PDF of Lecture Notes - School of Mathematical Sciences

More documents

Recommendations

Info

2. STATISTICAL INFERENCE Figure 21: The 3 tests Theorem. 2.4.1 (Neyman-Pearson Lemma) Consider the test H 0 : θ = θ 0 vs. H a : θ = θ a defined by the rule for some constant k. reject H 0 for f(x; θ 0) f(x; θ a ) ≤ k, Let α ∗ be the type I error rate and 1 − β ∗ be the power for this test. Then any other test with α ≤ α ∗ will have (1 − β) ≤ (1 − β ∗ ). Pro<strong>of</strong>. Let C be the critical region for the LR test and let D be the critical region (RR) for any other test. 111
2. STATISTICAL INFERENCE Let C 1 = C ∩ D and C 2 , D 2 be such that C = C 1 ∪ C 2 , C 1 ∩ C 2 = φ D = C 1 ∪ D 2 , C 1 ∩ D 2 = φ Figure 22: Neyman-Pearson Lemma 112
Page 1 and 2:
MATHEMATICAL STATISTICS III Lecture
Page 3 and 4:
1.9 Moments . . . . . . . . . . . .
Page 5 and 6:
1. DISTRIBUTION THEORY ⎧∑ h(x)p
Page 7 and 8:
n(1 − p) E(X) = , p n(1 − p) Va
Page 9 and 10:
1. DISTRIBUTION THEORY ) ( N ) p(x)
Page 11 and 12:
1. DISTRIBUTION THEORY that is, ⎧
Page 13 and 14:
1. DISTRIBUTION THEORY 3. Gamma (K,
Page 15 and 16:
1. DISTRIBUTION THEORY Figure 9: Ca
Page 17 and 18:
1. DISTRIBUTION THEORY F Y (y) = P
Page 19 and 20:
1. DISTRIBUTION THEORY Solution. Ob
Page 21 and 22:
1. DISTRIBUTION THEORY 1.6 Moments
Page 23 and 24:
1. DISTRIBUTION THEORY where φ(a)
Page 25 and 26:
1. DISTRIBUTION THEORY Examples 1.
Page 27 and 28:
1. DISTRIBUTION THEORY Possible val
Page 29 and 30:
1. DISTRIBUTION THEORY Remarks 1. O
Page 31 and 32:
1. DISTRIBUTION THEORY Figure 12: A
Page 33 and 34:
1. DISTRIBUTION THEORY Solution. Re
Page 35 and 36:
⎧ ⎪⎨ 1 0 < x 2 < 1 f 2 (x 2 )
Page 37 and 38:
1. DISTRIBUTION THEORY Just substit
Page 39 and 40:
1. DISTRIBUTION THEORY 2. Suppose Z
Page 41 and 42:
1. DISTRIBUTION THEORY Solution. St
Page 43 and 44:
1. DISTRIBUTION THEORY Recall stand
Page 45 and 46:
∂ ∂r g 1(r, θ) = ∂ r cos θ
Page 47 and 48:
1. DISTRIBUTION THEORY ⇒ det(G) =
Page 49 and 50:
1. DISTRIBUTION THEORY Definition.
Page 51 and 52:
1. DISTRIBUTION THEORY Proof. Consi
Page 53 and 54:
1. DISTRIBUTION THEORY Proof. 1. (C
Page 55 and 56:
1. DISTRIBUTION THEORY Proof. M X (
Page 57 and 58:
1. DISTRIBUTION THEORY Proof. M AX+
Page 59 and 60:
1. DISTRIBUTION THEORY Remark This
Page 61 and 62:
1. DISTRIBUTION THEORY 2. If Z 1 ,
Page 63 and 64: 1. DISTRIBUTION THEORY where X i ,
Page 65 and 66: 1. DISTRIBUTION THEORY ⎧ ⎪⎨
Page 67 and 68: 1. DISTRIBUTION THEORY If we take Z
Page 69 and 70: =⇒ Var(X) = Σ = =⇒ Σ −1 = =
Page 71 and 72: 1. DISTRIBUTION THEORY 2. Independe
Page 73 and 74: 1. DISTRIBUTION THEORY and (R 1 ) n
Page 75 and 76: 1. DISTRIBUTION THEORY Figure 16: D
Page 77 and 78: 1. DISTRIBUTION THEORY Remarks This
Page 79 and 80: 1. DISTRIBUTION THEORY Hence we hav
Page 81 and 82: 2. STATISTICAL INFERENCE Definition
Page 83 and 84: 2. STATISTICAL INFERENCE Figure 19:
Page 85 and 86: 2. STATISTICAL INFERENCE Remark If
Page 87 and 88: 2. STATISTICAL INFERENCE Finally ob
Page 89 and 90: 2. STATISTICAL INFERENCE Proof. (1)
Page 91 and 92: 2. STATISTICAL INFERENCE Hence the
Page 93 and 94: 2. STATISTICAL INFERENCE =⇒ f is
Page 95 and 96: 2. STATISTICAL INFERENCE Examples (
Page 97 and 98: 2. STATISTICAL INFERENCE According
Page 99 and 100: 2. STATISTICAL INFERENCE p = 2 =⇒
Page 101 and 102: 2. STATISTICAL INFERENCE To find ˆ
Page 103 and 104: 2. STATISTICAL INFERENCE If X ∼Ex
Page 105 and 106: 2. STATISTICAL INFERENCE (2) f is s
Page 107 and 108: 2. STATISTICAL INFERENCE Then, U n
Page 109 and 110: 2. STATISTICAL INFERENCE and the al
Page 111 and 112: 2. STATISTICAL INFERENCE =⇒ W =
Page 113: 2. STATISTICAL INFERENCE Recall Wal
Page 117 and 118: 2. STATISTICAL INFERENCE Figure 23:
show all

PDF of Lecture Notes - School of Mathematical Sciences

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?