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.     

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.     

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.     

Invariant distributions of the multivariate tests in the skew elliptical model

This paper gives the conditions for the invariance of the null distributions of the normal theory tests for the linear restriction on the location parameters in the family of the matrix variate skew elliptical distributions. Main properties of the Hotelling’s generalized statistic in this family are investigated.     

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.     

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.     

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.     

A non-adaptive search algorithm that identifies up to three defects

We give a simple and constructable non-adaptive search algorithm that identifies a defective subset of cardinality at most three in a sample of cardinality n when the testing procedure can detect the presence of an odd number of defects in any subset of the sample. Our algorithm uses at most

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many tests.     

A critical look at Lo's modified R/S statistic

We report on an empirical investigation of the modified rescaled adjusted range or R/S statistic that was proposed by Lo, 1991. Econometrica 59, 1279–1313, as a test for long-range dependence with good robustness properties under ‘extra’ short-range dependence. In contrast to the classical R/S statistic that uses the standard deviation S to normalize the rescaled range R, Lo's modified R/S-statistic V_q is normalized by a modified standard deviation S_q which takes into account the covariances of the first q lags, so as to discount the influence of the short-range dependence structure that might be present in the data. Depending on the value of the resulting test-statistic V_q, the null hypothesis of no long-range dependence is either rejected or accepted. By performing Monte-Carlo simulations with ‘truly’ long-range- and short-range dependent time series, we study the behavior of V_q, as a function of q, and uncover a number of serious drawbacks to using Lo's method in practice. For example, we show that as the truncation lag q increases, the test statistic V_q has a strong bias toward accepting the null hypothesis (i.e., no long-range dependence), even in ideal situations of ‘purely’ long-range dependent data.     

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.     

On characterizing distributions via regression of functions of generalized order statistics

Let X(1,n,m₁,k),X(2,n,m₂,k),…,X(n,n,m,k) be n generalized order statistics from a continuous distribution F which is strictly increasing over (a,b),−∞≤a<b≤∞, the support of F. Let g be an absolutely continuous and monotonically increasing function in (a,b) with finite g(a⁺),g(b⁻) and E(g(X)). Then for some positive integer s,1<s≤n, we give characterization of distributions by means of