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機率極限 & 機率收斂 Probability Limit and Convergence in Probability. Convergence Concepts. This section treats the somewhat fanciful idea of allowing the sample size to approach infinity and investigates the behavior of certain sample quantities as this happens. We are mainly
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機率極限&機率收斂Probability Limit and Convergence in Probability
Convergence Concepts • This section treats the somewhat fanciful idea of allowing the sample size to approach infinity • and investigates the behavior of certain sample quantities as this happens. We are mainly • concerned with three types of convergence, and we treat them in varying amounts of detail.
Sequences of Random Variables(x1,x2,…,x3) • Interested in behavior of functions of random variables such as means, variances, proportions • For large samples,exact distributions can be difficult/impossible to obtain • Limit Theorems can be used to obtain properties of estimators as the sample sizes tend to infinity • Convergence in Probability – Limit of an estimator • Convergence in Distribution – Limit of a CDF • Central Limit Theorem – Large Sample Distribution of the Sample Mean of a Random Sample
理論骰子平均出現點數是 1*1/6+2*1/6+…+6*1/6=21/6 骰子模擬1000次後的平圴出現點數是 5+2+ …+3+….+6/1000-21/6 dx = Law of Large Numbers dx
Convergence in Probability • The sequence of random variables, X1,…,Xn, is said to converge in probability to the constant c, if for every e>0, • Weak Law of Large Numbers (WLLN): Let X1,…,Xn be iid random variables with E(Xi)=m and V(Xi)=s2 < . Then the sample mean converges in probability to m:
Other Case/Rules • Binomial Sample Proportions • Useful Generalizations:
Convergence in Distribution • Let Yn be a random variable with CDF Fn(y). • Let Y be a random variable with CDF F(y). • If the limit as n of Fn(y) equals F(y) for every point y where F(y) is continuous, then we say that Ynconverges in distribution to Y • F(y) is called the limiting distribution function of Yn • If Mn(t)=E(etYn) converges to M(t)=E(etY), then Yn converges in distribution to Y
Example – Binomial Poisson • Xn~Binomial(n,p) Let l=np p=l/n • Mn(t) = (pet + (1-p))n = (1+p(et-1))n = (1+l(et-1)/n)n • Aside: limn (1+a/n)n = ea • limn Mn(t) = limn (1+l(et-1)/n)n = exp(l(et-1)) • exp(l(et-1)) ≡ MGF of Poisson(l) • Xn converges in distribution to Poisson(l=np)
Central Limit Theorem • Let X1,X2,…,Xn be a sequence of independently and identically distributed random variables with finite mean m, and finite variance s2. Then: • Thus the limiting distribution of the sample mean is a normal distribution, regardless of the distribution of the individual measurements
Proof of Central Limit Theorem (I) • Additional Assumptions for this Proof: • The moment-generating function of X, MX(t), exists in a neighborhood of 0 (for all |t|<h, h>0). • The third derivative of the MGF is bounded in a neighborhood of 0 (M(3)(t) ≤ B< for all |t|<h, h>0). • Elements of Proof • Work with Yi=(Xi-m)/s • Use Taylor’s Theorem (Lagrange Form) • Calculus Result: limn[1+(an/n)]n = ea if limnan=a
Asymptotic Distribution Obtaining an asymptotic distribution from a limiting distribution Obtain the limiting distribution via a stabilizing transformation Assume the limiting distribution applies reasonably well in finite samples Invert the stabilizing transformation to obtain the asymptotic distribution Asymptotic normality of a distribution.
趨近效率 Asymptotic Efficiency
Asymptotic Efficiency Comparison of asymptotic variances How to compare consistent estimators? If both converge to constants, both variances go to zero. Example: Random sampling from the normal distribution, Sample mean is asymptotically normal[μ,σ2/n] Median is asymptotically normal [μ,(π/2)σ2/n] Mean is asymptotically more efficient
Convergence in Probability Almost Sure
