All of Statistics: A Concise Course in Statistical Inference
Brief Contents
1. Introduction…………………………………………………………11 Part I Probability
2. Probability……………………………………………………………21 3. Random Variables…………………………………………………..37 4. Expectation…………………………………………………………..69 5. Equalities…………………………………………………………….85 6. Convergence of Random Variables…………………………………89 Part II Statistical Inference
7. Models, Statistical Inference and Learning………………………105 8. Estimating the CDF and Statistical Functionals…………………117 9. The Bootstrap………………………………………………………129 10. Parametric Inference……………………………………………..145 11. Hypothesis Testing and p-values…………………………………179 12. Bayesian Inference………………………………………………..205 13. Statistical Decision Theory……………………………………….227 Part III Statistical Models and Methods
14. Linear Regression…………………………………………………245 15. Multivariate Models………………………………………………269 16. Inference about Independence…………………………………..279 17. Undirected Graphs and Conditional Independence……………297 18. Loglinear Models…………………………………………………309 19. Causal Inference………………………………………………….327 20. Directed Graphs………………………………………………….343 21. Nonparametric curve Estimation……………………………….359 22. Smoothing Using Orthogonal Functions………………………..393 23. Classification……………………………………………………..425 24. Stochastic Processes………………………………………………473 25. Simulation Methods………………………………………………505 Appendix Fundamental Concepts in Inference
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6
Convergence of Random Variables
6.1 Introduction
The most important aspect of probability theory concerns the behavior of sequences of random variables. This part of probability is called large sample theory or limit theory or asymptotic theory. This material is extremely important for statistical inference. The basic question is this: what can we say about the limiting behavior of a sequence of random vari- ables X1, X2, X3, . . .? Since statistics and data mining are all about gathering data, we will naturally be interested in what happens as we gather more and more data.
In calculus we say that a sequence of real numbers xn con- verges to a limit x if, for every ε > 0, |xn − x| < ε for all large n. In probability, convergence is more subtle. Going back to calculus for a moment, suppose that xn = x for all n. Then, trivially, limn xn = x. Consider a probabilistic version of this example. Suppose that X1, X2, . . . is a sequence of random vari- ables which are independent and suppose each has a N(0,1) distribution. Since these all have the same distribution, we are
This is page 89 Printer: Opaque this
90 6. Convergence of Random Variables
tempted to say that Xn “converges” to X ∼ N(0,1). But this can’t quite be right since P(Xn = Z) = 0 for all n. (Two con- tinuous random variables are equal with probability zero.)
Here is another example. Consider X1, X2, . . . where Xi ∼ N (0, 1/n). Intuitively, Xn is very concentrated around 0 for large n. But P(Xn = 0) = 0 for all n. This chapter develops appro- priate methods of discussing convergence of random variables.
There are two main ideas in this chapter:
1. The law of large numbers says that sample average Xn = n−1 ni=1 Xi converges in probability to the ex- pectation μ = E(X).
2. The central limit theorem says that sample average has approximately a Normal distribution for large n. More precisely, √n(Xn − μ) converges in distribution to a Normal(0, σ2) distribution, where σ2 = V(X).
6.2 Types of Convergence
The two main types of convergence are defined as follows.
6.2 Types of Convergence 91
Definition 6.1 Let X1, X2, . . . be a sequence of random vari- ables and let X be another random variable. Let Fn de- note the cdf of Xn and let F denote the cdf of X.
P
1. Xn converges to X in probability, written Xn−→X, if, for every ε > 0,
P(|Xn −X|>ε)→0 (6.1) 2. Xn converges to X in distribution, written Xn X,
as n → ∞.
if,
at all t for which F is continuous.
lim Fn(t) = F(t) (6.2) n→∞
There is another type of convergence which we introduce mainly because it is useful for proving convergence in proba- bility.
Definition 6.2 Xn converges to X in quadratic mean qm
(also called convergence in L2), written Xn−→X, if, E(Xn − X)2 → 0 (6.3)
as n → ∞.
