PDF of Lecture Notes - School of Mathematical Sciences

More documents

Recommendations

Info

2. STATISTICAL INFERENCE =⇒ MSE T (θ) = Var(T ) = θ(1 − θ) . n Remark This example shows that MSE T (θ) must be thought of as a function of θ rather than just a number. E.g., Figure 18 Figure 18: MSE T (θ) as a function of θ Intuitively, a good estimator is one for which MSE is as small as possible. However, quantifying this idea is complicated, because MSE is a function of θ, not just a number. See Figure 19. For this reason, it turns out it is not possible to construct a minimum MSE estimator in general. To see why, suppose T ∗ is a minimum MSE estimator for θ. Now consider the estimator T a = a, where a ∈ R is arbitrary. Then for a = θ, T θ = θ with MSE Tθ (θ) = 0. Observe MSE Ta (a) = 0; hence if T ∗ exists, then we must have MSE T ∗(a) = 0. As a is arbitrary, we must have MSE T ∗(θ) = 0 ∀θ, ∈ Θ =⇒ T ∗ = θ with probability 1. Therefore we conclude that (excluding trivial situations) no minimum MSE estimator can exist. Definition. 2.1.5 The bias of the estimator T is defined by: b T (θ) = E(T ) − θ. 79
2. STATISTICAL INFERENCE Figure 19: MSE T (θ) as a function of θ An estimator T with b T (θ) = 0, i.e., E(T ) = θ, is said to be unbiased. Remarks (1) Although unbiasedness is an appealing property, not all commonly used estimates are unbiased and in some situations it is impossible to construct unbiased estimators for the parameter of interest. 80
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: 2. STATISTICAL INFERENCE Definition
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 and 114: 2. STATISTICAL INFERENCE Recall Wal
Page 115 and 116: 2. STATISTICAL INFERENCE Let C 1 =
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?