Statistics

Basic Properties

1. \(E(X) = \sum x p(x)\)
2. \(Var(X) = \sum (x-\mu)^2f(x)\)
3. X is around \(E(X)\), give or take \(SD(X)\)
4. \(E(aX + bY) = aE(X) + bE(Y)\)
5. \(Var(aX + bY) = a^2Var(X) + b^2Var(Y)\)
6. \(Var(X) = E(X^2) - [E(X)]^2\)
7. \(Cov(X_1, X_2) = E(X_1X_2) - E(X_1)E(X_2)\)
8. if \(X\), \(Y\) are independent:
   1. \(M_{X+Y}(t) = M_X(t)M_Y(t)\)
   2. \(E(XY)=E(X)E(Y)\), converse is true if \(X\) and \(Y\) are bivariate normal, extends to multivariate normal

Approximations

Law of Large Numbers

Let \(X_1, X_2, …, X_n\) be IID, with expectation \(\mu\) and variance \(\sigma^2\). \(\overline{X_n} = \frac{1}{n}\sum^{n}_{i=1}X_i\xrightarrow[n]{\infty}\mu\). Let \(x_1, x_2, …, x_n\) be realisations of the random variable \(X_1, X_2, …, X_n\), then \(\overline{x_n} = \frac{1}{n}\sum^{n}_{i=1}x_n \xrightarrow[n]{\infty} \mu\)

Central Limit Theorem

Let \(S_n = \sum^{n}_{i=1}X_i\) where \(X_1, X_2, …, X_n\) IID. \(\frac{S_n - n\mu}{\sqrt{n}\sigma} \xrightarrow[n]{\infty} \mathcal{N}(0,1)\)

Distributions

Poisson(\(\lambda\))

\(E(X) = Var(X) = \lambda\)

Normal \(X \sim \mathcal{N}(\mu, \sigma^2)\)

https://kfrankc.com/posts/2018/10/19/normal-dist-derivation

\(f(x) = \frac{1}{\sqrt{2\pi}\sigma} exp \left(-\frac{(x-\mu)^2}{2\sigma^2}\right), -\infty<x<\infty\)

1. When \(\mu = 0\), \(f(x)\) is an even function, and \(E(X^k) = 0\) where \(k\) is odd
2. \(Y = \frac{X-E(X)}{SD(X)}\) is the standard normal

Gamma \(\Gamma\)

\(g(t) = \frac{\lambda^\alpha}{\Gamma(\alpha)}t^{\alpha-1}e^{-\lambda t}, t \ge 0\)

\(\mu_1 = \frac{\alpha}{\lambda}, \mu_2 = \frac{\alpha(\alpha+1)}{\lambda^2}\)

\(\chi^2\) Distribution

Let \(\mathcal{Z} \sim \mathcal{N}(0,1)\), \(\mathcal{U} = \mathcal{Z}^2\) has a \(\chi^2\) distribution with 1 d.f.

\(f_{\mathcal{U}}(u) = \frac{1}{\sqrt{2\pi}} u^{-\frac{1}{2}} e^{-\frac{u}{2}}, u \ge 0\)

\(\chi_1^2 \sim \Gamma(\alpha=\frac{1}{2}, \lambda=\frac{1}{2})\)

Let \(U_1, U_2, …, U_n\) be \(\chi_1^2\) IID, then \(V=\sum^{n}_{i=1}U_i\) is \(\chi_n^2\) with n degree freedom, \(V \sim \Gamma(\alpha=\frac{n}{2}, \lambda=\frac{1}{2})\)

\(E(\chi_n^2) = n, Var(\chi_n^2) = 2n\)

\(M(t) = \left(1 - 2t\right)^{-\frac{n}{2}}\)

t-distribution

Let \(\mathcal{Z} \sim \mathcal{N}(0,1)\), \(\mathcal{U}_n \sim \chi_n^2\) be independent, \(t_n = \frac{\mathcal{Z}}{\sqrt{U_n / n}}\) has a t-distribution with n d.f.

\(f(t) = \frac{\Gamma([(n+1)/2])}{\sqrt{n}\pi\Gamma(n/2)}\left(1 + \frac{t^2}{n} \right)^{-\frac{n+1}{2}}\)

1. t is symmetric about 0
2. \(t_n \xrightarrow[n]{\infty} \mathcal{Z}\)

F-distribution

Let \(U \sim \chi_m^2, V \sim \chi_n^2\) be independent, \(W = \frac{U/m}{V/n}\) has an F distribution with (m,n) d.f.

