Elementary statistical inference1

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Elementary Statistical Inference 1
석 진 환
GROOT SEMINAR

Reference
• Introduction to Mathematical Statistics, Hogg, 7e

Which is better: machine learning or statistics?
• Prediction vs. Explanation
• Machine learning models are designed to ma...

목차
1. Estimation
1) Point estimation
2) Interval estimation
3) Order Statistics
2. Hypothesis Testing
3. The Method of Mon...

Sampling and Statistics
• We have a random variable 𝑋 of interest, but its pdf 𝑓 𝑥 or pmf 𝑝(𝑥) is
not known
1. 𝑓 𝑥 or 𝑝(𝑥)...

• We consider the second classification(parametric inference)
• 𝑋 has an exponential distribution, 𝐸𝑥𝑝 𝜃 , 𝜃 is unknown.
•...

• The sample observations have the same distribution as 𝑋,
and we denote them as the random variables 𝑋1, 𝑋2, … , 𝑋 𝑛,
whe...

• Definition 2 (Statistic)
• 𝑋1, 𝑋2, … , 𝑋 𝑛 sample on a random variable 𝑋. Let 𝑇 = 𝑇(𝑋1, 𝑋2, … , 𝑋 𝑛) be a
function of th...

Estimation
• Point estimation
• 표본평균이 15니까, 모평균도 15일것이다!
• MLE
• Interval estimation
• 표본평균이 15니까, 모평균은 (12.3, 14.7) 구간에 있...

Several Properties of point estimators
• Unbiasedness(비편향성), Consistency(일치성), efficiency(효율성)
• Definition 3 (Unbiasednes...

Example : Maximum Likelihood Estimator(MLE)
• 𝐿 𝜃 = 𝑖=1
𝑛
𝑓(𝑥𝑖, 𝜃), likelihood function of the random
sample.
• Often-used...

MLE with EXPONENTIAL DISTRIBUTION
• Random sample 𝑋1, 𝑋2, … , 𝑋 𝑛 is the Γ(1, 𝜃) density. The log of
likelihood function i...

MLE with NORMAL DISTRIBUTION
• 𝑋 ~ 𝑁 𝜇, 𝜎2
∗ 𝜽 = 𝜇, 𝜎
• 𝑓 𝑥; 𝜽 =
1
2𝜋𝜎
𝑒
−
𝑥−𝜇 2
2𝜎2
• If 𝑋1, 𝑋2, … , 𝑋 𝑛 is a random vari...

예제1(점추정)
베스킨라빈스 파인트의 무게는 정규분포를 따른다고 가정한다고 알고
있다. 베스킨라빈스에 10번을 가서, 무게를 재봤더니 다음과 같았다.
MLE에 의한 파인트 무게의 모평균의 추정값은??
표본 평균 : 32...

Confidence Intervals
• We discussed estimating 𝜃 by a statistic 𝜃 = 𝜃 𝑋1, 𝑋2, … , 𝑋 𝑛 ,
where 𝑋1, 𝑋2, … , 𝑋 𝑛 is a sample ...

Confidence Intervals
• Definition 4 (Confidence Interval)
• Let 𝑋1, 𝑋2, … , 𝑋 𝑛 be a sample on a random variable 𝑋, where ...

예제2(구간추정)
베스킨라빈스 파인트의 무게는 정규분포를 따른다고 알고있다.
베스킨라빈스에 10번을 가서, 무게를 재봤더니 다음과 같았다.
베스킨라빈스 파인트 무게의 평균에 대한 95% 신뢰구간은?
베스킨라빈스 파인트 ...

Confidence Interval for 𝜇 Under Normality
• Suppose 𝑋1, 𝑋2, … , 𝑋 𝑛 are a random sample from a 𝑁 𝜇, 𝜎2 . Let 𝑋 and 𝑆2 deno...

𝑡-interval and standard error
• Once the sample is drawn, let 𝑥 and 𝑠 denote the realized
values of the statistics 𝑋 and 𝑆...

예제2 풀이
1 2 3 4 5 6 7 8 9 10
310 315 330 320 325 325 310 315 320 340
𝑥 = 321, 𝑠 = 11.79 𝑛 = 10, 𝑡 𝛼
2,𝑛−1 = 2.262
95% 신뢰구간 ...

Central Limit Theorem(CLT)
• The last example depends on the normality of the sampled items
• However, this is approximate...

Confidence Intervals for Difference in Means
• We compare the means of 𝑋 and 𝑌, denoted by 𝜇1 and 𝜇2,
respectively. Assume...

Confidence Intervals for Difference in Means
• 𝑋 and 𝑌 are normal with the same variance; i.e., 𝜎2
= 𝜎1
2
= 𝜎2
2
.
• Estim...

