Law of iterated logarithm and almost sure limit points for random sums

Let {X_n,n≥1} be a sequence of independent identically distributed (i.i.d) random variables with a common distribution function F. When F belongs to the domain of partial attraction of a Semi-Stable law with index ,0<<2, we give complete solution to the results of R. Vasudeva and G. Divanji [Law of iterated logarithm for random subsequences, Statist. Probab. Lett. 12 (1991) 189–194], where they obtained Chover’s form of the law of iterated logarithm for random subsequences. Further, we extended the situation in obtaining almost sure limit points for random subsequences.     

A matrix variance inequality

In the course of solving a variational problem Chernoff (Ann. Probab. 9 (1981) 533) obtained what appears to be a specialized inequality for a variance, namely, that for a standard normal variable X, Var[g(X)]E[g^′(X)]². However, both the simplicity and usefulness of the inequality has generated a plethora of extensions, as well as alternative proofs. All previous papers have focused on a single function. We provide here an inequality for the covariance matrix of k functions, which leads to a matrix inequality in the sense of Loewner.     

Estimation of binomial parameters when both , are unknown

We revisit the classic problem of estimation of the binomial parameters when both parameters n,p are unknown. We start with a series of results that illustrate the fundamental difficulties in the problem. Specifically, we establish lack of unbiased estimates for essentially any functions of just n or just p. We also quantify just how badly biased the sample maximum is as an estimator of n. Then, we motivate and present two new estimators of n. One is a new moment estimate and the other is a bias correction of the sample maximum. Both are easy to motivate, compute, and jackknife. The second estimate frequently beats most common estimates of n in the simulations, including the Carroll–Lombard estimate. This estimate is very promising. We end with a family of estimates for p; a specific one from the family is compared to the presently common estimate and the improvements in mean-squared error are often very significant. In all cases, the asymptotics are derived in one domain. Some other possible estimates such as a truncated MLE and empirical Bayes methods are briefly discussed.     

Evaluation of asymptotic approximations for a two-stage Bernoulli bandit

We consider a two-stage procedure for allocating two treatments to yield a total of N dichoto-mous responses, where one of the treatments has a known probability of success. In the first stage, observations may be made on either of the treatments and observed successes are discounted by a factor β. One of the treatments must be chosen for the second stage, where observed successes are no longer discounted. We adopt a Bayesian approach and develop a continuous time approximation for this problem that turns out to be identical to one developed in Petkau (J. Amer. Statist. Assoc. 73 (1978) 328). Examination of both stopping boundaries and Bayes risks demonstrates that suboptimal strategies provided by the solution of the continuous time problem are excellent approximations to the optimal strategies for the discrete time problem. A “continuity correction” developed by Cheroff and Petkau (Ann. Probab. 4 (1976) 875) plays an important role in enhancing the naive approximation provided by the solution of the continuous time problem.     

A unified and elementary proof of serial and nonserial, univariate and multivariate, Chernoff–Savage results

We provide a simple proof that the Chernoff–Savage [H. Chernoff, I.R. Savage, Asymptotic normality and efficiency of certain nonparametric tests, Ann. Math. Statist. 29 (1958) 972–994] result, establishing the uniform dominance of normal-score rank procedures over their Gaussian competitors, also holds in a broad class of problems involving serial and/or multivariate observations. The non-admissibility of the corresponding everyday practice Gaussian procedures (multivariate least-squares estimators, multivariate t-tests and F-tests, correlogram-based methods, multivariate portmanteau and Durbin–Watson tests, etc.) follows. The proof, which generalizes to the multivariate—possibly serial—set-up the idea developed in J.L. Gastwirth, S.S. Wolff [An elementary method for obtaining lower bounds on the asymptotic power of rank tests, Ann. Math. Statist. 39 (1968) 2128–2130] in the context of univariate location problems, allows for avoiding technical convexity and variational arguments.     

Local information and the design of sequential hypothesis tests

Various results on sequential hypotheses testing are reviewed. Optimal stopping rules are related to a local measure of statistical information. In some cases, local information can be approximated by L-numbers discovered by Lorden, and simple rules based on these approximations are asymptotically optimal to better order than the cost for a single observation.     

An asymptotic approach to progressive censoring

Progressive Type-II censoring was introduced by Cohen (Technometrics 5(1963) 327) and has been the topic of much research. The question stands whether it is sensible to use this sampling plan by design, instead of regular Type-II right censoring. We introduce an asymptotic progressive censoring model, and find optimal censoring schemes for location-scale families. Our optimality criterion is the determinant of the 2×2 covariance matrix of the asymptotic best linear unbiased estimators. We present an explicit expression for this criterion, and conditions for its boundedness. By means of numerical optimization, we determine optimal censoring schemes for the extreme value, the Weibull and the normal distributions. In many situations, it is shown that these progressive schemes significantly improve upon regular Type-II right censoring.     

