Lecture notes. Written from the material of Berkeley Stat 210A, licensed CC BY 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

73. A Curved Gaussian Family¶

What this covers¶

This chapter preserves a single exam problem: part of Question 1 from the Fall 2018 final examination for STAT 210A (Prof. William Fithian), on a two-dimensional Gaussian family whose mean vector is traced out by a single real parameter. It assumes the reader already has the course's definitions of a sufficient statistic, a minimal sufficient statistic, and a complete statistic — the exam itself tells students they may use "any results from lecture or homework... without rederiving them." Those definitions are not repeated here because they are not in this source; see the course's own lecture notes for them.

The source materials for this task are two copies of the same converted PDF (from the fall-2024 and fall-2026 mirrors of the course repository), and the conversion breaks off after the first sub-part of the first problem. What follows is exactly what survived that conversion — no later parts of the problem, and no solutions, despite the file being named "Solution2018."

The family¶

The exam sets up \(n\) i.i.d. bivariate observations

\[X_1, \ldots, X_n = \begin{pmatrix} X_{1,1} \\ X_{1,2} \end{pmatrix}, \ldots, \begin{pmatrix} X_{n,1} \\ X_{n,2} \end{pmatrix} \stackrel{\text{i.i.d.}}{\sim} N_2(\mu(\theta), I_2),\]

for a single real parameter \(\theta \in \mathbb{R}\), where

\[\mu(\theta) = \begin{pmatrix} \theta \\ \theta^2 \end{pmatrix}.\]

So each observation is a bivariate normal with identity covariance, but the mean is not free to range over all of \(\mathbb{R}^2\): as \(\theta\) varies, \(\mu(\theta)\) traces out the parabola \(\{(t, t^2) : t \in \mathbb{R}\}\), a one-dimensional curve inside the two-dimensional mean space. That is the sense in which the exam calls this a "curved" Gaussian family — a single scalar parameter is being asked to do the work of a two-dimensional mean, and the family sits inside a larger, full two-parameter Gaussian model without filling it out.

The problem supplies the univariate Gaussian density as a reminder:

\[\frac{1}{\sqrt{2\pi\sigma^2}} \exp\left\{ -\frac{(x - \mu)^2}{2\sigma^2} \right\}.\]

Exercises¶

1(a). For the curved Gaussian family above, show that \(T(X) = \sum_i X_i \in \mathbb{R}^2\) is a minimal sufficient statistic, but is not complete.

(The exam problem is worth 20 points total, 4 points per part, and is explicitly marked as having further parts; the conversion available for this chapter stops here, so parts (b) onward are not reproduced.)

Sources¶

Both copies of the converted exam booklet are identical in the portion recovered: docs/statistics/berkeley/stat210a/fall-2024/old-exams/solution2018.md and docs/statistics/berkeley/stat210a/fall-2026/old-exams/solution2018.md, each converted from old-exams/solution2018.pdf in the respective course repository mirror (Fall 2018 final exam, Prof. William Fithian, STAT 210A, Berkeley; licensed CC BY 4.0).
Both files carry a fidelity note that the source PDF had no usable text layer and was reconstructed by a model, with every equation unverified; the conversion is truncated after part (a) of Problem 1 in both copies, so parts (b) and later, and any other problems on the exam, are not available in either input and are not reproduced here.
The exam instructions permit students to use, without proof, "any results from lecture or homework" — i.e. the course's definitions of sufficiency, minimal sufficiency and completeness — but the lecture material itself was not among the inputs for this chapter.

← 72. Score, Fisher Information, and CRLB (part 2) · Contents · 74. Sufficiency, Minimax, and Asymptotic Theory →