Order Statistics(순서통계량)
• 전구가 100개 있고, 𝑖 번째 전구의 수명을 확률변수 𝑋𝑖라고 하자.
첫 번째 꺼진 전구의 수명을 𝑌1, 두 번째 꺼진 전구의 수명을 𝑌2,
… , 100번째 꺼진 전구의...

Order Statistics(순서통계량)
• Theorem 2
• 𝑋 : continuous random variable whose pdf 𝑓(𝑥) has support 𝒮 =
𝑎, 𝑏 , where −∞ ≤ 𝑎 < ...

• Proof
• 𝑔 𝑦1, 𝑦2, … , 𝑦𝑛 = 𝑖=1
𝑛!
𝒥𝑖 𝑓 𝑦1 𝑓 𝑦2 ⋯ 𝑓(𝑦𝑛)
=
𝑛! 𝑓 𝑦1 𝑓 𝑦2 ⋯ 𝑓 𝑦𝑛 𝑎 < 𝑦1 < 𝑦2 < ⋯ < 𝑦𝑛 < 𝑏
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
∵ 𝒥𝑖 =...

Quantiles
• Let 𝑋 be a random variable with a continuous cdf 𝐹(𝑥). For
0 < 𝑝 < 1, define the 𝒑th quantile of 𝑋 to be 𝜉 𝑝 =...

• Let 𝑋1, 𝑋2, … , 𝑋 𝑛 be a random sample from the distribution of 𝑋
and let 𝑌1 < 𝑌2 < ⋯ < 𝑌𝑛 be the corresponding order st...

Application of Quantile : Five-number summary
• A five-number summary of the data consist of the following
sample quantile...

예제2
• 확률변수 𝑋의 샘플 15개 데이터가 다음과 같다.
• The minimum : 𝑦1 = 56
• 𝑄1 = 𝑦4 = 94
• 𝑄2 = 𝑦8 = 102
• 𝑄3 = 𝑦12 = 108
• The maximum : ...

Application of Quantile : Box-Whisker plot
• The five-number summary is the basis for a useful and quick plot of the
data....

56 70 89 94 96 101 102 102
102 105 106 108 110 113 116
The minimum : 𝑦1 = 56
𝑄1 = 𝑦4 = 94
𝑄2 = 𝑦8 = 102
𝑄3 = 𝑦12 = 108
The...

Application of Quantile : Diagnostic plot
• In practice, we often assume that the data follow a certain
distribution. Such...

q-q plot
• Suppose 𝑋 is a random variable with cdf 𝐹 𝑥 − 𝑎 𝑏 , where 𝐹(𝑥)
is known but 𝑎 and 𝑏 > 0 may not be.
• Let 𝑍 = (...

q-q plot
• 𝑋1, 𝑋2, … , 𝑋 𝑛 : Random sample from 𝑋 ⇒ 𝑌1 < ⋯ < 𝑌𝑛 the order statistics.
• For 𝑘 = 1, … , 𝑛, let 𝑝 𝑘 =
𝑘
𝑛+1
...

• Figure 4.1 contain q-q plots of three different
Distributions
• Panel C appears to be most linear. We may assume that th...

Confidence Intervals for Quantiles
• For a sample size 𝑛 on 𝑋, let 𝑌1 < 𝑌2 < ⋯ < 𝑌𝑛 be the order statistics. Let 𝑘 =
[ 𝑛 +...

Example:
Confidence Interval for the Median.
• 𝜉.50 ∶ median of 𝐹(𝑥), 𝑄2 ∶ sample median − point estimator 𝑜𝑓𝜉.50
• Select...

예제3
• 모기 4마리를 채집해서 모기 4마리의 수명을 관찰했다. 모기의
수명이 평균이 1인 지수분포를 따를 때, 제일 오래 산 모기의 수
명이 3 이상일 확률은?
• Let 𝑌1 < 𝑌2 < 𝑌3 < 𝑌4 be the...

예제4
• 모기 4마리를 채집해서 모기 4마리의 수명을 관찰했다. 모기의
수명이 0에서 1의 값을 가지는 균등분포를 따를 때, 제일 오래
산 모기와 제일 빨리 죽은 모기의 수명의 차이가 0.5 미만일 확
률은?
• Le...

• 𝑠𝑜𝑙) 𝑃 𝑌4 − 𝑌1 >
1
2
= 0
1
2
𝑦1+
1
2
1
𝑔1,4(𝑦1, 𝑦4)𝑑𝑦4 𝑑𝑦1
• 𝑔1,4 𝑦1, 𝑦4 = 𝑦1
𝑦4
𝑦2
𝑦4
𝑔 𝑦1, 𝑦2, 𝑦3, 𝑦4 𝑑𝑦3 𝑑𝑦2 = 12 𝑦4 ...