If \(X \sim t_n\), \(X^2 = \frac{\mathcal{Z}/1}{U_n/n}\) is an F distribution with (1,n) d.f, with \(w \ge 0\):

For \(n > 2\), \(E(W) = \frac{n}{n-2}\)

Sampling

Let \(X_1, X_2, …, X_n\) be IID \(\mathcal{N}(\mu, \sigma^2)\).

\(\text{sample mean, } \overline{X} = \frac{1}{n}\sum^{n}_{i=1}X_i\)

\(\text{sample variance, } S^2 = \frac{1}{n-1}\sum^{n}_{i=1}\left(X_i-\overline{X}\right)^2\)

Properties of \(\overline{X}\) and \(S^2\)

1. \(\overline{X}\) and \(S^2\) are independent
2. \(\overline{X} \sim \mathcal{N}(\mu, \frac{\sigma^2}{n})\)
3. \(\frac{(n-1)S^2}{\sigma^2} \sim \chi_{n-1}^2\)
4. \(\frac{\overline{X} - \mu}{S/\sqrt{n}} \sim t_{n-1}\)

Simple Random Sampling (SRS)

Assume \(n\) random draws are made without replacement. (Not SRS, will be corrected for later).

Summary of Lemmas

• \(P(X_i =\xi_j) = \frac{n_j}{N}\): Lemma A
• For \(i \ne j\), \(Cov(X_i, X_j) = - \frac{\sigma^2}{N-1}\): Lemma B

Estimation Problem

Let \(X_1, X_2, …, X_n\) be random draws with replacement. Then \(\overline{X}\) is an estimator of \(\mu\). and the observed value of \(\overline{X}\), \(\overline{x}\) is an estimate of \(\mu\).

Standard Error (SE)

SE of an \(\overline{X}\) is defined to be \(SD(\overline{X})\).

Without Replacement

SE is multiplied by \(\frac{N-n}{N-1}\), because \(s^2\) is biased for \(\sigma^2\): \(E(\frac{N-1}{N}s^2) = \sigma^2\), but N is normally large.

Confidence Interval

An approximate \(1-\alpha\) CI for \(\mu\) is

\((\overline{x} - z_{\alpha/2}\frac{s}{\sqrt{n}}, \overline{x} + z_{\alpha/2}\frac{s}{\sqrt{n}})\)

Biased Measurements

Let \(X = \mu + \epsilon\), where \(E(\epsilon) = 0\), \(Var(\epsilon) = \sigma^2\)

Suppose X is used to measure an unknown constant a, \(a \ne \mu\). \(X = a + (\mu - a) + \epsilon\), where \(\mu-a\) is the bias.

Mean square error (MSE) is \(E((X-a)^2) = \sigma^2 + (\mu - a)^2\)

with n IID measurements, \(\overline{x} = \mu + \overline{\epsilon}\)

\(E((x - a)^2) = \frac{\sigma^2}{n} + \left(\mu - a\right)^2\)

\(\text{MSE} = \text{SE}^2 + \text{bias}^2\), hence \(\sqrt{\text{MSE}}\) is a good measure of the accuracy of the estimate \(\overline{x}\) of a.

Estimation of a Ratio

Consider a population of \(N\) members, and two characteristics are recorded: \((X_1, Y_1), (X_2, Y_2), … , (X_n, Y_n)\), \(r = \frac{\mu_y}{\mu_x}\).

An obvious estimator of r is \(R = \frac{\overline{Y}}{\overline{X}}\)

\(Cov(\overline{X},\overline{Y}) = \frac{\sigma_{xy}}{n}\), where

\(\sigma_{xy} := \frac{1}{N}\sum^{N}_{i=1}(x_i-\mu_x)(x_i-\mu_y)\) is the population covariance.