If X is a point mass at c – that is P(X = c) = 1 – we write qm qm P
Xn−→c instead of X−→X. Similarly, we write Xn−→c and Xn c.
Example 6.3 Let Xn ∼ N (0, 1/n). Intuitively, Xn is concentrat- ingat0sowewouldliketosaythatXn 0.Let’sseeifthis is true. Let F be the distribution function for a point mass at
92 6. Convergence of Random Variables
0. Note that √nXn ∼ N (0, 1). Let Z denote a standard normal random variable. For t < 0, Fn(t) = P(Xn < t) = P(√nXn < √nt) = P(Z < √nt) → 0 since √nt → −∞. For t > 0, Fn(t)=P(Xn
E|Xn −X|2 P(|Xn −X|>ε)=P(|Xn −X|2 >ε2)≤ ε2 →0.
Proof of (b). This proof is a little more complicated. You may skip if it you wish. Fix ε > 0 and let x be a continuity point of F. Then
Fn(x) = P(Xn ≤x)=P(Xn ≤x,X ≤x+ε)+P(Xn ≤x,X >x+ε) ≤ P(X ≤x+ε)+P(|Xn −X|>ε)
= F(x+ε)+P(|Xn −X|>ε).
Also,
F(x−ε) = P(X ≤x−ε)=P(X ≤x−ε,Xn ≤x)+P(X ≤x+ε,Xn >x)
6.2 Types of Convergence 93
≤ Fn(x)+P(|Xn −X|>ε).
Hence,
F(x−ε)−P(|Xn−X| > ε) ≤ Fn(x) ≤ F(x+ε)+P(|Xn−X| > ε). Take the limit as n → ∞ to conclude that
F (x − ε) ≤ lim inf Fn(x) ≤ lim sup Fn(x) ≤ F (x + ε). n→∞ n→∞
This holds for all ε > 0. Take the limit as ε → 0 and use the fact that F is continuous at x and conclude that limn Fn(x) = F(x).
Proof of (c). Fix ε > 0. Then,
P(|Xn −c|>ε) = ≤ =
→
P(Xn
=
Let us now show that the reverse implications do not hold.
Convergence in probability does not imply con-
vergence in quadratic mean. Let U ∼ Unif(0,1) and let
Xn = √nI(0,1/n) (U ). Then P(|Xn | > ε) = P(√nI(0,1/n) (U ) > P
ε) = P(0 ≤U < 1/n) = 1/n → 0. Hence, Then Xn−→0. But E(X2)=n 1/ndu=1forallnsoX doesnotconvergein
n0n quadratic mean.
Convergence in distribution does not imply con- vergence in probability. Let X ∼ N(0,1). Let Xn = −X for n = 1,2,3,...; hence Xn ∼ N(0,1). Xn has the same distri- bution function as X for all n so, trivially, limn Fn(x) = F(x) for all x. Therefore, Xn →d X. But P(|Xn −X| > ε) = P(|2X| > ε) = P(|X| > ε/2) ̸= 0. So Xn does not tend to X in probability.
94
6. Convergence of Random Variables
point-mass distribution quadratic mean probability distribution
FIGURE 6.1. Relationship between types of convergence.
P
Warning! One might conjecture that if Xn−→ b then E(Xn) →
We can conclude b. This is not true. Let Xn be a random variable defined by
that E(Xn) → b P(Xn = n2) = 1/n and P(Xn = 0) = 1−(1/n). Now, P(|Xn| <
if X is uniformly
n ε) = P(Xn = 0) = 1−(1/n) → 1. Hence, Z−→0. However,
integrable. See the E(Xn)=[n2×(1/n)]+[0×(1−(1/n))]=n.Thus,E(Xn)→∞. technical appendix.
Summary. Stare at Figure 6.1.
Some convergence properties are preserved under transforma- tions.
Theorem 6.5 Let Xn,X,Yn,Y be random variables. Let g be a continuous function.
