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# 機率極限 & 機率收斂 Probability Limit and Convergence in Probability - PowerPoint PPT Presentation

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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.

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

1*1/6+2*1/6+…+6*1/6=21/6

5+2+ …+3+….+6/1000-21/6

dx

=

Law of Large Numbers

dx

• 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:

• 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)

Example – Scaled Poisson  N(0,1)

• 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 limnan=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.

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

Almost Sure