Properties

\(Var( R) \approx \frac{1}{\mu_x^2}\left(r^2\sigma_{\overline{X}}^2 + \sigma_{\overline{Y}}^2 - 2r\sigma_{\overline{X}\overline{Y}}\right)\)

Population coefficient \(\rho = \frac{\sigma_{xy}}{\sigma_{x}\sigma_{y}}\)

\(E( R) \approx r + \frac{1}{n}\left(\frac{N-n}{N-1}\right)\frac{1}{\mu_x^2}\left(r\sigma_x^2-\rho\sigma_x\sigma_y\right)\)

\(s_{xy} = \frac{1}{n-1}\sum^{n}_{i=1}\left(X_i - \overline{X}\right)\left(Y_i - \overline{Y}\right)\)

Ratio Estimates

\(\overline{Y}_R = \frac{\mu_x}{\overline{X}}\overline{Y} = \mu_xR\)

\(Var(\overline{Y}_R) \approx \frac{1}{n}\frac{N-n}{N-1}(r^2\sigma_x^2 + \sigma_y^2 -2r\rho\sigma_x\sigma_y)\)

\(E(\overline{Y}_R) - \mu_y \approx \frac{1}{n}\frac{N-n}{N-1}\frac{1}{\mu_x}\left(r\sigma_x^2 -\rho\sigma_x\sigma_y\right)\)

The bias is of order \(\frac{1}{n}\), small compared to its standard error.

\(\overline{Y}_R\) is better than \(\overline{Y}\), having smaller variance, when \(\rho > \frac{1}{2}\left(\frac{C_x}{C_y}\right)\), where \(C_i = \sigma_i/\mu_i\)

Variance of \(\overline{Y}_R\) can be estimated by

\(s_{\overline{Y}_R}^2 = \frac{1}{n}\frac{N-n}{N-1}\left(R^2s_x^2+s_y^2-2Rs_{xy}\right)\)

An approximate \(1-\alpha\) C.I. for \(\mu_y\) is \(\overline{Y}_R \pm z_{\alpha/2}s_{\overline{Y}_R}\)

Method of Moments

To estimate \(\theta\), express it as a function of moments \(g(\hat{\mu}_1,\hat{\mu}_2,…)\)

Monte Carlo

Monte Carlo is used to generate many realisations of random variable.

\(\overline{X} \xrightarrow[n]{\infty} \alpha/\lambda, \hat{\sigma}^2 \xrightarrow[n]{\infty}\alpha/\lambda^2\), MOM estimators are consistent (asymptotically unbiased).

\(\text{Poisson}(\lambda)\): \(\text{bias} = 0, SE \approx \sqrt{\frac{\overline{x}}{n}}\)

\(N(\mu, \sigma^2)\): \(\mu = \mu_1\), \(\sigma^2 = \mu_2 - \mu_1^2\)

\(\Gamma(\lambda, \alpha)\): \(\hat{\lambda} = \frac{\hat{\mu}_1}{\hat{\mu}_2-\hat{\mu}_1^2}=\frac{\overline{X}}{\hat{\sigma}^2}, \hat{\alpha} = \frac{\hat{\mu}_1^2}{\hat{\mu}_2-\hat{\mu}_1^2}=\frac{\overline{X}^2}{\hat{\sigma}^2}\)

Maximum Likelihood Estimator (MLE)

Poisson Case

\(L(\lambda) = \prod^n_{i=1}\frac{\lambda^{x_i}e^{-\lambda}}{x_i!} = \frac{\lambda\sum^n_{i=1}x_ie^{-n\lambda}}{\prod^{n}_{i=1}x_i!}\)

\(l(\lambda) = \sum^{n}_{i=1}x_i\log\lambda - n\lambda - \sum^{n}_{i=1}\log x_i!\)

ML estimate of \(\lambda_0\) is \(\overline{x}\). ML estimator is \(\hat{\lambda}_0 = \overline{X}\)

Normal case

\(l(\mu, \sigma) = -n\log\sigma - \frac{n\log 2\pi}{2} - \frac{\sum^{n}_{i=1}\left(X_i-\mu\right)^2}{2\sigma^2}\)

\(\frac{\partial l}{\partial \mu} = \frac{\sum \left(X_i - \mu\right)}{\sigma^2} \implies \hat{\mu} = \overline{x}\)

\(\frac{\partial l}{\partial \sigma} = \frac{\sum^{n}_{i=1}\left(X_i-\mu\right)^2}{\sigma^3} - \frac{n}{\sigma} \ \implies \hat{\sigma^2} = \frac{1}{n}\sum^{n}_{i=1}\left(X_i-\overline{X}\right)^2\)

Gamma case

\(l(\theta) = n\alpha\log\lambda + (\alpha -1)\sum^{n}_{i=1}\log X_i - \lambda\sum^{n}_{i=1} X_i - n\log\Gamma(\alpha)\)

