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KCL Statistical Modelling > Inference > Flashcards

Inference Flashcards

1
Q

fitted values?

A

Denoted by y^_i; points on FITTED REGRESSION LINE corresponding to values x_i

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2
Q

Residuals?

A
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3
Q

Residual sum of squares?

A
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4
Q

SS_T

A

Total sum of squares; SS_R + SS_E

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5
Q

SS_R =?

A
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6
Q

SS_E?

A
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7
Q

SS_T for constant model?

A

Given Y_i = β_0 + ε_i

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8
Q

Degrees of freedom for SS_T ? Why?

A

n-1
One degree of freedom is taken up by y-

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9
Q

Dof for SS_E? Why?p

A

n-2; because 2 estimated parameters

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10
Q

MS_R =?

A

SS_R / ν_R

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11
Q

ν_R =

A

1

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12
Q

ν_E =?

A

n-2

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13
Q

MS_E=?

A

SS_E / ν_E

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14
Q

ν_T

A

n-1

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15
Q

Variance Ratio=?

A

MS_R / MS_E

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16
Q

F test, F=?

A
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17
Q

H_0 for F test?

A
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18
Q

F test; we reject H_0 if?

A

H0 : β1 = 0

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19
Q

F_cal?

A

Value of variance ratio F calculated for given data set

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20
Q

F test; F_(α;1,n-2) is ?

A

Such that

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21
Q

What does rejecting H_0 in F test mean?

A
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22
Q
A

(n-2)σ^2

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23
Q
A
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24
Q

MS_E is biased? Estimator of?

A

Unbiased estimator of σ^2 ; often denoted S^2

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25
Q

Null model also known as?

A

Constant model

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26
Q

In null model, S^2 is? Isn’t in?

A

Sample variance; Full model

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27
Q

S^2 in full model?

A
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28
Q
A
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29
Q

Standardise SLRM β^_1

A
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30
Q

Form student t from

A
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31
Q

When converting normalised SLRM β^_1 to student t, U = ?

A
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32
Q

Form student t from SLRM β^_1

A
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33
Q

To find a CI for an unknown parameter θ ?

A

To find values of boundaries A and B which satisfy

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34
Q

Find CI for SLRM β^_1? (In terms of Probability)

A
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35
Q

Explicitly, CI for SLRM β_1?

A
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36
Q

Form T_cal under null from SLRM for β^_1?

A

Where null is β_1 = 0

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37
Q

For SLRM T-test reject H_0 if?

A

H_0: β_1=0

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38
Q
A

Standard error of β^_1 (sqrt of variance of β^_1)

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39
Q
A

Estimator of standard error of β^_1

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40
Q

Rewrite (1-α) 100% CI for β_1 in terms of standard error (T test for constant model)

A
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41
Q

Rewrite to include standard error: test statistic for H_0: β_1 = 0

A
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42
Q

in SLRM, μ_i = ?

A
43
Q

In SLRM, LSE of μ_i is?

A
44
Q

In full SLRM, distribution of LSE of μ_0 is?

A
45
Q

In full SLRM, CI for μ_0 is?

A
46
Q

In full SLRM, test H_0 : μ_0 = μ* given?

A
47
Q

se(μ0^)^

A
48
Q
A
49
Q

Hat matrix

A
50
Q

Special property of Hat matrix?

A

Idempotent:
- H=H^T
- HH = H

51
Q

If matrix A is idempotent then

A

(I-A) is idempotent

52
Q
A

Residual vector

53
Q

E(e) =?

A

0

54
Q

Var(e) = ?

A

σ2(I - H)

55
Q

Proof of Var(e)?

A

Var(e) = (I -H)Var(ε)(I - H)T =

56
Q

Proof of E(e) ?

A
57
Q

Vector of sum of squares of residuals?

A
58
Q
A

0

59
Q

Total sum of squares ? (describe)

A

Regression sum of squares and Residual sum of squares

60
Q

Proof of SS T in vectors

A
61
Q

SS R in vectors

A
62
Q

SSE in vectors

A
63
Q

H 0: for F-test for overall significance of regression

A
64
Q

F-test for Overall significance of regression; DF of overall regression?

A

p-1 (p=#parameters)

65
Q

F-test for Overall significance of regression; df of residual?

A

n-p

66
Q

F-test for Overall significance of regression; df of total?

A

n-1

67
Q

F-test for Overall significance of regression; sum of squares of regression?

A
68
Q

F-test for Overall significance of regression; sum of squares of residual

A
69
Q

F-test for Overall significance of regression; sum of squares for total

A
70
Q

In F-test for overall significance of regression; E(SSE) =

A

(n-p)σ 2

71
Q

The 2 test stats from F-test for overall significance of regression;

A
72
Q

For F-test of overall significance of regression; Reject H 0
If?

A

Reject at (1-α) 100% significance level if

73
Q

β ^vector takes ~

A
74
Q

β^j ~

A
75
Q

100(1-α)%CI for βj is ?

A
76
Q

Test stat for H0 : β j = 0 ?

A

Where c_k is jth diagonal element of (x-Tx-)-1 counting from 0 to p-1)

77
Q

T test for parameters β_j doesn’t tell us anything about comparisons between

A

Models E(Yi) = β0

And E(Yi) = β0+β jx j,i

Doesn’t tell us wether we can accept or reject constant model for linear

78
Q
A
79
Q

Point estimate?

A
80
Q

Prove normality of

A
81
Q

Prove

A
82
Q

Prove

A
83
Q
A
84
Q

Orthogonal matrix?

A

C has, CT C = I

85
Q

For symmetric idempotent A of rank r, There exists…

A

Orthogonal C

86
Q

Properties of trace for any matrices A and B (of appropriate dimensions) and scalar k

A
87
Q

For idempotent A, trace(A) =

A

Rank(A)

88
Q

Proof of reltionship between trace and rank of idempotent A

A
89
Q

Rank(I-H) = ? And prove

A
90
Q

E(ZTAZ) = ?

A
91
Q

Proof of E(ZTAZ) =

A
92
Q
A
93
Q

Proof of

A
94
Q
A

σ 2

95
Q

Lemmas needed to prove

A
96
Q

Prove

A
97
Q
A
98
Q
A
99
Q
A
100
Q

SSR in terms of

A
101
Q

SSR in terms of

A
102
Q

Prove

A
103
Q

What does the hat matrix do?

A

Maps observed values to predicted values:

Y^ = HY

104
Q

DoF of SS_R? Why?

A

KCL Statistical Modelling (9 decks)

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