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Hildreth–Houck 随机系数模型的极大似然估计AI 解读

We explore maximum likelihood (ML) estimation of the Hildreth-Houck random coefficients model. We show that the global ML estimator can be inconsistent. We develop an alternative LML (local ML) estimator and prove that it is consistent and asymptotically efficient for points in the interior of the parameters. Properties of the LML and comparisons with common method of moments (MM) estimates are done via Monte Carlo. Boundary parameters lead to nonstandard asymptotic…

平均随机函数上确界的渐近性质AI 解读

Suppose $X_i$ are i.i.d. random variables taking values in $\mathscr{X}, \Theta$ is a parameter space and $y: \mathscr{X} \times \Theta \rightarrow \mathbf{R}$ is a map. Define the averages $S_n(y, \theta) = (1/n)\sum^n_{i = 1}y(X_i, \theta)$ and the truncated expectations $T_m(y, \theta) = \mathbf{E} \max(y(X_1, \theta), - m)$. Under the hypothesis of global dominance [i.e., $\mathbf{E} \sup_\Theta y(X_1, \theta) < \infty$] and some regularity conditions, the main result of…

通过第二类不等式证明一致性：Wu 定理的推广AI 解读

Wu introduced a new technique for proving consistency of least-squares estimators in nonlinear regression. This paper extends his results in three directions. First, we consider the minimization of arbitrary functions (M-estimators instead of least squares). Second, we use an improved type 2 inequality. Third, an extension of Kronecker's lemma yields a more powerful result.

方向参数的最佳不变估计AI 解读

Let $X$ be an $n \times k$ random matrix whose coordinates are independently normally distributed with common variance $\sigma^2$ and means given by $EX = e\mu' + \theta\lambda',$ where $e$ is the vector in $R^n$ having all coordinates equal to $1, \theta \in R^n,$ and $\mu, \lambda \in R^k$ with $\sum^k_{j = 1} \lambda^2_j = 1.$ The problem is to estimate $\lambda$, say by $\hat{\lambda},$ with loss function $1 - (\lambda'\hat{\lambda})^2$ when $\mu, \theta,$ and $\sigma^2$…

控制问题中估计量可容许性与不可容许性的一般结果AI 解读

Let $\mathbf{X} = (X_1, \cdots, X_p)^t$ be an observation from a $p$-variate normal distribution with unknown mean $\mathbf{\theta} = (\theta_1, \cdots, \theta_p)^t$ and identity covariance matrix. We consider a control problem which, in canonical form, is the problem of estimating $\mathbf{\theta}$ under the loss $L(\mathbf{\theta, \delta}) = (\mathbf{\theta}^t \mathbf{\delta} - 1)^2$, where $\mathbf{\delta(x)} = (\delta_1(\mathbf{x}), \cdots, \delta_p(\mathbf{x}))^t$ is…

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