On the central limit theorem along subsequences of sums of i.i.d. random variables

Let \(\mathbb{N } = \{1, 2, 3, \ldots \}\) . Let \(\{X, X_{n}; n \in \mathbb N \}\) be a sequence of i.i.d. random variables, and let \(S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N \) . Then \( S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})\) for some \(\sigma ^{2} < \infty \) whenever, for a subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})\) . Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when \(\{\sqrt{n}; n \in \mathbb N \}\) is replaced by \(\{\sqrt{na_{n}};n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) . We show that, for given positive nondecreasing sequence \(\{a_{n}; n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) and \(\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)\) . In particular, for given \(0 < p < 2\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)\) .     

On the joint distribution of success runs of several lengths in the sequence of MBT and its applications

Let \(X_1 ,X_2 ,\ldots ,X_n \) be a sequence of Markov Bernoulli trials (MBT) and \(\underline{X}_n =( {X_{n,k_1 } ,X_{n,k_2 } ,\ldots ,X_{n,k_r } })\) be a random vector where \(X_{n,k_i } \) represents the number of occurrences of success runs of length \(k_i \,( {i=1,2,\ldots ,r})\) . In this paper the joint distribution of \(\underline{X}_n \) in the sequence of \(n\) MBT is studied using method of conditional probability generating functions. Five different counting schemes of runs namely non-overlapping runs, runs of length at least \(k\) , overlapping runs, runs of exact length \(k\) and \(\ell \) -overlapping runs (i.e. \(\ell \) -overlapping counting scheme), \(0\le \ell are considered. The pgf of joint distribution of \(\underline{X}_n \) is obtained in terms of matrix polynomial and an algorithm is developed to get exact probability distribution. Numerical results are included to demonstrate the computational flexibility of the developed results. Various applications of the joint distribution of \(\underline{X}_n \) such as in evaluation of the reliability of \(( {n,f,k})\!\!:\!\!G\) and \(\!:\!\!G\) system, in evaluation of quantities related to start-up demonstration tests, acceptance sampling plans are also discussed.     

EWMA charts: ARL considerations in case of changes in location and scale

Widely spread tools within the area of Statistical Process Control are control charts of various designs. Control chart applications are used to keep process parameters (e.g., mean \(\mu \) , standard deviation \(\sigma \) or percent defective \(p\) ) under surveillance so that a certain level of process quality can be assured. Well-established schemes such as exponentially weighted moving average charts (EWMA), cumulative sum charts or the classical Shewhart charts are frequently treated in theory and practice. Since Shewhart introduced a \(p\) chart (for attribute data), the question of controlling the percent defective was rarely a subject of an analysis, while several extensions were made using more advanced schemes (e.g., EWMA) to monitor effects on parameter deteriorations. Here, performance comparisons between a newly designed EWMA \(p\) control chart for application to continuous types of data, \(p=f(\mu ,\sigma )\) , and popular EWMA designs ( \(\bar{X}\) , \(\bar{X}\) - \(S^2\) ) are presented. Thus, isolines of the average run length are introduced for each scheme taking both changes in mean and standard deviation into account. Adequate extensions of the classical EWMA designs are used to make these specific comparisons feasible. The results presented are computed by using numerical methods.     

A unified method for constructing expectation tolerance intervals

Given a random sample of size \(n\) with mean \(\overline{X} \) and standard deviation \(s\) from a symmetric distribution \(F(x; \mu , \sigma ) = F_{0} (( x- \mu ) / \sigma ) \) with \(F_0\) known, and \(X \sim F(x;\; \mu , \sigma )\) independent of the sample, we show how to construct an expansion \( a_n^{\prime } = \sum _{i=0}^\infty \ c_i \ n^{-i} \) such that \(\overline{X} - s a_n^{\prime } < X < \overline{X} + s a_n^{\prime } \) with a given probability \(\beta \) . The practical value of this result is illustrated by simulation and using a real data set.     

The variable parameters T ^{2} chart with run rules

The Hotelling’s \(\textit{T}^{2 }\) control chart with variable parameters (VP \(T^{2})\) has been shown to have better statistical performance than other adaptive control schemes in detecting small to moderate process mean shifts. In this paper, we investigate the statistical performance of the VP \(T^{2}\) control chart coupled with run rules. We consider two well-known run rules schemes. Statistical performance is evaluated by using a Markov chain modeling the random shock mechanism of the monitored process. The in-control time interval of the process is assumed to follow an exponential distribution. A genetic algorithm has been designed to select the optimal chart design parameters. We provide an extensive numerical analysis indicating that the VP \(T^{2}\) control chart with run rules outperforms other charts for small sizes of the mean shift expressed through the Mahalanobis distance.     