PPP
(a) If Xn−→X and Yn−→Y, then Xn +Yn−→X +Y. qm qm qm
(b) If Xn−→X and Yn−→Y, then Xn +Yn−→X +Y. (c) If Xn X and Yn c, then Xn + Yn X + c.
PPP
(d) If Xn−→X and Yn−→Y, then XnYn−→XY. (e) If Xn X and Yn c, then XnYn cX.
6.3 The Law of Large Numbers
Now we come to a crowning achievement in probability, the law of large numbers. This theorem says that the mean of a large sample is close to the mean of the distribution. For example, the proportion of heads of a large number of tosses is expected to be close to 1/2. We now make this more precise.
PP
(f) If Xn−→ X then g(Xn)−→ g(X). (g) If Xn X then g(Xn) g(X).
P
6.3 The Law of Large Numbers 95
Let X1,X2,..., be an iid sample and let μ = E(X1) and
σ2=V(X1).RecallthatthesamplemeanisdefinedasXn=Notethatμ =
n−1 ni=1 Xi and that E(Xn) = μ and V(Xn) = σ2/n. E(Xi) is the same for all i so we can define μ = E(Xi) for any i. By con- vention, we often
Theorem 6.6 (The Weak Law of Large Numbers (WLLN).)
P
If X1,...,Xn are iid , then Xn−→μ.
Interpretation of WLLN: The distribution of Xn be- comes more concentrated around μ as n gets large.
Proof. Assume that σ < ∞. This is not necessary but it simplifies the proof. Using Chebyshev’s inequality,
P|Xn−μ|>ε≤V(Xn)= σ2 ε2 nε2
which tends to 0 as n → ∞.
Example 6.7 Consider flipping a coin for which the probability of heads is p. Let Xi denote the outcome of a single toss (0 or 1). Hence, p = P(Xi = 1) = E(Xi). The fraction of heads after n tosses is Xn. According to the law of large numbers, Xn converges to p in probability. This does not mean that Xn will numerically equal p. It means that, when n is large, the distribution of Xn is tightly concentrated around p. Suppose that p=1/2.HowlargeshouldnbesothatP(.4≤Xn ≤.6)≥.7? First, E(Xn) = p = 1/2 and V(Xn) = σ2/n = p(1−p)/n = 1/(4n). From Chebyshev’s inequality,
P(.4≤Xn ≤.6) = P(|Xn −μ|≤.1)
= 1−P(|Xn −μ|>.1)
≥ 1− 1 =1−25. 4n(.1)2 n
The last expression will be larger than .7 if n = 84.
There is a stronger theorem in the ap- pendix called the strong law of large numbers.
write μ = E(X1).
96 6. Convergence of Random Variables
6.4 The Central Limit Theorem
Suppose that X1,…,Xn are iid with mean μ and variance σ. The central limit theorem (CLT) says that Xn = n−1 i Xi has a distribution which is approximately Normal with mean μ and variance σ2/n. This is remarkable since nothing is assumed about the distribution of Xi, except the existence of the mean and variance.
Theorem 6.8 (The Central Limit Theorem (CLT).) Let X1, . . . , Xn be iid with mean μ and variance σ2. Let Xn = n−1 ni=1 Xi.
Then
where Z ∼ N (0, 1). In other words,
Zn ≡ √n(Xn − μ) Z σ
z 1 −x2/2 lim P(Zn ≤z)=Φ(z)= √2πe dx.
n→∞ −∞
Interpretation: Probability statements about X n can be approximated using a Normal distribution. It’s the probability statements that we are approximating, not the random variable itself.
