3110 Midterm Cheat Sheet

\begin{aligned} Change of variables & Distribution function technique \\ F_{Y} (y) & = P (Y \leq y) = P (h (X_{1}, \dots, X_{n}) \leq y), f_{Y} (y) = \frac{\partial F_{Y} (y)}{\partial y} \\ Transformation technique \\ f_{y} (y) & = f_{X} (h^{- 1} (y)) \cdot | \frac{\partial h^{- 1} (y)}{\partial y} | if only one region \\ f_{Y} (y) & = f_{X} (h_{1}^{- 1} (y)) | \frac{d h_{1}^{- 1} (y)}{d y} | + f_{x} (h_{2}^{- 1} (y)) | \frac{| d h_{2}^{- 1} (y)}{d y} | \\ f_{Y_{1}, . . ., Y_{n}} (y_{1}, . . ., y_{n}) & = f_{X_{1}, . . ., X_{n}} (g_{1} (y_{1}, . . ., y_{n}), . . ., g_{n} (y_{1}, . . ., y_{n})) \cdot | J | \\ | J | & = det [\begin{array}{c} \frac{\partial X_{1}}{\partial Y_{1}} & \frac{\partial X_{1}}{\partial Y_{2}} \\ \frac{\partial X_{2}}{\partial Y_{1}} & \frac{\partial X_{2}}{\partial Y_{2}} \end{array}] (integrate the dummy vars) \end{aligned}

\begin{aligned} Order Statistics \\ f_{X_{(1)}} & = n [1 - F_{X} (x)]^{n - 1} f_{X} (x) \\ f_{X_{(n)}} & = n [F_{X} (x)]^{n - 1} f_{X} (x) \\ f_{X_{(r)}} (x) & = \frac{n!}{(r - 1)! (n - r)!} [F_{X} (x)^{r - 1}] f_{X} (x) [1 - F_{X} (x)]^{n - r} \\ f_{\tilde{X}} (x) & = f_{X_{(n + 1)}} (x) = \frac{(2 n + 1)!}{n! \cdot n!} [F_{X} (x)]^{n} f_{X} (x) [1 - F_{X} (x)]^{n} \end{aligned}

\begin{aligned} Unbiasedness \\ b i a s (\hat{θ}) & = E (\hat{θ} - θ) = E (\hat{θ}) - θ \\ M S E (\hat{θ}) & = E [(\hat{θ} - θ)^{2}] = v a r (\hat{θ}) + [E (\hat{θ}) - θ]^{2} \end{aligned}

\begin{aligned} Efficiency (compare unbiased estimators) \\ efficiency (\hat{θ}) & = \frac{C R L B}{V a r (\hat{θ})} \leq 1 if V a r (\hat{θ}) = C R L B \to UMVUE \\ v a r (\hat{θ}) & \geq \frac{1}{n E [{(\frac{\partial \ln f (x; θ)}{\partial θ})}^{2}]} \end{aligned}

Distribution	PDF / PMF	(E(X))	E(X²)	(Var(X))
Uniform(a, b)	$\frac{1}{b - a}, a \leq x \leq b$	$\frac{a + b}{2}$	$\frac{a^{2} + a b + b^{2}}{3}$	$\frac{(b - a)^{2}}{12}$
Beta( $α$ , $β$ )	$\frac{Γ (α + β)}{Γ (α) Γ (β)} x^{α - 1} (1 - x)^{β - 1}, 0 < x < 1$	$\frac{α}{α + β}$	$\frac{α (α + 1)}{(α + β) (α + β + 1)}$	$\frac{α β}{(α + β)^{2} (α + β + 1)}$
Gamma(k, $θ$ )	$\frac{1}{Γ (k) θ^{k}} x^{k - 1} e^{- x / θ}, x > 0$	$k θ$	$k (k + 1) θ^{2}$	$k θ^{2}$
Poisson( $λ$ )	$\frac{λ^{x} e^{- λ}}{x!}, x = 0, 1, 2, \dots$	$λ$	$λ (λ + 1)$	$λ$
Binomial(n, p)	$(\binom{n}{x}) p^{x} (1 - p)^{n - x}, x = 0, 1, \dots, n$	$n p$	$n p (1 - p) + n p^{2}$	$n p (1 - p)$
Geometric(p)	$(1 - p)^{x - 1} p, x = 1, 2, 3, \dots$	$\frac{1}{p}$	$\frac{2 - p}{p^{2}}$	$\frac{1 - p}{p^{2}}$
$χ^{2} (k)$	$\frac{1}{2^{k / 2} Γ (k / 2)} x^{k / 2 - 1} e^{- x / 2}, x > 0$	$k$	$k (k + 2)$	$2 k$
Expntl( $λ$ )	$λ e^{- λ x}, x > 0$	$\frac{1}{λ}$	$\frac{2}{λ^{2}}$	$\frac{1}{λ^{2}}$
Normal( $μ, σ^{2}$ )	$\frac{1}{\sqrt{2 π} σ} e^{- \frac{1}{2 σ^{2}} (x - μ)^{2}}$	$μ$	$μ^{2} + σ^{2}$	$σ^{2}$

