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**Unformatted text preview: **Chapter 4: Sample Theory and Sample
Distributions
Joan Llull
Probability and Statistics.
QEM Erasmus Mundus Master. Fall 2015
joan.llull [at] movebarcelona [dot] eu Random Samples Chapter 3. Fall 2015 2 Simple random samples
Our population is described by a probabilistic model.
The data are a set of realizations from the probabilistic model.
The process of obtaining the data is called sampling
(e.g. what we did in Chapter 2 for nite sets) Simple random sampling: a collection of random variables (X1 , ..., XN )
is a simple random sample from FX if:
FX1 ...XN (x1 , ..., xN ) = N
Y FX (xi ), i=1 and thus:
fX1 ...XN (x1 , ..., xN ) = N
Y fX (xi ). i=1
Chapter 3. Fall 2015 3 Sample Mean and Variance Chapter 3. Fall 2015 4 Sample Mean
Statistic: single measure of some attribute of a sample.
Chapter 1, descriptive statistics. Now, we are using them to infer some characteristic of the population. A statistic is a random variable Sample mean: ¯N ≡
X 1
N ⇒ sample distribution. PN i=1 Xi . Some properties (regardless of the functional form of FX ): ¯ N ] = E[X].
E[X
¯N ) =
Var(X
Chapter 3. Fall 2015 Var(X)
(precision).
N 5 Sample variance
Sample variance:
N
1 X
¯ N )2 .
(Xi − X
σ
ˆ ≡
N
2 i=1 2 ]=
⇒ E[ˆ
σN N −1 2
N σ : expect less dispersion than in population. Corrected sample variance: N
≡
σ
ˆ2 =
N −1 N s2N PN ¯ N )2
−X
.
N −1 i=1 (Xi E[s2N ] = σ 2 .
Var(s2N ) =
Chapter 3. Fall 2015 2σ 4
N −1 + µ4 −3σ 4
.
N
6 Sample variance Ideal sample variance:
2
σ
˜N N
1 X
(Xi − µ)2 .
≡
N
i=1 2 ] = σ2.
E[˜
σN
2 )=
Var(˜
σN ⇒ 1
N [µ4 − σ 4 ] < Var(s2N ) This statistic cannot be computed without knowing Chapter 3. Fall 2015 µ. 7 Sampling form a normal population:
χ2 , t, and F distributions Chapter 3. Fall 2015 8 Distribution of the sample mean
Let X ∼ N (µ, σ 2 ). Then, Z≡ ¯ N ∼ N (µ, σ 2 /N ),
X ¯N − µ
X
√
∼ N (0, 1).
σ/ N This would help in making inference about ... but we don't know Use sN
2 instead and: µ if we knew σ2... σ2. ⇒ s2N is a random variable: we need to derive the distribution of the transformed random variable. Some intermediate steps rst. Chapter 3. Fall 2015 9 Intermediate steps I 1. Z˜ ≡ (Z˜1 , ..., Z˜K )0
˜i ∼ N (0, 1).
with Z Chi-squared: Let be a vector of random variables, Then: K i.i.d. 2
˜ = Z˜12 + ... + Z˜K
W
= Z˜ 0 Z˜ ∼ χ2K . The degrees of freedom (K ): number of independent squared
standard normals included.
The support of this distribution is R+ .
˜ ] = K and Var(W
˜ ) = 2K .
E[W
2. Let ˜ ∼ N N (0, Σ).
X Then: ˜ 0 Σ−1 X
˜ ∼ χ2
X
N Chapter 3. Fall 2015 10 Intermediate steps II 3. Let M be a size K ×K matrix that: is idempotent (satises M M = M ),
symmetric (satises M 0 = M ),
and has rank(M ) = R ≤ K .
Then: M is singular (with the only exception of M = I ).
M is diagonalizable, and its eigenvalues are either 0 or 1. It can always be diagonalized as M = C 0 ΛC such that C 0 C =
I , and Λ is a matrix that include ones in the rst R elements
of the diagonal and zeros elsewhere.
⇒ tr(M ) = rank(M ) Chapter 3. Fall 2015 (and thus always a natural number). 11 Intermediate steps III 4. Let Z˜ ∼ N K (0, I), and symmetric matrix with M be a size K × K
rank(M ) = R ≤ K . idempotent and
Then: Z˜ 0 M Z˜ ∼ χ2R
5. Let Z˜ ∼ N K (0, I), M be a size K × K idempotent and
rank(M ) = R ≤ K . Also let P be a
˜ 0 M Z˜ and P Z˜ are
that P M = 0. Then Z and symmetric matrix with Q×N matrix such independent. Chapter 3. Fall 2015 12 Student-t
Using these steps: (N − 1)s2N
∼ χ2N −1 .
σ2
2
Student-t: Let Z ∼ N (0, 1) and W ∼ χK , with Z and W being independent.
W ≡ Then: Z
t≡ q ∼ tK . W
K Some characteristics:
E[t] = 0.
K
for K > 2.
Var(t) = K−2
Symmetric with respect to zero, support is R.
When N → ∞ it is similar to a normal.
Thus we can make inference Z
q Chapter 3. Fall 2015 W
N −1 = q on µ without knowing σ :
¯
X−µ
√
σ/ N (N −1)s2
/σ 2
N
N −1 = ¯ − µ)
(X
√
∼ tN −1 .
s/ N 13 F distribution F -distribution: W1 and W2 be two independent
W1 ∼ χ2K and W2 ∼ χ2Q . Then: Let variables such that F ≡ random W1 /K
∼ FK,Q .
W2 /Q Some characteristics: E[F ] = Q
Q−2 for Q > 2. (tK )2 ∼ F1,K
Used to make Chapter 3. Fall 2015 inference about σ2. 14 Bivariate and Multivariate Sampling Chapter 3. Fall 2015 15 Bivariate and Multivariate Sampling In a multivariate random sample,
of a (X1 , ..., XN ) are N realizations random vector. All the results above apply. We can also construct corrected covariances. Chapter 3. Fall 2015 16 ...

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