Annals Of Mathematical Statistics Research Articles

A modified Huber loss function for continual reassessment methods in clinical trials

In dose-finding (DF) trials, methods for discovering an optimal criterion that controls toxicity while demonstrating a potential for efficacy have been the subject of statistical research. Although continual reassessment methods (CRMs) have been utilized, the consensus among practitioners and clinical trial specialists is that there is always room for improvement with CRMs. Within the paradigm of a full-Bayesian method for CRMs, we examine the performance of dose selection algorithms based on a family of loss functions defined in Huber (The Annals of Mathematical Statistics, vol. 35, issue 1, pp. 73–101, 1964), namely, the Huber loss function, with a special focus on the modified Huber loss function (MHLF). Our exploration suggests that, compared to Bayesian optimal interval design (BOIN) and toxicity interval loss function (TILF), the approach based on the MHLF has been able to select correct doses with fewer average number of patients across the spectrum of dose escalation schemes such as those seen in clinical settings.

One-two dependence and probability inequalities between one- and two-sided union-intersection tests.

In a paper published in 1939 in The Annals of Mathematical Statistics, Wald and Wolfowitz discussed the possible validity of a probability inequality between one- and two-sided coverage probabilities for the empirical distribution function. Twenty-eight years later, Vandewiele and Noé proved this inequality for Kolmogorov-Smirnov type goodness of fit tests. We refer to this type of inequality as one-two inequality. In this paper, we generalize their result for one- and two-sided union-intersection tests based on positively associated random variables and processes. Thereby, we give a brief review of different notions of positive association and corresponding results. Moreover, we introduce the notion of one-two dependence and discuss relationships with other dependence concepts. While positive association implies one-two dependence, the reverse implication fails. Last but not least, the Bonferroni inequality and the one-two inequality yield lower and upper bounds for two-sided acceptance/rejection probabilities which differ only slightly for significance levels not too large. We discuss several examples where the one-two inequality applies. Finally, we briefly discuss the possible impact of the validity of a one-two inequality on directional error control in multipletesting.

The Proximal Robbins–Monro Method

AbstractThe need for statistical estimation with large data sets has reinvigorated interest in iterative procedures and stochastic optimization. Stochastic approximations are at the forefront of this recent development as they yield procedures that are simple, general and fast. However, standard stochastic approximations are often numerically unstable. Deterministic optimization, in contrast, increasingly uses proximal updates to achieve numerical stability in a principled manner. A theoretical gap has thus emerged. While standard stochastic approximations are subsumed by the framework Robbins and Monro (The annals of mathematical statistics, 1951, pp. 400–407), there is no such framework for stochastic approximations with proximal updates. In this paper, we conceptualize a proximal version of the classical Robbins–Monro procedure. Our theoretical analysis demonstrates that the proposed procedure has important stability benefits over the classical Robbins–Monro procedure, while it retains the best known convergence rates. Exact implementations of the proximal Robbins–Monro procedure are challenging, but we show that approximate implementations lead to procedures that are easy to implement, and still dominate standard procedures by achieving numerical stability, practically without trade-offs. Moreover, approximate proximal Robbins–Monro procedures can be applied even when the objective cannot be calculated analytically, and so they generalize stochastic proximal procedures currently in use.

A pioneering study on discrete cosine transform

The discrete cosine transform is a widely applied linear transform not only in data compression but also in statistics. On the transform, John von Neumann's research article published in Annals of Mathematical Statistics in 1941 has not been recognized as a pioneering study. This note is intended to share the idea that his research is eligible to be regarded as such a study.

Independent, Tough Identical Results: The Class of Tweedie on Power Variance Functions and the Class of Bar-Lev and Enis on Reproducible Natural Exponential Families