In addition to Zn N (0, 1), there are several forms of nota- tion to denote the fact that the distribution of Zn is converging to a Normal. They all mean the same thing. Here they are:
Zn ≈ N(0,1)
Xn ≈Nμ,σ2 n
Xn−μ ≈ N 0,σ2 n
√n(Xn−μ) ≈ N0,σ2
6.4 The Central Limit Theorem 97
√n(Xn −μ) σ
Example 6.9 Suppose that the number of errors per computer program has a Poisson distribution with mean 5. We get 125 pro- grams. Let X1, . . . , X125 be the number of errors in the programs. We want to approximate P(X < 5.5). Let μ = E(X1) = λ = 5 and σ2 =V(X1)=λ=5. Then,
P(X <5.5)=P√n(Xn −μ) < √n(5.5−μ)≈P(Z <2.5)=.9938. σσ
The central limit theorem tells us that Zn = √n(X − μ)/σ is approximately N(0,1). However, we rarely know σ. We can estimate σ2 from X1,...,Xn by
1 n
S n2 = n − 1 ( X i − X n ) 2 .
i=1
This raises the following question: if we replace σ with Sn is the
central limit theorem still true? The answer is yes.
≈ N(0,1).
Theorem 6.10 Assume the same conditions as the CLT. Then, √n(Xn −μ) N(0,1).
Sn
You might wonder, how accurate the normal approximation
is. The answer is given in the Berry-Ess`een theorem.
Theorem 6.11 (Berry-Ess`een.) Suppose that E|X1|3 < ∞. Then
33 E|X1 − μ|3 sup|P(Zn ≤z)−Φ(z)|≤ 4 √nσ3
z
. (6.4)
There is also a multivariate version of the central limit theo- rem.
98 6. Convergence of Random Variables
Theorem 6.12 (Multivariate central limit theorem) Let X1, . . . , Xn
be iid random vectors where
X 1 i
X = X2i i .
Xki
μ1 E(X1i)
μ= μ2 = E(X2i) . .
μk E(Xki)
and variance matrix Σ. Let
with mean
X 1 X=X2 .
. Xk
where Xi = n−1 ni=1 X1i. Then,
√n(X − μ) N (0, Σ).
6.5 The Delta Method
If Yn has a limiting Normal distribution then the delta method allows us to find the limiting distribution of g(Yn) where g is any smooth function.
Theorem 6.13 (The Delta Method) Suppose that √n(Yn − μ) N (0, 1)
σ
and that g is a differentiable function such that g′(μ) ̸= 0. Then
√n(g(Yn) − g(μ)) N(0, 1). |g′(μ)|σ
In other words,
Yn ≈Nμ,σ2 =⇒ g(Yn)≈Ng(μ),(g′(μ))2σ2. nn
Example 6.14 Let X1,...,Xn be iid with finite mean μ and fi- nite variance σ2. By the central limit theorem, √n(Xn)/σ N(0,1). Let Wn = eXn. Thus, Wn = g(Xn) where g(s) = es. Since g′(s) = es, the delta method implies that Wn ≈ N(eμ, e2μσ2/n).
There is also a multivariate version of the delta method.
Theorem 6.15 (The Multivariate Delta Method) Suppose that Yn = (Yn1, . . . , Ynk) is a sequence of random vectors such that
√n(Yn −μ)N(0,Σ).
Letg:Rk →Ranelet
∂g
∂y1
∇ g ( y ) = . . . .
∂g ∂yk
Let ∇μ denote ∇g(y) evaluated at y = μ and assume that the elements of ∇μ are non-zero. Then
√n(g(Yn) − g(μ)) N 0, ∇Tμ Σ∇μ . Example 6.16 Let
X11 , X12 , ..., X1n X21 X22 X2n
be iid random vectors with mean μ = (μ1, μ2)T and variance Σ.
Let
1 n
X1 = n X1i,
i=1
1 n
X2 = n X2i
i=1
6.5 The Delta Method 99
100 6. Convergence of Random Variables
and define Yn = X1X2. Thus, Yn = g(X1, X2) where g(s1, s2) = s1s2. By the central limit theorem,
Now
and so
∇TμΣ∇μ = (μ2 Therefore,
∂g s2 ∂s1 =
√nX1−μ1 N(0,Σ). X2 −μ2
∇g(s) =
μ1) σ11 σ12 μ2 = μ2σ11 +2μ1μ2σ12 +μ21σ22.