\(\frac{\partial l}{\partial \alpha} = n\log\alpha + \sum^{n}_{i=1}\log X_i - \sum^{n}_{i=1}X_i - \frac{n}{\Gamma(\alpha)}\Gamma ‘(\alpha)\)

\(\frac{\partial l}{\partial \lambda} = \frac{n\alpha}{\lambda} - \sum^{n}_{i=1}X_i\)

\(\hat{\lambda} = \frac{\hat{\alpha}}{\hat{x}}\)

Multinomial Case

\(f(x_1, …, x_r) = {n \choose {x_1, x_2, … x_r}} \prod^{n}_{i=1} p_i^{X_i}\)

where \(X_i\) is the number of times the value occurs, and not the number of trials. and \(x_1, x_2, … x_r\) are non-negative integers summing to \(n\). \(\forall i\):

\(E(X_i) = np_i, Var(X_i)=np_i(1-p_i)\)

\(Cov(X_i,X_j) = -np_ip_j, \forall i \ne j\)

\(l(p) = \Kappa + \sum^{r-1}_{i=1}x_i\log p_i + x_r\log(1-p_1-…-p_{r-1})\)

\(\frac{\partial l}{\partial p_i} = \frac{x_i}{p_i} - \frac{x_r}{p_r} = 0 \text{ assuming MLE exists}\)

\(\frac{x_i}{\hat{p}_i} = \frac{x_r}{\hat{p}_r} \implies \hat{p}_i = \frac{x_i}{c}, c=\frac{x_r}{\hat{p}_r}\)

\(\sum^r_{i=1}\hat{p}_i = \sum^r_{i=1}\frac{x_i}{c} = 1 \ \implies c = \sum^{r}_{i=1}x_i = n \implies \hat{p}_i = \frac{\overline{x}_i}{n}\)

same as MOM estimator.

CIs in MLE

\(\frac{\hat{X} - \mu}{s/\sqrt{n}} \sim t_{n-1}\)

Given the realisations \(\overline{x}\) and \(s\), \(\overline{x} \pm t_{n-1, \alpha/2}\frac{s}{\sqrt{n}},\overline{x} + t_{n-1, \alpha/2}\frac{s}{\sqrt{n}}\) is the exact \(1-\alpha\) CI for \(\mu\).

\(\frac{n\hat{\sigma}^2}{\sigma^2} \sim \chi_{n-1}^\), \(\frac{n\hat{\sigma}^2}{\chi_{n-1,\alpha/2}^2}, \frac{n\hat{\sigma}^2}{\chi_{n-1,1-\alpha/2}^2}\) is the exact \(1-\alpha\) CI for \(\sigma\).

Fisher Information

\(I\left( \theta \right) = - E \left( \frac{\partial}{\partial \theta^2} \log f\left( x | \theta \right) \right)\)

General trinomial: \(\left(\frac{X_1}{n}, \frac{X_2}{n} \right)\)

\begin{equation*} \begin{bmatrix} p_1(1-p_1) & -p_1p_2 \ -p_1p_2 & p_2(1-p_2) \end{bmatrix} \frac{1}{n} \end{equation*}

In all the above cases, \(\text{var}(\hat{\theta}) = I(\theta)^{-1}\).

Asymptotic Normality of MLE

As \(n \rightarrow \infty\), \(\sqrt{nI(\theta)}(\hat{\theta} - \theta) \rightarrow N(0,1)\) in distribution, and hence \(\hat{\theta} \sim N\left(\theta, \frac{I\left( \theta \right)^{-1}}{n}\right)\)

As \(\hat{\theta} \xrightarrow[n]{\infty} \theta\), MLE is consistent.

SE of an estimate of \(\theta\) is the SD of the estimator \(\hat{\theta}\), hence \(SE = SD(\hat{\theta}) = \sqrt{\frac{I(\theta)^{-1}}{n}} \approx \sqrt{\frac{I(\hat{\theta})^{-1}}{n}}\)

\(1-\alpha \text{ CI } \approx \hat{\theta} \pm z_{\alpha/2}\sqrt{\frac{I(\theta)^{-1}}{n}}\)

Efficiency

Cramer-Rao Inequality: if \(\theta\) is unbiased, then \(\forall \theta \in \Theta\) , \(var(\hat{\theta}) \ge I(\hat{\theta})^{-1}/n\), if = then \(\hat{\theta}\) is efficient.