The Rao-Blackwell theorem has had a fundamental role in statistical theory. However, as opposed to what seems natural, Rao and Blackwell did not investigate and write the theorem jointly. In fact, they both published the same result independently, two years apart. Indeed, as C.R. Rao writes in Wikipedia: ”the result on one parameter case was published by Rao (1945) in the Bulletin of the Calcutta Mathematical Society and by Blackwell (1947) in The Annals of Mathematical Statistics. Only Lehmann and Sche ´e (1950) called the result as Rao-Blackwell theorem”. Forty years later, a situation very similar to the previous one seems to have happened. Tweedie (1984) in a paper published in a proceedings to a conference held in Calcutta and Bar-Lev and Enis (1986) in a paper published in The Annals of Statistics both presented for the first time, albeit two years apart, independently and in di erent contexts, the class of natural exponential families having power variance functions (NEF-PVFs). Tweedie’s results were then mentioned by Jorgensen (1987) in his fundamental paper on exponential dispersion models published in the Journal of the Royal Statistical Society, Series B. Jorgensen, however, mentioned also other researchers, including Bar-Lev and Enis, as dealt with the same problem. Nonetheless, Jorgensen (1987) stated in his paper that ”The most complete study” of NEF-PVFs was given by Tweedie (1984), a statement which has led to naming the class of NEF-PVFs as the Tweedie class. This statement of Jorgensen is entirely and utterly incorrect. Accordingly, one of the goals of this note is to 'prove' such incorrectness. Based on this 'proof' it will be evident, so I trust, that both Bar-Lev and Enis should have received the appropriate credit by re-naming the class of NEF-PVFs via the exploitation of the names of Tweedie, Bar-Lev and Enis. This would resemble the dignified and elegant manner Lehmann and Sche ´e acted on the Rao-Blackwell Theorem. Notwithstanding, the main aim of the note is to encourage young researchers to present their results with self-confidence and to get the credit they deserve.

Fluid limit for a genetic mutation model

AbstractWe trace the time evolution of the number Ut of nondeleterious mutations, present in a gene modeled by a word of length L and DNA fragments by characters labeled 0, 1,…, N. For simplification, deleterious mutations are codified as equal to 0. The discrete case studied in Grigorescu (Stochastic Models, 29, 2013 p. 328), is a modified version of the Pólya urn, where the two types are exactly the zeros and nonzeros. A random continuous‐time binary mutation model, where the probability of creating a deleterious mutation is 1/N, while the probability of recovery , γ continuous, is studied under a Eulerian scaling , L → ∞. The fluid limit ut, emerging due to the high frequency scale of mutations, is the solution of a deterministic generalized logistic equation. The power law γ(u) = cua captures important features in both genetical and epidemiological interpretations, with c being the intensity of the intervention, a the strength/virulence of the disease, and 1/N the decay rate/infectiousness. Among other applications, we obtain a quantitative study of ΔT, the maximal interval between tests. Several stochastic optimization problems, including a generalization of the Shepp urn (The Annals of Mathematical Statistics, 40, 1969 p. 993), are proposed.

The Marshall-Olkin generalized gamma distribution

Attempts have been made to define new classes of distributions that provide more flexibility for modelling skewed data in practice. In this work we define a new extension of the generalized gamma distribution (Stacy, The Annals of Mathematical Statistics, 33, 1187-1192, 1962) for Marshall-Olkin generalized gamma (MOGG) distribution, based on the generator pioneered by Marshall and Olkin (Biometrika, 84, 641-652, 1997). This new lifetime model is very flexible including twenty one special models. The main advantage of the new family relies on the fact that practitioners will have a quite flexible distribution to fit real data from several fields, such as engineering, hydrology and survival analysis. Further, we also define a MOGG mixture model, a modification of the MOGG distribution for analyzing lifetime data in presence of cure fraction. This proposed model can be seen as a model of competing causes, where the parameter associated with the Marshall-Olkin distribution controls the activation mechanism of the latent risks (Cooner et al., Statistical Methods in Medical Research, 15, 307-324, 2006). The asymptotic properties of the maximum likelihood estimation approach of the parameters of the model are evaluated by means of simulation studies. The proposed distribution is fitted to two real data sets, one arising from measuring the strength of fibers and the other on melanoma data.

A Conversation with G. A. F. Seber

In a career spanning more than 50 years, George Arthur Frederick Seber has made significant contributions in many areas of statistics as well as foundational work in capture–recapture modeling. Seber was born in 1938 in Australia to New Zealand parents and moved back to New Zealand where he completed almost all of his schooling. After graduating with bachelors and masters degrees from the University of Auckland, Seber studied for his Ph.D. at Manchester on a Commonwealth Scholarship. In his first year, he was supervised by John Darroch with David Silvey taking over supervision for the final 18 months of the degree. Seber’s Ph.D. thesis was remarkably fruitful leading to three papers in Biometrika, two in the Annals of Mathematical Statistics and one in the Journal of the Royal Statistical Society B. The topics covered were broad mixing foundational work on linear model theory with capture–recapture problems. His solution to the multi-sample single recapture problem anticipated the general solution to the open population problem that Seber published in 1965 concurrent with the work of George Jolly. Seber has been a prolific writer of books including classical texts on the linear model and nonlinear regression. His comprehensive book The Estimation of Animal Abundance appeared in two editions. He has recently completed four books and is working on a book on open population capture recapture. Seber took up an invited personal Chair in Biometrics at the University of Otago in 1971, and then the foundational Chair in Statistics at the University of Auckland in 1973. Seber developed a remarkable statistics group built around people such as Jeffrey Hunter, Alan Lee, Alastair Scott, Chris Triggs and Chris Wild, all of whom went on to hold Chairs in Statistics and to head Departments of Statistics in New Zealand. Seber was elected a Fellow of the Royal Society of New Zealand (RSNZ) in 1997 and in 1999 was awarded the Hector Medal by the RSNZ.