∂g s1 ∂s2
σ12 σ22 μ1
√n(X1X2 −μ1μ2)N0,μ2σ11 + 2μ1μ2σ12 + μ21σ22. 6.6 Bibliographic Remarks
Convergence plays a central role in modern probability the- ory. For more details, see Grimmet and Stirzaker (1982), Karr (1993) and Billingsley (1979). Advanced convergence theory is explained in great detail in van der Vaart and Wellner (1996) and van der Vaart (1998).
6.7 Technical Appendix
6.7.1 Almost Sure and L1 Convergence
We say that Xn converges almost surely to X, written
as
L1
We say that Xn converges in L1 to X, written Xn−→X, if
E|Xn − X| → 0
Xn−→X, if
P({s : Xn(s) → X(s)}) = 1.
6.7 Technical Appendix 101
as n → ∞.
Theorem 6.17 Let Xn and X be random vaiables. Then:
as P
(a) Xn−→ X implies that Xn−→ X. qm L1
(b) Xn−→ X implies that Xn−→ X. L1 P
(c) Xn−→ X implies that Xn−→ X.
The weak law of large numbers says that Xn converges to EX1 in probability. The strong law asserts that this is also true almost surely.
Theorem 6.18 (The strong law of large numbers.) Let X1, X2, . . . as
be iid. If μ = E|X1| < ∞ then Xn−→μ.
A sequence Xn is asymptotically uniformly integrable if
lim limsupE(|Xn|I(|Xn|>M))=0. M→∞ n→∞
P
6.7.2 Proof of the Central Limit Theorem
If X is a random variable, define its moment generating func- tion (mgf) by ψX(t) = EetX. Assume in what follows that the mgf is finite in a neighborhood around t = 0.
Lemma 6.19 Let Z1, Z2, . . . be a sequence of random variables. Let ψn the mgf of Zn. Let Z be another random variable and denote its mgf by ψ. If ψn(t) → ψ(t) for all t in some open interval around 0, then Zn Z.
proof of the central limit theorem. Let Yi = (Xi − μ)/σ. Then, Zn = n−1/2 i Yi. Let ψ(t) be the mgf of Yi. The mgf of i Yi is (ψ(t))n and mgf of Zn is [ψ(t/√n)]n ≡ ξn(t).
If Xn−→b and Xn is asymptotically uniformly integrable, then E(Xn) → b.
102 6. Convergence of Random Variables
Now ψ′(0) = E(Y1) = 0, ψ′′(0) = E(Y12) = V ar(Y1) = 1. So, ′ t2 ′′ t3 ′′
Now,
= 1+2+3!ψ(0)+··· ξn(t) = ψ√t n
ψ(t) = ψ(0)+tψ(0)+2!ψ (0)+3!ψ (0)+··· t2 t3 ′′
= 1+0+ 2 + 3!ψ (0)+··· t2 t3 ′′
= =
1+ 2n + 3!n3/2ψ (0)+··· t2 + t3 ψ′′(0)+···n
n n t2 t3 ′′
1+ 2 3!n1/2 → et2/2
n
which is the mgf of a N(0,1). The result follows from the previous Theorem. In the last step we used the fact that, if an → a then
1+ann →ea. n
1. Let X1,…,Xn be iid with finite mean μ = E(X1) and finite variance σ2 = V(X1). Let Xn be the sample mean and let Sn2 be the sample variance.
6.8 Excercises
(a) Show that E(Sn2) = σ2.
2P2 2 −1n2
(b)ShowthatSn−→σ .Hint:ShowthatSn =cnn i=1Xi− dnX2n where cn → 1 and dn → 1. Apply the law of large numbers to n−1 ni=1 Xi2 and to Xn. Then use part (e) of Theorem 6.5.
