2025-08-08

<< 2025-08-07 | 2025-08-11 >>

---

UAS statmat 2

Distributions

Memorize pdf, mean, variance of:

• Gamma distribution : pdf, $\alpha \beta$, $\alpha {\beta }^2$
• Chi-square distribution : $\Gamma (r/2,2)$, $r$, $2r$
• Normal distribution : pdf
• Bernoulli distribution : $p^x(1-p{)}^{1-x}$, $p$, $p(1-p)$
• Binomial distribution : $\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$, $np$, $np(1-p)$
• Poisson distribution : pdf, $\lambda$, $\lambda$

$$\begin{array}{rl}& \frac{1}{\Gamma (\alpha ){\beta }^{\alpha }}{x}^{\alpha -1}{e}^{-x/\beta }\\ & \frac{1}{\sigma \sqrt{2\pi }}\exp \left[-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right]\\ & \frac{e^{-\lambda }{\lambda }^x}{x!}\end{array}$$

Distribution relationships

$$\begin{array}{rl}\bar{X}& =\frac{1}{n}\sum X_i\sim N\left(\mu ,\frac{{\sigma }^2}{n}\right)\\ S^2& =\frac{1}{n-1}\sum (X_i-\bar{X}{)}^2\end{array}$$

Random variables	Transformation	Distribution of transformation
$X\sim N(\mu ,{\sigma }^2)$	$\frac{n{S}^2}{\sigma }$	${\chi }^2(n)$
$Z\sim N(0,1)$	${\sum }^k{Z}_i$	${\chi }^2(k)$
$U\sim N(0,1)$ $V\sim {\chi }^2(r)$	$\frac{U}{\sqrt{V/r}}$	$t(r)$
$U\sim {\chi }^2(r_1)$ $V\sim {\chi }^2(r_2)$	$\frac{U/r_1}{V/r_2}$	$F(r_1,r_2)$
$X\sim N(\mu ,{\sigma }^2)$	$\frac{\bar{X}-\mu }{S/\sqrt{n}}$	$t(n-1)$
$N({\mu }_1,{\sigma }_1^2)$ $N({\mu }_2,{\sigma }_2^2)$	$\frac{S_1^2/{\sigma }_1^2}{S_2^2/{\sigma }_2^2}$	$F(n_1-1,n_2-1)$

Definitions & theorems

Sufficient Statistic

$\frac{{\prod }_{i=1}^{n}f(x_i,\theta )}{f_{Y_1}[u_1(\mathbf{x});\theta ]}\text{independent of }\theta$

Neyman Theorem:

• $Y_1=u_1(\mathbf{x})$ Statistic for $\theta$
• $\exists k_1,k_2\ni \prod f(x_i;\theta )=k_1[u_1(\mathbf{x});\theta ]\cdot k_2(\mathbf{x})$
• $k_2(\mathbf{x})$ independent of $\theta$

Regular Exponential Class

$$f(x;\theta )=\{\begin{array}{ll}\exp [p(\theta )K(x)+H(x)+q(\theta )]& x\in \mathcal{S}\\ 0& \text{elsewhere}\end{array}$$

• $\mathcal{S}$ independent of $\theta$
• $p(\theta )$ nontrivial
• $X$ continuous:
  • $K^{\prime }(x)\ne 0$
  • $H(x)$ continuous
• $X$ discrete:
  • $K(x)$ nontrivial

---

Fisher Information

$$\begin{array}{rl}I(\theta )& =E\left[{\left(\frac{\partial }{\partial \theta }\ln f(X;\theta )\right)}^2\right]\\ & =-E\left[\frac{{\partial }^2}{\partial {\theta }^2}\ln f(X;\theta )\right]\\ I_n(\theta )& =nI(\theta )\end{array}$$

Rao-Cramer & Efficient Estimator

$\mathrm{Var}(Y)\ge \frac{[k^{\prime }(\theta ){]}^2}{I_n(\theta )}$

1. $Y$ is Unbiased Estimator for $\theta$, i.e., $E(Y)=\theta$
2. $\mathrm{Var}(Y)=\frac{1}{nI(\theta )}$

Confidence interval

Pivot variable: Function of estimator of $\theta$, where dist and param of dist of the pivot variable is known

1. Find estimator for $\theta$, e.g., $\hat{\theta }=\bar{X}$ or $\hat{\theta }=S^2$
2. Find pivot variable $Z$, i.e., function of $\hat{\theta }$, where dist and param of dist of $Z$ is known
3. Compute confidence interval $1-\alpha =P_{\mu }[\Phi <Z(\hat{\theta })<-\Phi ]$
4. Find realization of confidence interval

Warning

Best test & UMPT incomplete

Best test

1. Determine hypotheses: $H_0:\theta ={\theta }^{\prime }$ vs $H_1:\theta ={\theta }^{\prime \prime }$
2. Construct $\lambda =\frac{L({\theta }^{\prime };\mathbf{x})}{L({\theta }^{\prime \prime };\mathbf{x})}$
3. Apply Neyman-Pearson Theorem. Best test has the form:
   • Reject $H_0$ if $\lambda \le k$
   • Accept $H_0$ if $\lambda >k$ Where $k>0$
4. Simplify test statistic:
   1. Write $\lambda \le k$ in terms of $C=\{\mathbf{x}:u(\mathbf{x};{\theta }^{\prime },{\theta }^{\prime \prime })\le \text{(or }\ge )c\}$. $c$ usually constant obtained from $k$
   2. Use $\ln \lambda =\ln \frac{L({\theta }^{\prime };\mathbf{x})}{L({\theta }^{\prime \prime };\mathbf{x})}$ or both
5. Get $c$ (critical value) such that $\alpha =P_{{\theta }^{\prime }}(\mathbf{x}\in C)$

UMPT

1. Determine hypotheses: $H_0:\theta ={\theta }^{\prime }$ vs $H_1:\theta \in {\Theta }_1$
2. Construct ${\lambda }_{{\theta }_1}=\frac{L({\theta }^{\prime };\mathbf{x})}{L({\theta }_1;\mathbf{x})},\forall {\theta }_1\in {\Theta }_1$
3. Apply Neyman-Pearson Theorem for each ${\theta }_1$. Show critical region is the same:
   • Reject $H_0$ if ${\lambda }_{{\theta }_1}\le k_{{\theta }_1}$
   • Accept $H_0$ if ${\lambda }_{{\theta }_1}>k_{{\theta }_1}$ Where $k_{{\theta }_1}>0$
4. Simplify test statistic:
   1. Write ${\lambda }_{{\theta }_1}\le k_{{\theta }_1}$ in terms of $C=\{\mathbf{x}:u(\mathbf{x})\le \text{(or }\ge )c\}$. Show $C$ is independent of ${\theta }_1$
   2. Use $\ln {\lambda }_{{\theta }_1}=\ln \frac{L({\theta }^{\prime };\mathbf{x})}{L({\theta }_1;\mathbf{x})}$ or both
5. Get $c$ (critical value) such that $\alpha =P_{{\theta }^{\prime }}(\mathbf{x}\in C)$