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密度函数的序贯置信带AI 解读

This paper proposes a fully sequential procedure for constructing a fixed-width confidence band for an unknown density on a finite interval and shows the procedure has the desired coverage probability asymptotically as the width of the band approaches zero. The procedure is based on a result of Bickel and Rosenblatt. Its implementation in the sequential setting cannot be obtained using Anscombe's theorem, because the normalized maximal deviations between the kernel estimate…

逻辑回归模型参数的固定大小置信域AI 解读

Let $(\mathbf{X}_i, Y_i)$ be independent, identically distributed observations that satisfy a logistic regression model; that is, for each $i, \log \lbrack P(Y_i = 1 | \mathbf{X}_i)/P(Y_i = 0 |\mathbf{X}_i)\rbrack = \mathbf{X}^T_i \beta_0$, where $Y_i \in \{0, 1\}, \mathbf{X}_i \in \mathbf{R}^p$ and $\beta_0 \in \mathbf{B}^p$ is the unknown parameter vector of the model. The marginal distribution of the covariate vectors $\mathbf{X}_i$ is assumed to be unknown. Sequential…

利用停止规则约束密度估计的平均积分平方误差AI 解读

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d. with unknown density $f$. There is a well-known expression for the asymptotic mean integrated squared error (MISE) in estimating $f$ by a kernel estimate $\hat{f}_n$, under certain conditions on $f$, the kernel and the bandwidth. Suppose that one would like to choose a sample size so that the MISE is smaller than some preassigned positive number $w$. Based on the asymptotic expression for the MISE, one can identify an appropriate…

负遗憾、可选停止与异常值的消除AI 解读

This article examines the phenomenon of negative regret in sequential point estimation, in which a stopping rule used in ignorance of the variance of the distribution sometimes performs better than the best fixed sample size when the variance is known. An explanation in terms of a certain robustifying effect of the stopping rule is given. Results of simulation studies illustrating both negative regret and its explanation are also given. Suppose X 1 X 2, … are iid with mean μ…

估计量与样本量的序贯联合选择AI 解读

For certain point and interval estimation problems, sequential procedures are proposed for choosing an estimator (from the class of trimmed means with trimming proportion in a specified range) as well as a sample size on which to base the estimator. It is shown that these procedures, which do not require knowledge of the best trimming proportion or the asymptotic variance of the corresponding trimmed mean, are asymptotically efficient with respect to the procedure that uses…

序贯程序风险的二阶近似AI 解读

Given $X_1, X_2, \cdots$, i.i.d. with mean $\mu$ and variance $\sigma^2$, suppose that at stage $n$ one wishes to estimate $\mu$ by the sample mean $\bar{X}_n$, subject to the loss function $L_n = A(\bar{X}_n - \mu)^2 + n, A > 0$. If $\sigma$ is known, the optimal fixed sample size $n_0 \approx A^{1/2}\sigma$ can be used, with corresponding risk $R_{n_0}$, but if $\sigma$ is unknown there is no fixed sample size procedure that will achieve the risk $R_{n_0}$. For the…

序贯概率比检验停止时间的方差与高阶中心矩札记AI 解读

Results are obtained on the limiting distribution and corresponding moment convergence for the (normalized) stopping time of an SPRT, as the boundaries of the continuation region become infinite. These asymptotic results suggest a certain approximation for the variance of the stopping time, which is compared with some Monte Carlo results of Cox and Roseberry (1966) and an approximation due to Ghosh (1969), in the case of testing a normal mean with known variance. The…

序贯概率比检验停止时间的方差与高阶中心矩札记AI 解读

Results are obtained on the limiting distribution and corresponding moment convergence for the (normalized) stopping time of an SPRT, as the boundaries of the continuation region become infinite. These asymptotic results suggest a certain approximation for the variance of the stopping time, which is compared with some Monte Carlo results of Cox and Roseberry (1966) and an approximation due to Ghosh (1969), in the case of testing a normal mean with known variance. The…