(d-ii) (3%) Find the limiting distribution of √n (pU - p) and √n (pB - p), respectively.

題組內容

3. (30%) Let X₁, X₂, ..., Xₙ be a random sample from Bernoulli(p). Define R(p, δ(X)) = Eₚ[(δ(X) - p)²] (the Mean-Square Error).
(d) Consider three estimators

$$\hat{p}_U = \text{UMVUE found in (b)},$$

$\hat{p}_B$ = the posterior mean with constant risk found in (c-iii),

and

$$\hat{p}T = \begin{cases} \hat{p}U \text{ with probability } \frac{n}{n+1}, \\ 1/2 \text{ with probability } \frac{1}{n+1}. \end{cases}$$

二、下面這段論文來自 David Jones 的《犯罪學歷史》，History of Criminology，請做內容說明與評論。(25 分) In his 1970 book, *The Social Reality of Crime*, Richard Quinney went on to recite six conflict-oriented propositions concerning the nature and etiology of crime: 1. Crime is a definition of human conduct that is created by authorized agents in a politically organized society. 2. Criminal definitions describe behaviors that conflict with the interests of the segments of society that have the power to shape public policy. 3. Criminal definitions are applied by the segments of society that have the power to shape the enforcement and administration of criminal law. 4. Behavior patterns are structured in segmentally organized society in relation to criminal definitions and within this context persons engage in actions that have relative probabilities of being defined as criminal. 5. Conceptions of crime are constructed and diffused in the segments of society by various means of communication. 6. The social reality of crime is constructed by the formulation and application of criminal definitions, the development of behavior patterns related to criminal definitions, and the construction of criminal conceptions. 　Quinney drew upon several earlier criminological theories in formulating his six propositions. His first proposition, for instance, has origins in the pragmatic perspective of the "interactionists," particularly labeling theorists, while his second proposition stems in large measure from Vold, as does his third, and his fourth proposition comes from Sutherland's differential association theory. The last two of Quinney's propositions seem to flow at least partially from a 1966 book by sociologists Peter L. Berger and Thomas Luckmann, *The Social Construction of Reality*, wherein they observed the subjectivity of society and its dependency upon both individual perception and ideology.

題組內容

(d-ii) \( (3\%) \) Find the limiting distribution of \( \sqrt{n} (\hat{p}U - p) \) and \( \sqrt{n} (\hat{p}B - p) \), respectively.

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