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機率極限 & 機率收斂 Probability Limit and Convergence in Probability

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機率極限&機率收斂Probability Limit and Convergence in Probability

- 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.

- 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:

- 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

- 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

- 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

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

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 Mean Square

Convergence in Probability

Almost Sure