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Problem 1. Sufficiency. Find a two-dimensional sufficient statistic for each of the following models. (a) $Y_1, \dots, Y_n \stackrel{iid}{\sim} \text{Unif}(\theta - a, \theta + a)$, $\theta \in \mathbb{R}$ and $a > 0$ both unknown. (b) $Y_1, \dots, Y_n \stackrel{iid}{\sim} \text{Gamma}(\alpha, \lambda)$, both parameters unknown. (c) $Y_i \stackrel{ind}{\sim} \text{Bern}(p_i)$, $i = 1, \dots, n$, with $p_i = \frac{\exp(\beta_0 + \beta_1 x_i)}{1 + \exp(\beta_0 + \beta_1 x_i)}$. Both $\beta_0$ and $\beta_1$ are unknown, and the $x_i$ are treated as known, fixed constants.

          Problem 1. Sufficiency. Find a two-dimensional sufficient statistic for each of the following models.
(a) $Y_1, \dots, Y_n \stackrel{iid}{\sim} \text{Unif}(\theta - a, \theta + a)$, $\theta \in \mathbb{R}$ and $a > 0$ both unknown.
(b) $Y_1, \dots, Y_n \stackrel{iid}{\sim} \text{Gamma}(\alpha, \lambda)$, both parameters unknown.
(c) $Y_i \stackrel{ind}{\sim} \text{Bern}(p_i)$, $i = 1, \dots, n$, with
$p_i = \frac{\exp(\beta_0 + \beta_1 x_i)}{1 + \exp(\beta_0 + \beta_1 x_i)}$.
Both $\beta_0$ and $\beta_1$ are unknown, and the $x_i$ are treated as known, fixed constants.

Problem 1. Sufficiency. Find a two-dimensional sufficient statistic for each of the following models.
(a) Y1, …, Yn iid∼Unif(θ - a, θ + a), θ∈ℝ and a > 0 both unknown.
(b) Y1, …, Yn iid∼Gamma(α, λ), both parameters unknown.
(c) Yi ind∼Bern(pi), i = 1, …, n, with
pi = ((β0 + β1 xi))/(1 + (β0 + β1 xi)).
Both β0 and β1 are unknown, and the xi are treated as known, fixed constants.

Added by Edgar W.

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