Homework Assignments (LaTeX Notation) STAT 700 Fall 2022 ======================================================== Homework Assignments, including any changes and hints, will continually be posted here. The most current form of the assignment will be posted also on ELMS. Selected problem solutions in the course-page directory HWslns/. -------------------------------- HW1, due Monday Sept.12, 11:59pm (upload to ELMS) Read Chapters 1, Sec.1.1 and Appendices A. 10-A.14 and B.7 of Bickel and Doksum. In Bickel and Doksum, do problems # 1.1.1(d), 1.2.(b)-(c), 1.1.15, and B.7.10, along with 3 additional problems: (A) Suppose that {\it iid\/} real random variables $X_1, \ldots, X_n$ are observed and can be assumed to follow one of the densities $~f(x,\theta)~$ from a family with real-valued unknown parameter $\theta$. Suppose that there is a function $~r(x)~$ such that $~R(\theta) \equiv \int \, r(x) \, f(x,\theta)\, dx~$ exists, is finite, and is strictly increasing in $~\theta$. Show that the parameter $~\theta~$ is {\it identifiable\/} from the data. (B) In the setting of problem (A), explain (as constructively as possible) why there is a consistent (in probability) estimator $~g_n((X_1, \ldots,X_n)~$ of $~\theta$. {\it Hint:\/} Start from $~n^{-1} \sum_{j=1}^n \, r(X_j)$, ~and assume that $~R(\theta)~$ is continuous if you have to. (C) In the setting of {\it iid\/} vector-valued data $~Y_1, \ldots, Y_n$~ with vector-valued parameter $~\theta \in \Theta \subset \mathbb{R}^k$, ~suppose that there exists a consistent (in probability) estimator $~g_n(Y_1, \ldots, Y_n)~$ of $~\theta$. Then show that $\theta$~ is identifiable from the density family $~f(y,\theta)$. All 7 problems are to be handed in (uploaded) Monday Sept. 12 in ELMS. -------------------------------- HW2, due Tuesday September 27, 11:59pm (7 problems total) Read Chapter 1 Sections 1.2-1.3 of Bickel and Doksum and continue to review Appendix B.7. In Bickel and Doksum, do problems # 1.2.2, 1.2.8, 1.2.12, 1.3.2, 1.3.3, 1.3.4(a) plus one additional problem that I will assign on a Bayesian view of the 2-component mixture model.