Multistage point estimation methodologies for a negative exponential location under a modified linex loss function: Illustrations with infant mortality and bone marrow data

ABSTRACTWe have designed Stein-type (Stein, 1945, Annals of Mathematical Statistics) two-stage, modified two-stage (Mukhopadhyay and Duggan, 1997, Sankhya, Series A), and purely sequential strategies (Chow and Robbins, 1965, Annals of Mathematical Statistics) to estimate an unknown location parameter of a negative exponential distribution having an unknown scale parameter under a newly defined and modified Linex loss function. We aim at controlling the associated risk function per unit cost by bounding it from above with a fixed preassigned positive number, ω, and we emphasize both asymptotic first-order and asymptotic second-order properties for the modified two-stage and purely sequential estimation strategies. In developing asymptotic second-order properties for the modified two-stage methodology, we have heavily relied upon basic ideas rooted in Mukhopadhyay and Duggan (1997). In developing asymptotic second-order properties for the purely sequential methodology, however, we have heavily relied upon nonlinear renewal theory (Lai and Siegmund, 1977, 1979, Annals of Statistics; Woodroofe, 1977, Annals of Statistics). Then, we take to extensive data analysis carried out via computer simulations when requisite sample sizes range from small to moderate to large. We find that the Stein-type two-stage estimation methodology oversamples significantly and yet the achieved risk is not close to preset goal ω. On the other hand, both modified two-stage and purely sequential estimation strategies perform remarkably well. We have validated their main theoretical first-order and second-order properties through simulated data. The latter methodologies have been illustrated and implemented using two real data sets from health studies, namely, infant mortality data and bone marrow data.

Family of the generalised gamma kernels: a generator of asymmetric kernels for nonnegative data

Unlike symmetric kernels, so far exploring asymptotics on asymmetric kernels has relied on diversified approaches. This paper proposes a family of the generalised gamma (GG) kernels that is built on the probability density function of the GG distribution [Stacy, E.W. (1962), ‘A Generalization of the Gamma Distribution’, Annals of Mathematical Statistics, 33, 1187–1192] and a few common conditions. The family can generate asymmetric kernels that share appealing properties with the modified gamma kernel [Chen, S.X. (2000), ‘Probability Density Function Estimation Using Gamma Kernels’, Annals of the Institute of Statistical Mathematics, 52, 471–480]. Asymptotics on the kernels generated from the family can be delivered by manipulating the conditions directly, as with symmetric kernels.

Existence of an equilibrium for infinite horizon economies with and without complete information

This work proves the existence of an equilibrium for an infinite horizon economy where trade takes place sequentially over time. There exist two types of agents: the first correctly anticipates all future contingent endogenous variables with complete information as in Radner [Radner, R. (1972). Existence of equilibrium of plans, prices and price expectations in a sequence of markets. Econometrica, 289–303] and the second has exogenous expectations about the future environment as in Grandmont [Grandmont, J. M. (1977). Temporary general equilibrium theory. Econometrica, 535–572] and information based on the current and past aggregate variables including those which are private knowledge. Agents with exogenous expectations may have inconsistent optimal plans but have predictive beliefs in the context of Blackwell and Dubbins [Blackwell, D., Dubins, L. (1962). Merging of opinions with increasing information. The Annals of Mathematical Statistics, 882–886] with probability transition rules based on all observed variables. We provide examples of this framework applied to models of differential information and environments exhibiting results of market selection and convergence of an equilibrium. The existence result can be used to conclude that, by adding the continuity assumption on the probability transition rules, we obtain the existence of an equilibrium for some models of differential information and incomplete markets.