2. Let X1,X2,… be a sequence of random variables. Show qm
that Xn−→ b if and only if
lim E(Xn) = b and lim V(Xn) = 0.
n→∞ n→∞
3. Let X1,…,Xn be iid and let μ = E(X1). Suppose that qm
the variance is finite. Show that Xn−→μ.
4. Let X1,X2,… be a sequence of random variables such
that
PXn=1=1−1 and P(Xn=n)=1.
nn2 n2 Does Xn converge in probability? Does Xn converge in
quadratic mean?
5. Let X1, . . . , Xn ∼ Bernoulli(p). Prove that
1 n
n
i=1
6. Suppose that the height of men has mean 68 inches and standard deviation 4 inches. We draw 100 men at ran- dom. Find (approximately) the probability that the aver- age height of men in our sample will be at least 68 inches.
P
1 n qm 2P2
Xi −→p and n Xi −→p. i=1
7. Let λn = 1/n for n = 1,2,…. Let Xn ∼ Poisson(λn). P
(a) Show that Xn−→ 0.
(b) Let Yn = nXn. Show that Yn−→ 0.
8. Suppose we have a computer program consisting of n =
100 pages of code. Let Xi be the number of errors on the ith
′
1 and that they are independent. Let Y = ni=1 Xi be the total number of errors. Use the central limit theorem to
page of code. Suppose that the Xis are Poisson with mean
approximate P(Y < 90).
6.8 Excercises 103
104 6. Convergence of Random Variables
9. Suppose that P(X = 1) = P(X = −1) = 1/2. Define X = X with probability 1 − n1
n en with probability n1 .
Does Xn converge to X in probability? Does Xn converge
to X in distribution? Does E(X − Xn)2 converge to 0? 10. LetZ∼N(0,1).Lett>0.
(a) Show that, for any k > 0, P(|Z|>t)≤ tk .
E|Z |k (b) (Mill’s inequality.) Show that
21/2 e−t2/2 P(|Z|>t)≤ π t .
Hint. Note that P(|Z| > t) = 2P(Z > t). Now write out what P(Z > t) means and note that x/t > 1 whenever x > t.
11. Suppose that Xn ∼ N(0,1/n) and let X be a random variable with distribution F(x) = 0 if x < 0 and F(x) = 1 if x ≥ 0. Does Xn converge to X in probability? (Prove or disprove). Does Xn converge to X in distribution? (Prove or disprove).
12. Let X,X1,X2,X3,... be random variables that are posi- tive and integer valued. Show that Xn X if and only if
lim P(Xn =k)=P(X =k) n→∞
for every integer k.
6.8 Excercises 105
13. Let Z1,Z2,... be i.i.d., random variables with density f. SupposethatP(Zi >0)=1andthatλ=limx↓0f(x)>0. Let
Xn = n min{Z1,…,Zn}.
Show that Xn Z where Z has an exponential distribu-
tion with mean 1/λ.
14. Let X1,…,Xn ∼ Uniform(0,1). Let Yn = X2n. Find the
limiting distribution of Yn.
15. Let
be iid random vectors with mean μ = (μ1, μ2) and variance
Σ. Let
1 n
X1 = n X1i,
i=1
1 n
X2 = n X2i
i=1
X11 ,X12 ,…,X1n X21 X22 X2n
and define Yn = X1/X2. Find the limiting distribution of Yn.
