A maximum-likelihood estimator with infinite error

The standard error of the maximum-likelihood estimator for 1/μ based on a random sample of size N from the normal distribution N(μ,σ²) is infinite. This could be considered to be a disadvantage.Another disadvantage is that the bias of the estimator is undefined if the integral is interpreted in the usual sense as a Lebesgue integral. It is shown here that the integral expression for the bias can be interpreted in the sense given by the Schwartz theory of generalized functions. Furthermore, an explicit closed form expression in terms of the complex error function is derived. It is also proven that unbiased estimation of 1/μ is impossible.Further results on the maximum-likelihood estimator are investigated, including closed form expressions for the generalized moments and corresponding complete asymptotic expansions. It is observed that the problem can be reduced to a one-parameter problem depending only on , and this holds also for more general location-scale problems. The parameter can be interpreted as a shape parameter for the distribution of the maximum-likelihood estimator.An alternative estimator is suggested motivated by the asymptotic expansion for the bias, and it is argued that the suggested estimator is an improvement. The method used for the construction of the estimator is simple and generalizes to other parametric families.The problem leads to a rediscovery of a generalized mathematical expectation introduced originally by Kolmogorov [1933. Foundations of the Theory of Probability, second ed. Chelsea Publishing Company (1956)]. A brief discussion of this, and some related integrals, is provided. It is in particular argued that the principal value expectation provides a reasonable location parameter in cases where it exists. This does not hold generally for expectations interpreted in the sense given by the Schwartz theory of generalized functions.     

城镇化进程中大型商业银行对农户信贷支持的现状、存在问题和对策建议

文章分析了在城镇化进程中大型商业银行对农户进行信贷支持的必要性,大型商业银行支持农户融资需求的现状,以及农户贷款工作中存在的网点覆盖不足、客户经理不够、自助设备作用发挥不充分、考核压力大等制约因素,并提出了对策建议：宏观方面的建议包括,要充分利用农村的信息渠道,不断加强农村信用体系建设;夯实农户金融基础服务,向监管机构反映对差异化监管的需求;积极争取财政补贴和优惠,协助政府创新成本补偿机制;创新金融产品,间接防范信贷风险;使网点布局与渠道开发并举,扩大金融服务覆盖;重视对农民进行素质和技能培训,满足农户的＂融智＂需求。微观方面的建议是,通过严把客户准入关保证信贷质量;推行差异化放贷和定价策略有效管理风险;运用精细化管理防范信用风险;采取有效担保转移信贷风险;借助涉农保险等方式分散贷款风险;建立风险补偿机制防范信贷风险。     

新中国三代领导人与西部大开发

西部是中国最辽阔的疆域 ,也是中国最大的欠发达地区。新中国三代领导人始终关注着这片热土的开发建设 ,并把发展西部提到关系各民族共同繁荣和中国全面实现现代化的战略高度加以认识 ,逐步形成了西部大开发的战略构想 ,成为新世纪指导中华民族走向伟大复兴的强大推动力量。     

权力支配与汉代豪族的大地产性

豪族集多种特征于一体,大地产性是豪族重要特征之一。豪族大地产的形成与权力支配密不可分。豪族占有土地的方式中,都直接或间接地表现为权力占有,豪族的大地产是权力支配的结果。     

线性Volterra积分微分大系统的稳定性

研究了线性Volterra积分微分大系统的稳定性,对于实际应用中最为感兴趣的指数稳定性问题,建立了简洁、实用的指数稳定性充分准則。     

Contingency tables with prescribed marginals

For an adjustment of contingency tables to prescribed marginal frequencies Deming and Stephan (1940) minimize a Chi-square expression. Asymptotically equivalently, Ireland and Kullback (1968) minimize a Leibler-Kullback divergence, where the probabilistical arguments for both methods remain vague. Here we deduce a probabilistical model based on observed contingency tables. It shows that the two above methods and the maximum likelihood approach in Smith (1947) yield asymptotically the ‘most probable’ adjustment under prescribed marginal frequencies. The fundamental hypothesis of statistical mechanics relates observations to ‘most probable’ realizations. ‘Most probable’ is going to be used in the sense of so-called large deviations. The proposed adjustment has a significant product form and will be generalized to contingency tables with infinitely many cells.     

On an improved Erdő-Rényi-type law for increments of partial sums

Let X₁, X₂,… be an independently and identically distributed sequence with ξX₁ = 0, ξ exp (tX₁ < ∞ (t ≧ 0) and partial sums S_n = X₁ + … + X_n. Consider the maximum increment D₁ (N, K) = max_{0≤n ≤ N - K} (S_{n + K} - S_n)of the sequence (S_n) in (0, N) over a time K = K_N, 1 ≦ K ≦ N. Under appropriate conditions on (K_N) it is shown that in the case K_N/log N → 0, but K_N/(log N)^1/2 → ∞, there exists a sequence (α_N) such that K_-1/2 D₁ (N, K) - α_N converges to 0 w. p. 1. This result provides a small increment analogue to the improved Erd?s-Rényi-type laws stated by Csörg? and Steinebach (1981).     

一类时变调节大系统的稳定性

本文利用参数变易法来研究时变调节大系统的全局指数渐近稳定性,得到了时变调节大系统的全局指数渐近稳定性的充分条件,从而改进了文献[3]的结果。利用本文所得到的充分条件来研究文献[1～3,5]所讨论的同一例子,得到了比文献[1～3,5]更宽的指数渐近稳定性的参数区域。     

时变直接调节大系统的稳定性

本文利用参数变易法和不等式技巧来研究时变直接调节大系统的全局指数渐近稳定性,得到了形式简洁、易于验证的代数判据。     

Relative efficiencies of goodness-of-fit procedures with truncated data

Efficiency properties of the Kolmogorov-Smirnov, Kuiper, Cramér-von Mises, and Anderson-Darling statistics with truncated data are investigated. For these statistics, the approximate slopes are determined, and the equivalence of ratios of limiting approximate slopes to limiting Pitman efficiencies is established. Efficiencies for certain parametric families of distributions incorporating both location and scale alternatives are evaluated as the amount of truncation increases.