Stationarity of the transition probabilities in the Markov chain formulation of owner payments on projects

Stationarity of the transition probabilities in the Markov chain formulation of owner payments on projects

Goodness-of-fit testing: the thresholding approach

The classical Pearson's chi-square test for goodness-of-fit has found extensive applications in areas such as contingency tables and, recently, multiple testing. Mann and Wald [(1942), ‘On the Choice of the Number of Class Intervals in the Application of the Chi Square Test’, The Annals of Mathematical Statistics, 13, 306–317] were the first to establish the power advantages of letting the number n bin of bins tend to infinity with n, and found n bin=n 2/5 to be the optimal rate. For a corresponding development in the area of contingency tables, see Holst [(1972), ‘Asymptotic Normality and Efficiency for Certain Goodness-of-Fit Tests’, Biometrika, 59, 137–145], Morris [(1975), ‘Central Limit Theorems for Multinomial Sums’, The Annals of Statistics, 3, 165–188], and Koehler and Larntz [(1980), ‘An Empirical Investigation of Goodness-of-Fit Statistics for Sparse Multinomials’, Journal of the American Statistical Association, 75, 336–344]. In this paper, we consider the use of thresholding methods to further improve on the power of Pearson's chi-square test. An alternative statistic, based on the cell averages, is also studied. The Fourier or wavelet transformation is used to ensure power enhancement in both high- and low-signal-to-noise ratio alternatives. Simulations suggest that application of order thresholding (Kim, M.H., and Akritas, M.G. (2010), ‘Order Thresholding’, The Annals of Statistics, 38, 2314–2350) achieves accurate type I error rates, and competitive power.

On Two-Stage Confidence Interval Procedures and Their Comparisons for Estimating the Difference of Normal Means

We consider two independent N(μ1, σ2) and N(μ2, a 2σ2) populations where μ1, μ2, σ2 are unknown parameters with −∞ < μ1, μ2 < ∞, 0 < σ < ∞. We assume that a (>0) is known! The problem is one of estimating Δ = μ1 − μ2 by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient at least 1 − α. Here, d (>0) and 0 < α < 1 are both preassigned numbers. First, a two-stage procedure 𝒫1 is designed in the spirit of Stein (1945, Annals of Mathematical Statistics 16: 243–258). Then, another two-stage procedure 𝒫2 is tried in the spirit of Chapman (1950, Annals of Mathematical Statistics 21: 601–606). We report that the Stein-type two-stage procedure 𝒫1 performs better than 𝒫2. Next, a variant of a two-stage procedure due to Aoshima et al. (1996, Sequential Analysis 15: 61–70) is included as our third procedure, 𝒫3. In a variety of situations, we find that 𝒫1 also comes out ahead of 𝒫3. A new alternative sampling design 𝒫4 is then introduced by incorporating a two-stage sampling technique from one population alone followed by a single-stage sampling strategy from the other population. We observe that 𝒫4 compares favorably with 𝒫1 with regard to the achieved confidence level whereas the margin of oversampling in the case of 𝒫4 is justifiably smaller than that associated with 𝒫1. Additionally, 𝒫4 has a significant operational edge over 𝒫1 and hence we suggest implementing 𝒫4 in practice. In the end, we illustrate superiority of the new procedure 𝒫4 with an example using a real data set from horticulture (Mukhopadhyay et al., 2004, Journal of Agricultural, Biological and Environmental Statistics 38: 1384–1391).

Univariate linear calibration via replicated errors-in-variables model

In this paper, we deal with the comparative calibration problem, i.e. with the situation when one instrument or measurement technique is calibrated against another, each of which is subject to the measurement error. We propose an approximate, small sample, calibration confidence interval of the unknown true value of the measured substance in units of the more precise instrument, given measurement in units of the less precise instrument. Here we deal with the simplest case—single-use linear univariate calibration, i.e. the case in which we assume linear relationship between the two measurement techniques (instruments), and, further, that the calibration procedure is conducted in order to obtain one value for an unknown, reported together with an interval estimate. The method for deriving the approximate confidence interval is based on estimation of the calibration line via the replicated errors-in-variables model. The model is locally linearized and the Wald-type F-statistic is constructed. An essential point in this approach is the use of the F-approximation of the distribution of the F-statistic suggested by Kenward and Roger [Kenward, M.G. and Roger, J.H., 1997, Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53, 983–997]. The statistical properties of the proposed interval estimator are verified by simulations for wide spectrum of experimental designs and compared with two standard univariate interval estimates—the classical approach for deriving the calibration interval was proposed by Eisenhart [Eisenhart, C., 1939, The interpretation of certain methods and their use in biological and industrial research. Annals of Mathematical Statistics, 10, 162–186] and the inverse method was proposed by Krutchkoff [Krutchkoff, R.G., 1967, Classical and inverse methods of calibration. Technometrics, 9, 425–439].