Optimal robust influence functions in semiparametric regression

Robust statistics allows the distribution of the observations to be any member of a suitable neighborhood about an ideal model distribution. In this paper, the ideal models are semiparametric with finite-dimensional parameter of interest and a possibly infinite-dimensional nuisance parameter.In the asymptotic setup of shrinking neighborhoods, we derive and study the Hampel-type problem and the minmax MSE-problem. We show that, for all common types of neighborhood systems, the optimal influence function can be approximated by the optimal influence functions for certain parametric models.For general semiparametric regression models, we determine in case of error-in-variables and in case of error-free-variables.Finally, the results are applied to Cox regression where we compare our approach to that of Bednarski [1993. Robust estimation in Cox's regression model. Scand. J. Statist. 20, 213–225] in a small simulation study and on a real data set.     

Effect of adding regressors on the equality of the BLUEs under two linear models

In this paper we consider the estimation of regression coefficients in two partitioned linear models, shortly denoted as , and , which differ only in their covariance matrices. We call and full models, and correspondingly, and small models. We give a necessary and sufficient condition for the equality between the best linear unbiased estimators (BLUEs) of X₁β₁ under and . In particular, we consider the equality of the BLUEs under the full models assuming that they are equal under the small models.     

Distribution of the C statistic with applications to the sample mean of Poisson data

The C statistic, also known as the Cash statistic, is often used in astronomy for the analysis of low-count Poisson data. The main advantage of this statistic, compared to the more commonly used χ2 statistic, is its applicability without the need to combine data points. This feature has made the C statistic a very useful method to analyze Poisson data that have small (or even null) counts in each resolution element. One of the challenges of the C statistic is that its probability distribution, under the null hypothesis that the data follow a parent model, is not known exactly. This paper presents an effort towards improving our understanding of the C statistic by studying (a) the distribution of C statistic for a fully specified model, (b) the distribution of C_min resulting from a maximum-likelihood fit to a simple one-parameter constant model, i.e. a model that represents the sample mean of N Poisson measurements, and (c) the distribution of the associated ΔC statistic that is used for parameter estimation. The results confirm the expectation that, in the high-count limit, both C statistic and C_min have the same mean and variance as a χ2 statistic with same number of degrees of freedom. It is also found that, in the low-count regime, the expectation of the C statistic and C_min can be substantially lower than for a χ2 distribution. The paper makes use of recent X-ray observations of the astronomical source PG 1116+215 to illustrate the application of the C statistic to Poisson data.     

Estimating the parameter in the Pauling equation

Four procedures are suggested for estimating the parameter ‘a’ in the Pauling equation:

e-X/a+e ? Y/a = 1.

The procedures are: using the mean of individual solutions, least squares with Y the subject of the equation, least squares with X the subject of the equation and maximum likelihood using a statistical model. In order to compare these estimates, we use Efron's bootstrap technique (1979), since distributional results are not available. This example also illustrates the role of the bootstrap in statistical inference.     

Remarks on an integral functional driven by sub-fractional Brownian motion

This paper studies the functionals where is a one-dimension sub-fractional Brownian motion with index H∈(0,1). It shows that there exists a constant p_H∈(1,2) such that p-variation of the process (j=1,2) is equal to 0 if p>p_H, where ?_j, j=1,2, are the local time and weighted local time of S^H, respectively. This extends the classical results for Brownian motion.     

Stochastic modelling of regional annual rainfall anomalies in East Africa

ARIMA (p, d, q) models were fitted to areal annual rainfall of two homogeneous regions in East Africa with rainfall records extending between the period 1922–80. The areal estimates of the regional rainfall were derived from the time series of the first eigenvector, which was significantly dominant at each of the two regions. The first eigenvector accounted for about 80% of the total rainfall variance in each region.

The class of ARIMA (p, d, q) models which best fitted the areal indices of relative wetness/dryness were the A R M A (3, 1) models. Tests of forecasting skill however indicated low skill in the forecasts given by these models. In all cases the models accounted for less than 50% of the total variance.

Spectral analysis of the indices time series indicated dominant quasi-periodic fluctuations around 2.2–2.8 years, 3–3.7 years, 5–6 years and 10–13 years. These spectral bands however accounted for very low proportion of the total rainfall variance.     

Is there an alternative to the european farm typology?

The 1978 European Community Typology for Agricultural Holdings is described in this paper and contrasted with a data based, polythetic-multivariate classification based on cluster analysis.

The requirement to reduce the size of the variable set employed in an optimisation-partition method of clustering suggested the value of principal components and factor analysis for the identification of major ‘source’ dimensions against which to measure farm differences and similarities.

The Euclidean cluster analysis incorporating the reduced dimensions quickly converged to a stable solution and was little influenced by the initial number or nature of ‘seeding’ partitions of the data.

The assignment of non-sampled observations from the population to cluster classes was completed using classification functions.

The final scheme, based on a sample of over 2,000 observations, was found to be both capable of interpretation and meaningful in terms of agricultural structure and practice and much superior in its explanatory power when compared with a version of the principal activity typology.     

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.