\(eff(\hat{\theta}) = \frac{I(\hat{\theta})^{-1}/n}{var(\hat{\theta})}< 1\)

Sufficiency

Characterisation

Let \(S_t = {x: T(x) = t}\). The sample space of \(X\), \(S\) is the disjoint union of \(S_t\) across all possible values of \(T\).

\(T\) is sufficient for \(\theta\) if \(\exists q() \text{ s.t. } \forall x \in S_t, f_{\theta}(X\x|T=t) = q(x)\).

Factorisation Theorem

\(T\) is sufficient for \(\theta\) iff \(\exists g(t,\theta), h(x) \text{ s.t. } \forall \theta \in \Theta, f_\theta(x) = g(T(x), \theta) h(x) \forall x\)

Rao-Blackwell Theorem

Let \(\hat{\theta}\) be an estimator of \(\theta\) with finite variance, \(T\) be sufficient for \(\theta\). Let \(\tilde{\theta} = E[\hat{\theta}|T]\). Then for every \(\theta \in \Theta\), \(E\left(\hat{\theta} - \theta\right)^2 \le E\left(\hat{\theta}-\theta\right)^2\). Equality holds iff \(\hat{\theta}\) is a function of \(T\).

Random Conditional Expectation

1. \(E(X) = E(E(X|T))\)
2. \(var(X) = var(E(X|T)) + E(var(X|T))\)
3. \(var(Y|X) =E(Y^2|X) - E(Y|X)^2\)
4. \(E(Y) = Y, var(Y) =0\) iff \(Y\) is a constant

Hypothesis Testing

Let \(X_1… X_n\) be IID with density \(f(x|\theta)\). null \(H_0: \theta = \theta_0\), \(H-1 : \theta = \theta_1\). Critical region is \(R\subset R_n\). \(size = P_0(X \in R)\) and \(power = P_1(X\in R)\).

\(\Lambda(x) = \frac{f_0(x_1)…f_0(x_n)}{f_1(x_1)…f_1(x_n)}\). Critical region \({x : \Lambda(x) < c_\alpha}\), and among all tests with this size, it has the maximum power (Neyman-Pearson Lemma).

A hypothesis is simple if it completely specifies the distibution of the data.

\(H_1 : \mu > \mu_0\): Critical region \(\{\bar{x} > \mu_0 + z_\alpha\frac{\sigma}{\sqrt{n}}\}\), the power is a function of \(\mu\), and this is uniformly the most powerful test for size \(\le \alpha\).

\(H_1 : \mu \ne \mu_0\): Critical region \(\{|\bar{x}-\mu_0| > c\}, c = z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}\), but not uniformly most powerful.

The \((1-\alpha)\) CI for \(\mu\) consists of precisely the values \(\mu_0\) for which \(H_0: \mu = \mu_0\) is not rejected against \(H_1: \mu \ne \mu_0\). Exact for normal with known variance, approx. in others.

p-value

the probability under \(H_0\) that the test statistic is more extreme than the realisation. (A, B): \(p = p_0(\bar{X} > \bar{x}) = P(Z>\frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}})\). (C): \(p = P_0(|\bar{X} - \mu_0| > |\bar{x} - \mu_0|)\). The smaller the p-value, the more suspicious one should be about \(H_0\). If size is smaller than p-value, do not reject \(H_0\).

Generalized Likelihood Ratio

\(\Lambda^* = \frac{\text{max}_{\theta \in \omega_0}L(\theta)}{\text{max}_{\theta\in\Omega}L(\theta)}\), \(\Omega = \omega_0 \cup \omega_1\). The closer \(\Lambda\) is to 0, the stronger the evidence for \(H_1\).

Large-sample null distribution of \(\Lambda\)

Under \(H_0\), when n is large, \(-2\log\Lambda = \chi_k^2\), where \(k = \text{dim}(\Omega) - \text{dim}(\omega_0)\).

Normal (C): \(p = P\left(\chi_1^2 > \frac{(\bar{x} - \mu_0)^2}{\sigma^2/n}\right)\)

Multinomial: \(\Lambda = \prod_{i=1}^{r} \left(\frac{E_i}{X_i}\right)^{X_i}\) where \(E_i = np_i(\hat{\theta})\) is the expected frequency of the ith event under \(H_0\). \(-2\log\Lambda \approx \sum_{i=1}^{r}\frac{(X_i-E_i)^2}{E_i}\), which is the Pearson chi-square statistic, written as \(X^2\).