Asymptotics for a Population Size Estimator of a Partially Uncatchable Population

We present the asymptotic distribution for an estimator of the population size for the case of s partially catchable populations. Our approach is useful for capture–recapture studies with photo-identification data where part of the population does not have any distinctive characteristic which allows unique identification of the individuals. This work represents an extension of Theorem 4 in Sanathanan (1972, The Annals of Mathematical Statistics, 43, 142–152).

Programs to Compute Distribution Functions and Critical Values for Extreme Value Ratios for Outlier Detection

A set of FORTRAN subprograms is presented to compute density and cumulative distribution functions and critical values for the range ratio statistics of Dixon (1951, The Annals of Mathematical Statistics ) These statistics are useful for detection of outliers in small samples.

A comparative study of monotone nonparametric kernel estimates

In this paper we present a detailed numerical comparison of three monotone nonparametric kernel regression estimates, which isotonize a nonparametric curve estimator. The first estimate is the classical smoothed isotone estimate of Brunk [Brunk, H.D., 1955, Maximum likelihood estimates of monotone parameters. The Annals of Mathematical Statistics, 26, 607–616.]. The second method has recently been proposed by Hall and Huang [Hall, P. and Huang, L.-S., 2001, Nonparametric kernel regression subject to monotonicity constraints. The Annals of Statistics, 29, 624–647.] and modifies the weights of a commonly used kernel estimate such that the resulting estimate is monotone. The third estimate was recently proposed by Dette et al. [Dette, H., Neumeyer, N. and Pilz, K.F., 2003, A simple non-parametric estimator of a monotone regression function. Technical report, Department of Mathematics. Available online at: http://www.ruhr-uni-bochum.de/mathematik3/preprint.htm] and combines density and regression estimation techniques to obtain a monotone curve estimate of the inverse of the isotone regression function. The three concepts are briefly reviewed and their finite sample properties are studied by means of a simulation study. Although all estimates are first-order asymptotically equivalent (provided that the unknown regression function is isotone) some differences for moderate sample sizes are observed.

Selfinformative limits of Bayes estimates and generalized maximum likelihood

A definition of selfinformative Bayes carriers or limits is given as a description of an approach to non-informative Bayes estimation in non- and semiparametric models. It takes the posterior w.r.t. a prior as a new prior and repeats this procedure again and again. A main objective of this article is to clarify the relation between selfinformative carriers or limits and maximum likelihood estimates (MLEs). For a model with dominated probability distributions, we state sufficient conditions under which the set of MLEs is a selfinformative carrier or in the case of a unique MLE its selfinformative limit property. Mixture models are covered. The result on carriers is extended to more general models without dominating measure. Selfinformative limits, in the case of estimation, of hazard functions based in censored observations and in the case of normal linear models with possibly non-identifiable parameters are shown to be identical to the generalized MLEs in the sense of Gill [Gill, R.D., 1989, Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. Scandinanian Journal of Statistics, 16(2), 97–128.] and Kiefer and Wolfowitz [Kiefer, J. and Wolfowitz, J., 1956, Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Annals of Mathematical Statistics, 27, 887–906.]. Selfinformative limits are given for semiparametric linear models. For a location model, they are identical to generalized MLEs, while this is not true in general.

A Burr Type X Chain-of-Links Model for Carbon Fibers

A model for application to carbon fibrous composites is proposed based on the Burr type X (BurrX) distribution first proposed by Burr (Burr, I.W. (1942). Annals of Mathematical Statistics, 13: 215-222) with an included parameter to account for fiber length. The model is further generalized similar to the generalization of the Weibull proposed by Padgett et al. (Padgett, W.J., Durham, S.D. and Mason, A.M. (1995). Journal of Composite Materials, 29: 1873-1884). Maximum likelihood estimation is briefly discussed, and the model is applied to the single-filament data of Bader and Priest (Bader, M.G. and Priest, A.M. (1982). In: Hayashi, T., Kawata, K. and Umekawa, S. (eds), Progress in Science and Engineering of Composites, pp. 1129-1136. ICCM-IV, Tokyo).