍ෼〣⿼《〾⿶ 3 Theorem 3.1ɿぶ゚の๏ 4 Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ 16 / 39
18. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Theorem 3.1 (ぶ゚の๏) ϕ : Dϕ (⊂ Rk) → RmɿRk 〣෦෼ू߹ Dϕ ্〜ఆٛ《ぁ぀ࣸ૾〜⿴〿ɺ θ ∈ Dϕ ্〜ඍ෼Մೳ Tn ɿϕ 〣ఆٛҬ〠஋ぇ〝぀֬཰ม਺よぜぷ゚ rn → ∞ 〟぀਺ྻ rn 〠ର「ɺrn (Tn − θ) ⇝ T 〈〣〝　ɺrn (ϕ(Tn ) − ϕ(θ)) ⇝ ϕ′ θ (T) ⿿੒〿ཱ〙ɻ《〾〠ɺrn (ϕ(Tn ) − ϕ(θ)) 〝 ϕ′ θ (rn (Tn − θ)) 〣ࠩ〤 0 〠֬཰ऩଋ『぀ɻ 17 / 39
19. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Theorem 3.1 (ぶ゚の๏) ϕ : Dϕ (⊂ Rk) → RmɿRk 〣෦෼ू߹ Dϕ ্〜ఆٛ《ぁ぀ࣸ૾〜⿴〿ɺ θ ∈ Dϕ ্〜ඍ෼Մೳ Tn ɿϕ 〣ఆٛҬ〠஋ぇ〝぀֬཰ม਺よぜぷ゚ rn → ∞ 〟぀਺ྻ rn 〠ର「ɺrn (Tn − θ) ⇝ T 〈〣〝　ɺrn (ϕ(Tn ) − ϕ(θ)) ⇝ ϕ′ θ (T) ⿿੒〿ཱ〙ɻ《〾〠ɺrn (ϕ(Tn ) − ϕ(θ)) 〝 ϕ′ θ (rn (Tn − θ)) 〣ࠩ〤 0 〠֬཰ऩଋ『぀ɻ ূ໌ํ਑ɿޙ൒〣ओுˠલ൒〣ओு 〣ॱ〜ࣔ『ʂ ϕ 〤 θ ্ඍ෼Մೳ〕⿾〾ɺϕ(θ + h) − ϕ(θ) = ϕ′ θ (h) + o( h ) (h → 0) R(h) = ϕ(θ + h) − ϕ(θ) − ϕ′ θ (h) 〝⿼　ɺLemma 2.12 ぇద༻「〛 R(Tn − θ) = op ( Tn − θ ). ্ࣜ⿾〾 rn R(Tn − θ) = op (rn Tn − θ ) 〜⿴〿ɺop (Op (1)) = op (1) 〽〿 rn (ϕ(Tn ) − ϕ(θ)) − ϕ′ θ (rn (Tn − θ)) = op (1). と゘びずが〣ఆཧぇద༻「〛લ൒〣ओுぇࣔ『ɻ 17 / 39
20. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Theorem 3.1 〣ূ໌ rn (Tn − θ) ⇝ T 〝 rn → ∞(n → ∞) 〽〿ɺTn − θ P − → 0. ɹ ্ه〣ূ໌ rn → ∞ 〽〿ɺ⿴぀ࣗવ਺ N ⿿ଘࡏ「ɺn ≥ N 〟〾〥 rn > 0 〜⿴぀ɻ〈ぁ〽 〿ɺn ≥ N 〟぀ n 〠ର「ɺ Tn − θ = 1 rn · rn (Tn − θ) 〝ද【぀ɻ 1 rn ⇝ 0, rn (Tn − θ) ⇝ T 〝と゘びずが〣ఆཧ (Theorem 2.8 (ii)) 4 ɺTheorem 2.7 (iii) 5 〽〿ɺ Tn − θ ⇝ 0 · T = 0. ∴ Tn − θ P − → 0. 4と゘びずが〣ఆཧ (Theorem 2.8 (ii))ɿXn ⇝ Xɺ⿾〙 Yn ⿿⿴぀ఆ਺ c 〠෼෍ऩଋ『぀〟〾〥ɺ YnXn ⇝ cX. 5Theorem 2.7 (iii)ɿXn ⿿⿴぀ఆ਺ c 〠෼෍ऩଋ『぀〟〾〥ɺXn P − → c. 18 / 39
21. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Theorem 3.1 〣ূ໌ 〈〈〜ɺϕ 〤 θ ্ඍ෼Մೳ〕⿾〾ɺ⿴぀ઢܗࣸ૾ ϕ′ θ (h) = ϕ′ θ h ⿿ଘࡏ「〛ɺ ϕ(θ + h) − ϕ(θ) = ϕ′ θ (h) + o( h ) (h → 0). 〈ぁ〽〿ɺؔ਺ R ぇ R(h) = ϕ(θ + h) − ϕ(θ) − ϕ′ θ (h) 〝⿼。〝ɺ R(h) = o( h ) (h → 0). (1) 〳〔ɺ R(0) = ϕ(θ + 0) − ϕ(θ) − ϕ′ θ (0) = 0. (2) 19 / 39
22. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Theorem 3.1 〣ূ໌ ࣜ (1)ɾ(2) 〝ɺTn − θ P − → 0 〽〿ɺิ୊ 2.12 ⿿ద༻〜　〛ɺ R(Tn − θ) = op ( Tn − θ ). (3) (෮श) Lemma 2.12 (p.13) R : Rk → R, R(0) = 0. Xn ɿR 〣ఆٛҬ্〠஋ぇ〝〿ɺXn P − → 0 ぇຬ〔『֬཰ม਺ྻ. 〈〣〝　ɺ೚ҙ〣 p > 0 〠ର「ɺ࣍⿿੒〿ཱ〙ɿ R(h) = o ( h p) (h → 0) =⇒ R(Xn ) = op ( Xn p) ࣜ (3) 〤ɺop 〣ఆٛ〽〿ɺ R(Tn − θ) = Yn Tn − θ , Yn P − → 0, ∴ rn R(Tn − θ) = Yn · rn Tn − θ , ∴ rn R(Tn − θ) = op (rn Tn − θ ). 20 / 39
23. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Theorem 3.1 〣ূ໌ rn (Tn − θ) 〤෼෍ऩଋ『぀⿾〾ɺTheorem 2.4 (i) 6 〽〿֬཰༗քɻ〽〘〛ɺ rn (Tn − θ) = Op (1) 〜⿴぀⿾〾ɺ rn R(Tn − θ) = op (Op (1)). ∴ rn R(Tn − θ) = op (1). (∵ op (Op (1)) = op (1)) ҰํɺR 〣ఆٛ⿾〾ࠨลぇมܗ『぀〝ɺ rn R(Tn − θ) = rn ϕ(θ + (Tn − θ)) − ϕ(θ) − ϕ′ θ (Tn − θ) = rn ϕ(Tn ) − ϕ(θ) − ϕ′ θ (Tn − θ) = rn (ϕ(Tn ) − ϕ(θ)) − ϕ′ θ (rn (Tn − θ)). Ҏ্〽〿ɺ rn (ϕ(Tn ) − ϕ(θ)) − ϕ′ θ (rn (Tn − θ)) = op (1). 〽〘〛ɺఆཧ〣ޙ൒〣ओு⿿ࣔ《ぁ〔ɻ 6Theorem 2.4 (i):Xn ⇝ X 〟〾〥ɺXn 〤Ұ༷ۓີ (֬཰༗ք)ɻ 21 / 39
24. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Theorem 3.1 〣ূ໌ ࣍〠ɺఆཧ〣લ൒ぇࣔ『ɻrn (Tn − θ) ⇝ T 〝ɺઢܗࣸ૾ ϕ′ θ 〣࿈ଓੑ〽〿ɺ࿈ ଓࣸ૾ఆཧ⿿ద༻〜　〛ɺ ϕ′ θ (rn (Tn − θ)) ⇝ ϕ′ θ (T). 〈〈〜ɺ rn (ϕ(Tn ) − ϕ(θ)) = rn (ϕ(Tn ) − ϕ(θ)) − ϕ′ θ (rn (Tn − θ)) + ϕ′ θ (rn (Tn − θ)) 〜⿴぀〈〝〝ɺ rn (ϕ(Tn ) − ϕ(θ)) − ϕ′ θ (rn (Tn − θ)) ⇝ 0, ϕ′ θ (rn (Tn − θ)) ⇝ ϕ′ θ (T) 〽〿ɺと゘ びずが〣ఆཧ (Theorem 2.8 (i)) 7 ぇద༻〜　ɺ rn (ϕ(Tn ) − ϕ(θ)) ⇝ ϕ′ θ (T). 〽〘〛ɺఆཧ〣લ൒〣ओு〷ࣔ《ぁ〔ɻ 7と゘びずが〣ఆཧ (Theorem 2.8 (i)): Xn ⇝ Xɺ⿾〙 Yn ⿿⿴぀ఆ਺ c 〠෼෍ऩଋ『぀〟〾〥ɺ Xn + Yn ⇝ X + c. 22 / 39
25. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Theorem 3.1 〣ద༻ྫ Theorem 3.1 ぇద༻〜　぀ঢ়گ〣Ұྫ〝「〛ɺ √ n(Tn − θ) ⇝ Nk (µ, Σ) ⿿੒〿ཱ〙৔߹⿿ڍ〆〾ぁ぀ɻϕ : Rk → Rm ぇ Theorem 3.1 〣Ծఆぇຬ〔『 ؔ਺〝『ぁ〥ɺ √ n(ϕ(Tn ) − ϕ(θ)) ⇝ Nm (ϕ′ θ µ, ϕ′ θ Σ(ϕ′ θ )⊤) ⿿੒〿ཱ〙ɻ ଟมྔਖ਼ن෼෍〣ઢܗม׵ ([3], ԋश 24, p.109) T ∼ Nk (µ, Σ) 〝「ɺϕ′ θ ぇ m × k ߦྻ〝「〔〝　ɺ ϕ′ θ T ∼ Nm (ϕ′ θ µ, ϕ′ θ Σ(ϕ′ θ )⊤). 23 / 39
26. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ ໨࣍ 1 ぶ゚の๏〣֓ཁ 2 શඍ෼〣⿼《〾⿶ 3 Theorem 3.1ɿぶ゚の๏ 4 Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ 24 / 39
27. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) Example 3.2 〣֓ཁɿඪຊ෼ࢄ〣઴ۙ࿦ ⿴぀ 1 ࣍ݩ֬཰෼෍ Q ⿾〾ɺn ݸ〣ඪຊ X1 , . . . , Xn ⿿ಠཱ〠ੜ੒《ぁ〛⿶぀ ঢ়گぇߟ⿺぀ɻ 〔〕「ɺ෼෍ Q 〤ɺ4 ࣍ゑがゐアぷ⿿ଘࡏ『぀〷〣〝『぀ɻ 〈〣〝　ɺඪຊ෼ࢄぇ S2 〝『぀〝ɺ࣍⿿੒〿ཱ〙ɿ √ n(S2 − µ2 ) ⇝ N(0, µ4 − µ2 2 ). 〔〕「ɺµk 〤ɺ֬཰෼෍ Q 〣ฏۉप〿〣 k ࣍ゑがゐアぷぇҙຯ『぀ɻ 〳〔ɺෆภඪຊ෼ࢄ ˜ S2 〠⿼⿶〛〷ಉ༷〣݁Ռ⿿੒〿ཱ〙ɻ Theorem3.1 〹ɺத৺ۃݶఆཧɺਖ਼ن෼෍〣ੑ࣭〟〞ぇ༻⿶〛্هぇࣔ『ɻ 25 / 39
28. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) n ݸ〣ඪຊ X1 , . . . , Xn ⿿ಘ〾ぁ〛⿶぀〝「ɺ౷ܭྔ ¯ X, ¯ X2 ぇ ¯ X = 1 n n i=1 Xi , (ඪຊฏۉ) ¯ X2 = 1 n n i=1 X2 i 〝『぀ɻ〳〔ɺඪຊ෼ࢄぇ S2 = 1 n n i=1 (Xi − ¯ X)2. 〝⿼。ɻ 26 / 39
29. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) ඪຊ෼ࢄ〤ɺؔ਺ ϕ(x, y) = y − x2 ぇ༻⿶〛ɺS2 = ϕ( ¯ X, ¯ X2) 〝ද【぀ɻ ɹ ্ه〣ূ໌ S2 = 1 n n i=1 (Xi − ¯ X)2 = 1 n n i=1 X2 i − 2 ¯ X n i=1 Xi + n ¯ X2 = 1 n n ¯ X2 − 2n ¯ X2 + n ¯ X2 = ¯ X2 − ¯ X2 = ϕ( ¯ X, ¯ X2) 27 / 39
30. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) αk (k = 1, 2, 3, 4) ぇݪ఺प〿〣 k ࣍ゑがゐアぷ〝『぀ɿ αk = E[Xk 1 ] (k = 1, 2, 3, 4) ଟมྔਖ਼ن෼෍〠ؔ『぀த৺ۃݶఆཧぇ༻⿶぀〈〝〜ɺ √ n ¯ X ¯ X2 − α1 α2 ⇝ N2 0 0 , α2 − α2 1 α3 − α1 α2 α3 − α1 α2 α4 − α2 2 (4) ⿿੒〿ཱ〙〈〝ぇࣔ【぀ɻ Example 2.18 ଟมྔਖ਼ن෼෍〣த৺ۃݶఆཧ (p.16) Z1 , Z2 , . . . : i.i.d in Rk, µ = E[Z1 ], Σ = E (Z1 − µ)(Z1 − µ)⊤ 〣〝　ɺ √ n ¯ Zn − µ ⇝ Nk (0, Σ) . 28 / 39
31. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) ࣜ (4) 〣ূ໌ Zn = Xn X2 n (n = 1, 2, . . . ) 〝『぀〝ɺ ¯ Z = ¯ X ¯ X2 〜⿴぀ɻZ1 〣ظ଴஋ µ 〤 µ = α1 α2 〜⿴〿ɺ෼ࢄڞ෼ࢄߦྻ Σ 〤ɺ Σ = E (Z1 − µ)(Z1 − µ)⊤ = E X1 − α1 X2 1 − α2 X1 − α1 X2 1 − α2 ⊤ = E X2 1 − 2α1 X1 + α2 1 X3 1 − α1 X2 1 − α2 X1 + α1 α2 X3 1 − α1 X2 1 − α2 X1 + α1 α2 X4 1 − 2α2 X2 1 + α2 2 = α2 − α2 1 α3 − α1 α2 α3 − α1 α2 α4 − α2 2 . 〽〘〛ɺଟมྔਖ਼ن෼෍〣த৺ۃݶఆཧ〽〿ɺ √ n( ¯ Z − µ) ⇝ N2 (0, Σ). 29 / 39
32. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) 〈〈〜ɺؔ਺ ϕ(x, y) = y − x2 〤ඍ෼Մೳ〜⿴〿ɺθ = (α1 , α2 )⊤ 〠⿼々぀ඍ෼ ܎਺〤ɺ ϕ′ θ = ∂ϕ ∂x (θ), ∂ϕ ∂y (θ) = (−2α1 , 1). T = (T1 , T2 )⊤ ぇ N2 (0, Σ) 〠「〔⿿⿸֬཰ม਺〝『぀〝ɺ √ n( ¯ Z − µ) ⇝ T 〜 ⿴぀⿾〾ɺTheorem 3.1 ぇద༻『぀〝ɺ √ n(ϕ(Zn ) − ϕ(θ)) ⇝ ϕ′ θ (T), ∴ √ n(ϕ(Zn ) − ϕ(θ)) ⇝ −2α1 T1 + T2 . 30 / 39
33. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) √ n(ϕ(Zn ) − ϕ(θ)) ⇝ −2α1 T1 + T2 . ऩଋઌ〣֬཰ม਺ −2α1 T1 + T2 〤ฏۉ 0ɺ෼ࢄڞ෼ࢄߦྻ⿿ α1 , . . . , α4 〜ද 《ぁ぀ਖ਼ن෼෍〜⿴぀ɻ 〈〈〜ɺα1 = 0 〣৔߹ɺϕ(θ) = α2 , Σ = α2 α3 α3 α4 − α2 2 〜⿴぀⿾〾ɺଟมྔ ਖ਼ن෼෍〣पล෼෍〣ੑ࣭ 8 〽〿ɺ T2 ∼ N(0, α4 − α2 2 ). 〽〘〛ɺ √ n(S2 − α2 ) ⇝ N(0, α4 − α2 2 ). 8ଟมྔਖ਼ن෼෍〣पล෼෍〷ਖ਼ن෼෍ɿX = (X1, . . . , Xk) ∼ Nk(µ, Σ) 〣〝　ɺ Xi ∼ N (µk, σkk) 〜⿴぀ɻ[3] ྫ 3.3.8, p.110 31 / 39
34. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) ࣍〠ɺҰൠ〣৔߹ (α1 = 0 〝〤ݶ〾〟⿶৔߹) 〣ۃݶ෼෍ぇٻ〶぀ɻ〒〣〔〶 〠ɺXn ぇத৺Խ「〔֬཰ม਺ Yn = Xn − α1 ぇߟ⿺ɺલらがで〣 α1 = 0 〣ٞ ࿦〠ؼண「〛ࣔ『ɻ 〳』ɺY1 , . . . , Yn 〣ඪຊ෼ࢄ S2 Y 〤ɺX1 , . . . , Xn 〣ඪຊ෼ࢄ S2 〝౳「⿶ɻ ɹ ্ه〣ূ໌ S2 Y = 1 n n i=1 (Yi − ¯ Y )2 = 1 n n i=1 (Xi − α1 ) − 1 n n j=1 (Xj − α1 ) 2 = 1 n n i=1 Xi − ¯ X 2 = S2. 32 / 39
35. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) µk ぇɺY1 〣ݪ఺प〿〣 k ࣍ゑがゐアぷ〝『぀ɿ µk = E[Y k 1 ] (= E[(X1 − α1 )k]) (k = 1, . . . , 4) S2 = S2 Y = ϕ( ¯ Y , ¯ Y 2) 〜⿴〿ɺY1 = X1 − α1 〽〿 µ1 = 0 〜⿴぀ɻ〳〔ɺ ϕ(µ1 , µ2 ) = µ2 − µ2 1 = µ2 〜⿴぀ɻ Ҏ্〽〿ɺα1 = 0 〣৔߹〣ٞ࿦〠⿼⿶〛ɺα2 , α4 ぇ µ2 , µ4 〠ɺZ ぇ Y 〠ஔ　 ׵⿺〛ಉ༷〠ߟ⿺぀〈〝⿿〜　ɺ √ n(S2 − µ2 ) ⇝ N(0, µ4 − µ2 2 ). 33 / 39
36. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 (ඪຊ෼ࢄ) ࠷ޙ〠ɺඪຊ෼ࢄ S2 ぇෆภඪຊ෼ࢄ ˜ S2 = 1 n − 1 n i=1 (Xi − ¯ X)2 〠ஔ　׵⿺〔 ৔߹ぇߟ⿺぀ɻ √ n( ˜ S2 − µ2 ) = √ n n − 1 n S2 − µ2 = √ n n − 1 n S2 − n − 1 n µ2 − 1 n µ2 = √ n · n − 1 n (S2 − µ2 ) − 1 √ n µ2 = n − 1 n →1 · √ n(S2 − µ2 ) ⇝N (0,µ4−µ2 2 ) + − 1 √ n µ2 →0 . 〽〘〛ɺと゘びずが〣ఆཧ〽〿ɺ √ n( ˜ S2 − µ2 ) ⇝ N(0, µ4 − µ2 2 ). 34 / 39
37. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 〣਺஋࣮ݧ (࠶ܝ) Example 3.2 〣֓ཁɿඪຊ෼ࢄ〣઴ۙ࿦ ⿴぀ 1 ࣍ݩ֬཰෼෍ Q ⿾〾ɺn ݸ〣ඪຊ X1 , . . . , Xn ⿿ಠཱ〠ੜ੒《ぁ〛⿶぀ ঢ়گぇߟ⿺぀ɻ 〔〕「ɺ෼෍ Q 〤ɺ4 ࣍ゑがゐアぷ⿿ଘࡏ『぀〷〣〝『぀ɻ 〈〣〝　ɺඪຊ෼ࢄぇ S2 〝『぀〝ɺ࣍⿿੒〿ཱ〙ɿ √ n(S2 − µ2 ) ⇝ N(0, µ4 − µ2 2 ). 〔〕「ɺµk 〤ɺ֬཰෼෍ Q 〣ฏۉप〿〣 k ࣍ゑがゐアぷぇҙຯ『぀ɻ ࠓճ〤ɺ֬཰෼෍ Q ぇਖ਼ن෼෍ N(µ, σ2) 〝「〔ࡍ〠ɺ √ n(S2 − µ2 ) 〣෼෍⿿ ਖ਼ن෼෍ N(0, µ4 − µ2 2 ) 〠ۙ〚。༷ࢠぇɺつアゆ゚つぐど n ぇ਺௨〿ม⿺〛਺ ஋࣮ݧぇߦ〘〔ɻ 35 / 39
38. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 〣਺஋࣮ݧ ਺஋࣮ݧ〣खॱ 1 ぶがの〣ੜ੒ݯ N(µ, σ2) 〣ฏۉ µɺ෼ࢄ σ2 ぇఆ〶぀ɻ ࠓճ〤 µ = 0, σ = 10 〝「〔ɻ 2 つアゆ゚つぐど n ぇఆ〶぀ɻ ࠓճ〤 n = 10, 50, 100, 1000 〣 4 ௨〿ぇࢼ「〔ɻ 3 ऩଋઌ〣෼෍ N(0, µ4 − µ2 ) ⿾〾 M ݸ〣఺ぇੜ੒『぀ɻ ਖ਼ن෼෍〣৔߹ɺµ4 = 3σ4 ⿿੒〿ཱ〙〈〝ぇ༻⿶぀ɻ ࠓճ〤 M = 100000 〝「〔ɻ 4 ҎԼぇ M ճ܁〿ฦ『ɿ 1 ਖ਼ن෼෍ N(µ, σ2) ⿾〾 n ݸ〣ぶがのぇੜ੒「ɺ √ n(S2 − µ2) ぇܭࢉ『぀ɻ 5 ऩଋઌ〣෼෍⿾〾ੜ੒「〔 M ݸ〣఺〝ɺM ݸ〣֤つアゆ゚⿾〾ܭࢉ「〔 √ n(S2 − µ2 ) 〠ؔ「〛めとぷそ゘わぇඳը「ɺ෼෍ぇൺֱ『぀ɻa aৼ〿ฦ〿ɿऩଋઌ〣෼෍〤めとぷそ゘わぇඳը『぀〽〿〷ີ౓〣ۂઢぇゆ゜ひぷ『぀ํ⿿෼⿾〿 〹『⿶〜『〢ɻ 36 / 39
39. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Example 3.2 〣਺஋࣮ݧ݁Ռ 37 / 39 400 200 0 200 400 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 n=10 N(0, 4 2 2 ) n(S2 2 ) 400 200 0 200 400 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 n=50 N(0, 4 2 2 ) n(S2 2 ) 400 200 0 200 400 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 n=100 N(0, 4 2 2 ) n(S2 2 ) 400 200 0 200 400 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 n=1000 N(0, 4 2 2 ) n(S2 2 )
40. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ ⿼い〿〠 ࠓճ〣ษڧձ〜〤ɺԼه〣߲໨ぇѻ⿶〳「〔ɻ ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ɿଟม਺ؔ਺〣ඍ෼ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿぶ゚の๏ぇ༻⿶〔ඪຊ෼ࢄ〣઴ۙ࿦〣ղੳ ɹ ࣍ճɿぶ゚の๏〣〒〣ଞ〣Ԡ༻ྫ 38 / 39
41. ぶ゚の๏〣֓ཁ શඍ෼〣⿼《〾⿶ Theorem 3.1ɿぶ゚の๏ Example 3.2ɿඪຊ෼ࢄ Example 3.2 Example 3.2 〣਺஋࣮ݧ Reference I [1] Aad W Van der Vaart. Asymptotic statistics, Vol. 3. Cambridge university press, 2000. [2] Ճ౻จݩ. ਺ݚߨ࠲でがどɹେֶڭཆɹඍ෼ੵ෼. ਺ݚग़൛. [3] ౷ܭֶ〭〣֬཰࿦, 〒〣ઌ〭: に゜⿾〾〣ଌ౓࿦తཧղ〝઴ۙཧ࿦〭〣Ս々ڮ. ಺ా࿝௽ะ, 2021. [4] ਿӜޫ෉. جૅ਺ֶ 2 ղੳೖ໳ 1, ୈ 2 ר. ౦ژେֶग़൛ձ. [5] स໦௚ٱ. ֬཰࿦. ߨ࠲਺ֶ〣ߟ⿺ํ / ൧ߴໜ [〰⿾] ฤ. ே૔ॻళ, 2004. 39 / 39