Introduction to Hypothesis Testing
4 Introduction to Hypothesis Testing
Statistical Inference
Estimation
Hypothesis Testin...

예제5(점추정/가설검정)
베스킨라빈스 파인트의 무게는 정규분포를 따른다고 가정한다. 베스킨라빈스에 10
번을 가서, 무게를 재봤더니 다음과 같았다.
MLE에 의한 파인트 무게의 모평균의 추정값은??
표본 평균 : 321...

• 귀무가설 : 파인트 무게는 평균 310g이다.
• 연구가설 : 파인트 무게는 평균 310g이 아니다.
• 모평균의 추정값 : 321g -> 귀무가설 기각 (X)
• 기각역 설정 -> 추정값이 기각역에 포함??(O)
...

모분포 : 평균 100, 분산 30
Sampling
표본 평균 : 90
50개
표본 평균 : 80
표본 평균 : 40
표본 평균 : 10
표본 평균 : 1
표본 평균 : 150
귀무가설
귀무가설의 모평균에서 멀어질수록
...

Hypothesis
• 𝐻0 : Null hypothesis(귀무가설)
• Represents no change or no difference from the past
• 𝐻1 : Alternative hypothesi...

Decision Rule
• 𝑋1, … , 𝑋 𝑛 : random sample from the distribution of 𝑋.
• Consider testing the hypotheses :
• 𝐻0 ∶ 𝜃 ∈ 𝜔0 ...

• the decision could wrong.
• Type 1, Type 2 error
• Goal : Select a critical region from all possible critical regions wh...

• Definition 5 (Critical region of size 𝛼)
• We say a critical region 𝐶 is of size 𝛼 if
𝛼 = max
𝜃∈𝜔0
𝑃 𝜃[ 𝑋1, … 𝑋 𝑛 ∈ 𝐶]
•...

Better critical region
• Given two critical regions 𝐶1 and 𝐶2 which are both of size 𝛼,
𝐶1 is better than 𝐶2 if 𝛾 𝐶1
𝜃 ≥ 𝛾...

Test for a Binomial Proportion of Success
• 𝑋~𝐵𝑒𝑟(𝑝), 𝑝0 : probability of dying with some standard treatment
• We want to ...

4 Introduction to Hypothesis Testing

Nomenclature
• 𝐻0 ∶ 𝑝 = 𝑝0 completely specifies the underlying distribution, it
is called a simple hypothesis. ( 𝜔0 = 1)
•...

𝒕-test : Test for 𝜇 Under Normality
• Let 𝑋 have a 𝑁(𝜇, 𝜎2) distribution. Consider the hypotheses
𝐻0 ∶ 𝜇 = 𝜇0 𝑣𝑒𝑟𝑠𝑢𝑠 𝐻1 ∶ ...

• Large sample rule ??
• In practice, we may not be willing to assume that the
population is normal. Usually 𝑡-critical va...

예제6(가설검정 𝑡-test)
베스킨라빈스 파인트의 무게는 모평균이 310g 인 정규분포를 따
른다고 알고있다. 베스킨라빈스에 10번을 가서, 무게를 재봤더니
다음과 같았다.
가설 설정 :
• 귀무가설(𝐻0) : 𝜇 =...

• 기각역 설정(유의수준(𝛼) = 5%)
• Intuitive rejection rule :
• 𝑋 much larger than 310 or much smaller than 310
• 𝑇 =
𝑋−310
𝑆 10
≥ 𝑡...

Randomized tests and 𝑝-value
• Let 𝑋1, … , 𝑋10 be random sample of size 𝑛=10 from a Poisson
distribution with mean 𝜃.
• A ...

• Let 𝑊 have a Bernoulli distribution with probability of success
equal to
𝑃 𝑊 = 1 =
0.050 − 0.019
0.080 − 0.019
=
31
61
•...

Observed Significance Level
• Not many statisticians like randomized tests in practice,
because the use of them means that...

모분포 : 평균 100, 분산 30
Sampling
50개
표본 평균 : 30
표본 평균 : 180
귀무가설
두 표본이 얼마나 귀무가설을
벗어나는지 측정하고 싶다.
표본평균 30이 뽑히는 것만큼
극단적으로 관찰될 확률은...

• 𝑝-value is the observed “tail” probability of a statistic being at
least as extreme as the particular observed value whe...

예제7
• 𝑋1, 𝑋2, … 𝑋25 a random sample from 𝑁(𝜇, 𝜎2 = 4). 𝑋~𝑁(𝜇, 0.16)
• Test 𝐻0 ∶ 𝜇 = 77 against the one-sided alternative h...

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