Poisson Dispersion Test

For \(i = 1 … n\) let \(X_i \sim Poisson(\lambda_i)\) are independent.

\(w_0 = \{ \tilde{\lambda} | \lambda_1 = \lambda_2 = … = \lambda_n\}\)

\(w_1 = \{\tilde{\lambda} | \lambda_i \ne \lambda_j \text{ for some } i,j\}\)

\(-2\log\Lambda \approx \frac{\sum_{i=1}^{n}(X_i-\bar{X})^2}{\bar{X}}\). For large n, the null distribution of \(-2\log\Lambda\) is approximately \(\chi_{n-1}^2\)

Comparing 2 samples

Normal Theory: Same Variance

\(X_1, …, X_n\) be i.i.d \(N(\mu_X,\sigma^2)\) and \(Y_1,…,Y_m\) be i.i.d \(N(\mu_Y, \sigma^2)\), independent. \(H_0: \mu_X - \mu_Y = d\)

Known Variance

\(Z := \frac{\bar{X} - \bar{Y} - (\mu_X - \mu_Y)}{\sigma{\sqrt{\frac{1}{n} + \frac{1}{m}}}}\) and reject \(H_0\) when \(|Z| > z_{\alpha/2}\)

Unknown Variance

\(s_p^2 = \frac{(n-1)s_X^2 + (m-1)s_Y^2}{m+n-2}\) where \(s_X^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2\). \(s_p^2\) is an unbiased estimator of \(\sigma^2\). \(s_X\) within factor of 2 from \(s_Y\).

\(t := \frac{\bar{X} - \bar{Y} - (\mu_X - \mu_Y)}{s_p{\sqrt{\frac{1}{n} + \frac{1}{m}}}}\) follows a t distribution with \(m+n-2\) d.f.

If two-sided: reject \(H_0\) when \(|t| > t_{n+m-2,\alpha/2}\). If one-sided, e.g \(H_1: \mu_X > \mu_Y\), reject \(H_0\) when \(t > t_{n+m-2,\alpha}\).

CI

\(\frac{\bar{X}-\bar{Y}}\pm z_{\alpha/2} \cdot \sigma \sqrt{\frac{1}{n} + \frac{1}{m}}\) if \(\sigma\) is known, or \(\frac{\bar{X}-\bar{Y}}\pm t_{m+n-2, \alpha/2} \cdot s_p \sqrt{\frac{1}{n} + \frac{1}{m}}\) if \(\sigma\) is unknown.

Unequal Variance

\(Z := \frac{\bar{X} - \bar{Y} - (\mu_X - \mu_Y)}{{\sqrt{\frac{\sigma_X^2}{n} + \frac{\sigma_Y^2}{m}}}}\)

\(t := \frac{\bar{X} - \bar{Y} - (\mu_X - \mu_Y)}{{\sqrt{\frac{s_X^2}{n} + \frac{s_Y^2}{m}}}}\), with \(df = \frac{(a+b)^2}{\frac{a^2}{n-1} + \frac{b^2}{m-1}}\) where \(a = \frac{s_X^2}{n}\) and \(b = \frac{s_Y^2}{m}\)

Mann-Whitney Test

We take the smaller sample of size \(n_1\), and sum the ranks in that sample. \(R’ = n_1(m+n+1) -R\), and \(R* = min(R’,R)\), we reject \(H_0: F = G\) if \(R*\) is too small.

Test works for all distributions, and is robust to outliers.

Paired Samples

\((X_i, Y_i)\) are paired and related to the same individual. \((X_i, Y_i)\) is independent from \((X_j, Y_j)\). Compute \(D_i = Y_i - X_i\), To test \(H_0 : \mu_D = d\), \(t = \frac{\bar{D} - \mu_D}{s_D/\sqrt{n}}\).

\(1-\alpha\) CI: \(\bar{D}\pm t_{n-1,\alpha/2}S_D/\sqrt{n}\)

Ranked Test

\(W_+\) is the sum of ranks among all positive \(D_i\) and \(W_i\) is the sum of ranks among all negative \(D_i\). We want to reject \(H_0\) if \(W = min(W_+, W_-)\) is too large.