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7 8 9 10 11 12 log Bush
7 8 9 10 11 12 log Bush
log Buchanon 2345678
Buchanon
0 1000 3000
Residuals
-1.0 0.0 1.0
Residuals -400 0 200
Æ
Æ
0.2
0.4 0.6
0.8
0.2
0.4 0.6
0.8
−10 10 30 800 1200
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ Æ Æ
Æ
Æ
Æ
Æ Æ
Æ
Æ Æ
Æ
Æ
Æ
Æ
Bias squared
Optimal Smoothing
Risk
Variance
0.1 0.2 0.3 0.4 0.5 more smoothing −−−>
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.05 0.10 Oversmoothed
0.15 0.20
0.00 0.05
0.10 0.15 0.20 Just Right
0.00
0.05 0.10 Undersmoothed
0.15 0.20
0 200
400 600 800 1000 number of bins
0 20 40 60 0 2 4 6 8 10 12 14
cross validation score
−14 −12 −10 −8 −6 0 10 20 30 40 50 60
0 10 20 30 40 50
0.00 0.05 0.10
0.15 0.20
−10 −5 0 5
0.00 0.05 0.10 0.15
0.00
0.05 0.10
oversmoothed
0.15 0.20
0.00 0.05 0.10 0.15 0.20
just right
0.00
0.05 0.10
undersmoothed
0.15 0.20
0.002 0.004 0.006 0.008 0.010 0.012 bandwidth
0 200 400 600 800 1000 1200 1400 0 2 4 6 8 10 12
cross−validation score
−7.8 −7.6 −7.4 −7.2 −7.0 0 10 20 30 40 50 60 70
0 10 20 30 40
0.00 0.05 0.10 0.15 0.20
200 400 600 800 1000 200 400 600 800 1000
Undersmoothed Oversmoothed
200 400 600 800 1000
Just Right (Using cross−valdiation)
20 40 60 80 100 120 bandwidth
0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000
estimated risk
3e+05 4e+05 5e+05 6e+05 7e+05 8e+05 0 1000 2000 3000 4000 5000 6000
200 400 600 800 1000
−1000 0 1000 2000 3000 4000 5000 6000
0.0
0.2
0.4 0.6
0.8 1.0
0.0 0.2 0.4
0.6 0.8 1.0
0.0
0.2
0.4 0.6
0.8 1.0
0.0 0.2 0.4
0.6 0.8 1.0
0.0
0.2
0.4 0.6
0.8 1.0
0.0 0.2 0.4
0.6 0.8 1.0
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 0.6 0.8 1.0 1.2 1.4
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
Æ
−0.4 −0.2 0.0 0.2 0.4 −0.4
−0.2
0.0
0.2 0.4
−0.4 −0.2 0.0 0.2 0.4
−0.1
0.0
0.1
0.2
0.0 0.2 0.4
0.6
0.8 1.0
0.0 0.2
0.4 0.6
0.8 1.0
0.0 0.2 0.4
Æ
0.6
0.8 1.0
0.0 0.2
0.4 0.6
0.8 1.0
Æ
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
01234 01234
Æ
0.0 0.2 0.4
0.6
0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.02
Estimated Risk
0.04 0.06 0.08 −0.5 0.0 0.5
−0.4
−0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 0.4
0.0 0.2 0.4
0.6
0.8 1.0
0 500 1000 J
1500 2000
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6 0.8 1.0
0.0
0.2
0.4
0.6 0.8 1.0
0.0
0.2 0.4
0.6 0.8 1.0
0.4 0.6 0.8 1.0 1.2 1.4
Æ Æ
0.0 0.2 0.4 0.6
0.8 1.0
0.0 0.2
0.4 0.6
0.8 1.0
0.0 0.2 0.4 0.6
0.8 1.0
0.0 0.2
0.4 0.6
0.8 1.0
−4 −2 0 2 4 −4 −2 0 2 4
−4 −2 0 2 4 −4 −2 0 2 4
Æ
Æ
f
−0.4 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 0.4
−0.4 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2
0.4
0.0
0.2
0.4
0.6
0.8 1.0
0.0 0.2 0.4
0.6 0.8 1.0
x
