• One-sample tests:
  • Sign test
  • Signed-Rank Wilcoxon
• Estimators, confidence intervals, and robustness to outliers
• Bootstrap
  • Error of estimators and significance for hypothesis testing
  • Complete enumerations
  • Tail probability
• Observations from continuous distributions

• Observations from discrete distributions
• Proportion problems
• \(\chi^2\) Tests

• Discrete variables
• The random variable \(X\) consists of categories
• For now, we focus on binary categories failure (0) and success (1)
• \(X\) is a random variable distributed according to the Bernoulli distribution
  • with success probabiliy \(p\) and
  • failure probabililty \(1-p\)
• We know that \(\operatorname{E}(X) = p\) and \(\operatorname{Var} = p(1-p)\)

• Statistical problems can be
  • estimating \(p\)
  • forming confidence intervals around estimate \(\widehat{p}\)
  • and testing hypothesis \[H_0: p = p_0 \text{ versus } H_A: p \ne p_0\]
• Let \(X_1,\dots,X_n\) iid Bernoulli with success probability \(p\) and \(S\) be the total number of successes
• Then \(S\) follows a binomial distribution with \(n\) trials and success probability \(p\)
• If \(p\) is unkown, we estimate \(p\) with \(\widehat{p} = \frac{S}{n}\)

n = 10; p = 1/2; nsim = 10000; obsv = rbinom(nsim, size = n, prob = p)

n = 100; p = 1/2; nsim = 10000; obsv = rbinom(nsim, size = n, prob = p)

• As \(n \to \infty\) while \(p\) is fixed:
  • The de Moivre–Laplace theorem (a special case of the central limit theorem) says \(S\) approach a normal distribution
  • Easy confidence interval (just evaluate cdf of the normal)
  • Approximation of binomial distribution with \(n\) trials and \(p\) success by \(N(np,np(1-p))\)

• Reject null hypotheis if \(|z|\) is large \[z = \frac{\widehat{p}-p_0}{\sqrt{p_0(1-p_0)/n}} \sim N(0,1)\]
• \(z\) is asymptotically standard normal
• An equivalent test statistic of \(|z|\) is \(z^2\)
• Squared normal is distributed as \(\chi^2\) distribution

• Theory:
  • Professional baseball players have different proportion of left handed player than left-handed people in genral population
  • From previous study, we know that general public is has a proportion of \(p_0 = 0.15\)
• Hypothesis testing:
  • \(H_0: p = 0.15 \text{ versus } H_A: p \ne 0.15\)

library(Rfit)
head(baseball)

##   height weight bat throw field average
## 1     74    218   R     L     0   3.330
## 2     75    185   R     R     1   0.286
## 3     77    219   L     L     0   3.040
## 4     73    185   R     R     1   0.271
## 5     69    160   S     R     1   0.242
## 6     73    222   R     R     0   3.920

ind = with(baseball,throw=='L')
n = length(ind)
phat = sum(ind)/n
phat

## [1] 0.2542373

p0 = 0.15
z = (phat-p0)/(sqrt(p0*(1-p0)/n))
pvalue = 1-pchisq(z^2,df=1)
pvalue

## [1] 0.02494189

• In all three nonparametric test (sign, signed-rank, \(\chi^2\)) no assumption on variances of observations
• In contrast, in the \(t\)-test variances are estimated

Since we know that \(S\) follows a binomial distribution, why don't we use it?

## 
##  Exact binomial test
## 
## data:  sum(ind) and n
## number of successes = 15, number of trials = 59, p-value = 0.04192
## alternative hypothesis: true probability of success is not equal to 0.15
## 95 percent confidence interval:
##  0.1498208 0.3844241
## sample estimates:
## probability of success 
##              0.2542373

• Finite sample \(p\)-values have upper bounded significance level \(\alpha\)
• Asymptotic \(p\)-values may be above above \(\alpha\)

• Extension from two categories to multiple caterogries
• Consider discrete RV \(X\) with \(1,2,\dots,c\) categories
• Let \(p(j) = P(X = j)\) define the probabiliy mass function
• We wish to test: \[H_0: p(j) = p_0(j), j = 1,\dots,c\] \[H_A: p(j) \ne p_0(j), \text{ for some } j\]

• \(X_1,\dots,X_n\) is a random sample on \(X\)
• Let \(O_j = \#\{ X_i = j \}\)
• Observed frequencies are constrained \(\sum_{j=1}^c O_j = n\)
• The expected frequency for category \(j\) is \(\operatorname{E}_j = \operatorname{E}_{H_0}(O_j)\)
• Two cases for \(H_0\)

• Case 1:
  • All \(p_0(j)\) are specified
  • So we get \(E_j = np_0(j)\)
  • Test stastitics is \[\chi^2 = \sum_{j=1}^c \frac{(O_j-E_j)^2}{E_j}\]
• Hypothesis \(H_0\) is rejected in favor of \(H_A\) for large values of \(\chi^2\)
• Observed frequencies, \((O_1,\dots,O_c)^T\) has a multinomial distribution, so exact distribution can be obtained
• Asymptotically \(\chi^2\) distribution with \(c−1\) degrees of freedom
• If we know \(c-1\) frequencies, we can calculate the \(c\)th from total \(n\)

Assymptotically equal to \(\chi^2\) with \(c-1\) degress of freedom

pvalue = 1-pchisq(Chi2_0,df=6-1); pvalue

## [1] 0.8624375

Thus there is no evidence to support the dice being unfair

• Case 2:
  • Only form of pmf is known
  • Have to estimate \(p\)
  • Same test stastitics but now with estimate \(\widehat{p}\) \[\chi^2 = \sum_{j=1}^c \frac{(O_j-E_j)^2}{E_j}\]
• Hypothesis \(H_0\) is rejected in favor of \(H_A\) for large values of \(\chi^2\)

• Alternatively to testing for deviation from a model
• We can get a confidence interval on pairwise difference in proportions \(\widehat{p}_j - \widehat{p}_k\)
• The confidence intervals are easy again because, we assume asymptotic normallity

• Goal is to compare several discrete RV, which have same range \(\{ 1,2,\dots,c \}\)
• Consider hypothesis test:
  • \(H_0:\) \(X_1,\dots,X_r\) have the same distribution
  • \(H_A:\) Distributions of \(X_i\) and \(X_j\) differ for some \(i \ne j\)
• Total number of samples \(n = \sum_{i=1}^r n_i\)
• Observed frequencies: \[O_{ij} = \#\{ \text{sample items in sample drawn on } X_i \text{ such that } X_i = j\},\]
• for \(i = 1,\dots,r\) and \(j = 1,\dots,c\)
• \(O_{ij}\) is a \(r \times c\) matrix of observed frequency
• They are called contingency tables

• Compare observed frequencies to the expected frequencies under \(H_0\)
• Estimate the common distribution \((p_1,\dots,p_c)^T\), where \(p_j\) is the probability that category \(j\) occurs
• Estimate probability of category \(j\) overall \[ \widehat{p}_j = \frac{\sum_{i=1}^r O_{ij}}{n}, j = 1,\dots,c\]
• Estimate expected frequencies \(\widehat{E}_{ij} = n_i \widehat{p}_j\)
• Notice that the sample size can vary between variables

• Test statistics \[\chi^2 = \sum_{i=1}^r \sum_{j=1}^c \frac{(O_{ij}-\widehat{E}_{ij})^2}{\widehat{E}_{ij}}\]
• Asymptotically \(\chi^2\) with \((r−1)(c−1)\) degrees of freedom
• This is called test for homogeneity