Chapter4.pdf - Chapter 4 Sample Theory and Sample Distributions Joan Llull Probability and Statistics QEM Erasmus Mundus Master Fall 2015 joan.llull[at

Chapter4.pdf - Chapter 4 Sample Theory and Sample...

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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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