0.0
0.2
0.4
0.6
0.8 1.0
0.0 0.2 0.4
0.6 0.8 1.0
−2 0 2 4
0246
−0.2 0.2 0.6 1.0 −1.0 0.0 0.5 1.0 1.5
0.0 0.2
0.4
0.6
0.8 1.0
0.0 0.2
0.4
0.6
0.8 1.0
−0.4 −0.2 0.0 0.2 0.4 −0.5 0.0 0.5
Æ Æ
0.0
0.2 0.4
0.6
0.8 1.0
x1
0.0 0.2 0.4
0.6 0.8 1.0
x2
100 120 140 160 180 200 220 Systolic Blood Pressure
Tobacco
0 5 10 15 20 25 30
Æ
Æ
Æ
Æ
Æ Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
tobacco < 0.51
age < 31.5 |
typea < 68.5
age < 50.5
alcohol < 11.105
famhist:a
0
tobacco < 7.605
ldl < 6.705
0001
typea < 42.5 adiposity < 28.955
tobacco < 4.15
adiposity < 28 0111
typea < 48 000
01
adiposity < 24.435
1
age < 31.5 |
tobacco < 0.51
age < 50.5
000
tobacco < 7.47
01
0
0
1
0
0
20 30 40 50 60 age
tobacco
0 5 10 15 20 25 30
5 10 15 Number of terms in the model
error rate
0.26 0.28 0.30 0.32 0.34
2 4 6 8 10 12 14 size
score
130 135 140 145 150 155 160
0
age < 31.5 |
age < 50.5
typea < 68.5 famhist:a
01
tobacco < 7.605
01
1
Æ
Æ
+ +
+ +
+
+
+
++
++ +
+
+
++ +
+
+
++ +
Æ
0 2 4 6 8 10 time
price
7 8 9 10 11 12 13
0
200
400
600 800 1000
time
0
time
200
400
600 800 1000
xx
−2 0 2 4 6 8 10 −150 −100 −50 0
Æ
Æ
0.6 0.8 1.0 1.2 1.4 0.6 0.8 1.0 1.2 1.4
0246 time
0
200 400 600 800 1000 1200 time
24/09
24/09
24/09
time (seconds)
24/09
24/09
24/09
24/09
24/09
25/09 time (seconds)
25/09
25/09
X(t) X(t)
0 200 600 1000 5 10 15 20
Histogram of w/60
density(x = w/60)
0
200 400 600 800 1000 1200 w/60
density(x = w/60, bw = "ucv")
0
500 1000 1500
0
200 400 600 800 1000 1200
Density Frequency
0.000 0.001 0.002 0.003 0.004 0 50 100 150 200
Density
0.0000 0.0005 0.0010 0.0015
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ Æ
Æ Æ
Æ
Æ
Æ
Æ
Æ
−0.8 −0.6 −0.4 −0.2 0.0 0.2 delta = p2 − p1
0.4 0.6
Æ
f(delta | data)
0.0 0.5 1.0 1.5 2.0
Æ
Æ
Æ
Æ
Æ Æ
Æ
Æ
Æ
Æ
Æ
6
ne4s mNwe4xj Nwobakcfl qn1zsc 01 Nhba5kl 2
mwp q Na3zcj Nheb5s Nung81xvktl
od72 Nu0tf
Nod092yrfi N74
N6
N Nh8 N mNg97yr Ng9yri Nupdi Np83v
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Nd Nph05r 0 Nq2xv
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2 4 6 8 10 x
345 LD50
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0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4
0.6 0.8 1.0
0.0 0.2 0.4
0.6 0.8 1.0
t(p)
Æ
Æ
Æ
0
200
400
600 800 1000
0
200
400
600 800 1000
0
200
400
600 800 1000
−8 −4 0 2 4 −2 0 2 4 −1.0 0.0 1.0 2.0
−5 0 5
−5 0 5
−5 0 5
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
mu p1
−0.6 −0.2 0.2 0.4 0.6 0.8
0
200
400 600 Simulated values of p1
800 1000
0
200
400 600 Simulated values of mu
800 1000
−0.6
−0.4
−0.2
0.0
0.2 0.4
mu
0.2
0.8 1.0
0.0
0.4
0.6
raw estimates and Bayes estimates
0.0 1.0 2.0 3.0 0 50 100 150 200
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