Consistency
$\hat{θ}$ is a consistent estimator of the parameter $θ$ if and only if $\exists ϵ > 0$

lim_{n \to \infty} P (| \hat{θ} - θ | < ϵ) = P (- ϵ \leq \hat{θ} - θ \leq ϵ) = P (θ - ϵ \leq \hat{θ} \leq θ + ϵ) = 1

If $\hat{θ}$ is an unbiased estimator of the parameter $θ$ and the $V a r (\hat{θ}) \to 0$ as $n \to \infty$ , then $\hat{θ}$ is a consistent estimator of $θ$

\begin{aligned} Chebyshev’s Theorem \\ P (| X - μ | < k σ) & \geq 1 - \frac{1}{k^{2}}, or P (| X - μ | \geq k σ) \leq \frac{1}{k^{2}} \end{aligned}

Sufficiency
If the conditional given $\hat{θ}$ depends on $θ$ , $\hat{θ}$ is not sufficient

f (X_{1} = x_{1}, \dots, X_{n} = x_{n} | \hat{θ}) = \frac{f (X_{1} = x_{1}, \dots, X_{n} = x_{n}, \hat{θ})}{g (\hat{θ}) \leftarrow pdf of \hat{θ}} = \frac{f (X_{1} = x_{1}, \dots, X_{n} = x_{n})}{g (\hat{θ})}

$\hat{θ}$ is a sufficient estimator iff the joint can be factorized (fact. theorem)

f (X_{1} = x_{1}, \dots, X_{n} = x_{n}; θ) = g (\hat{θ}, θ) \cdot h (x_{1}, \dots, x_{n})

\begin{aligned} Method of Moments (k moments for k parameters) \\ E (X) & = \bar{X}, E (X^{2}) = {\bar{X}}^{2} \\ Method of maximum likelihood estimator \\ L (θ) & = L (θ; x_{1}, \dots, x_{n}) = f (x_{1}, \dots, x_{n}; θ) = \prod_{i = 1}^{n} f (x_{i}; θ) \\ l (θ) & = \ln L (θ) = \ln \prod f (x_{i}; θ) = \sum_{i = 1}^{n} \ln f (x_{i}; θ) \\ \frac{d l (θ)}{d θ} & = 0 to find crit point at \hat{θ} \\ {\frac{d^{2} l (θ)}{d θ^{2}} |}_{θ = \hat{θ}} & = to check maximum, should be < 0 \\ case 2+ parameters: \\ if & [\begin{array}{c} \frac{\partial^{2} ℓ}{\partial θ_{1}^{2}} & \frac{\partial^{2} ℓ}{\partial θ_{1} \partial θ_{2}} \\ \frac{\partial^{2} ℓ}{\partial θ_{2} \partial θ_{1}} & \frac{\partial^{2} ℓ}{\partial θ_{2}^{2}} \end{array}] < 0, then our \hat{θ_{1}}, \hat{θ_{2}} are MLEs of θ_{1}, θ_{2} \end{aligned}

\begin{aligned} Bayesian Estimation \\ (Prior distribution) g (θ) & = Prior belief about θ \\ (Likelihood) L (θ) & = f (x; θ) = Data’s likelihood given θ \\ (Posterior distribution) h (θ | x) & = \frac{f (x, θ)}{f (x)} = \frac{f (x; θ) g (θ)}{f (x)} = \frac{L (θ) g (θ)}{f (x)} \\ L (θ) g (θ) & = Unnormalized posterior \\ f (x) & = \int_{all θ} L (θ) g (θ) d θ = Marginal likelihood \\ h (θ | x) & = \frac{L (θ) g (θ)}{f (x)} = Normalized posterior \\ {\hat{θ}}_{B} & = E (θ | x) = \int_{all θ} θ h (θ | x) d θ = Bayesian estimate \end{aligned}