Equalities between OLSE,BLUE and BLUP in the linear model

We consider equalities between the ordinary least squares estimator ( $\mathrm {OLSE} $ ), the best linear unbiased estimator ( $\mathrm {BLUE} $ ) and the best linear unbiased predictor ( $\mathrm {BLUP} $ ) in the general linear model $\{ \mathbf y , \mathbf X \varvec{\beta }, \mathbf V \}$ extended with the new unobservable future value $ \mathbf y _{*}$ of the response whose expectation is $ \mathbf X _{*}\varvec{\beta }$ . Our aim is to provide some new insight and new proofs for the equalities under consideration. We also collect together various expressions, without rank assumptions, for the $\mathrm {BLUP} $ and provide new results giving upper bounds for the Euclidean norm of the difference between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {BLUE} ( \mathbf X _{*}\varvec{\beta })$ and between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {OLSE} ( \mathbf X _{*}\varvec{\beta })$ . A remark is made on the application to small area estimation.     

Evidence synthesis through a degradation model applied to myocardial infarction

We propose an evidence synthesis approach through a degradation model to estimate causal influences of physiological factors on myocardial infarction (MI) and coronary heart disease (CHD). For instance several studies give incidences of MI and CHD for different age strata, other studies give relative or absolute risks for strata of main risk factors of MI or CHD. Evidence synthesis of several studies allows incorporating these disparate pieces of information into a single model. For doing this we need to develop a sufficiently general dynamical model; we also need to estimate the distribution of explanatory factors in the population. We develop a degradation model for both MI and CHD using a Brownian motion with drift, and the drift is modeled as a function of indicators of obesity, lipid profile, inflammation and blood pressure. Conditionally on these factors the times to MI or CHD have inverse Gaussian ( ${\mathcal{IG}}$ ) distributions. The results we want to fit are generally not conditional on all the factors and thus we need marginal distributions of the time of occurrence of MI and CHD; this leads us to manipulate the inverse Gaussian normal distribution ( ${\mathcal{IGN}}$ ) (an ${\mathcal{IG}}$ whose drift parameter has a normal distribution). Another possible model arises if a factor modifies the threshold. This led us to define an extension of ${\mathcal{IGN}}$ obtained when both drift and threshold parameters have normal distributions. We applied the model to results published in five important studies of MI and CHD and their risk factors. The fit of the model using the evidence synthesis approach was satisfactory and the effects of the four risk factors were highly significant.     

An interesting property of the Poisson operating characteristic function

In elementary probability theory, as a result of a limiting process the probabilities of aBi(n, p) binomial distribution are approximated by the probabilities of aPo(np) Poisson distribution. Accordingly, in statistical quality control the binomial operating characteristic function \(\mathcal{L}_{n,c} (p)\) is approximated by the Poisson operating characteristic function \(\mathcal{F}_{n,c} (p)\) . The inequality \(\mathcal{L}_{n + 1,c + 1} (p) > \mathcal{L}_{n,c} (p)\) forp∈(0;1) is evident from the interpretation of \(\mathcal{L}_{n + 1,c + 1} (p)\) , \(\mathcal{L}_{n,c} (p)\) as probabilities of accepting a lot. It is shown that the Poisson approximation \(\mathcal{F}_{n,c} (p)\) preserves this essential feature of the binomial operating characteristic function, i.e. that an analogous inequality holds for the Poisson operating characteristic function, too.     

Consistency of a nonparametric conditional mode estimator for random fields

Given a stationary multidimensional spatial process $\left\{ Z_{\mathbf{i}}=\left( X_{\mathbf{i}},\ Y_{\mathbf{i}}\right) \in \mathbb R ^d\right. \left. \times \mathbb R ,\mathbf{i}\in \mathbb Z ^{N}\right\} $ , we investigate a kernel estimate of the spatial conditional mode function of the response variable $Y_{\mathbf{i}}$ given the explicative variable $X_{\mathbf{i}}$ . Consistency in $L^p$ norm and strong convergence of the kernel estimate are obtained when the sample considered is a $\alpha $ -mixing sequence. An application to real data is given in order to illustrate the behavior of our methodology.     

Let us do the twist again

Krämer (Sankhy $\bar{\mathrm{a }}$ 42:130–131, 1980) posed the following problem: “Which are the $\mathbf{y}$ , given $\mathbf{X}$ and $\mathbf{V}$ , such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors $\mathbf{y}$ for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector $\varvec{\beta }$ coincide under the general Gauss–Markov model $\mathbf{y} = \mathbf{X} \varvec{\beta } + \mathbf{u}$ . The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of $\varvec{\beta }$ to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of $\varvec{\beta }$ , we consider the estimation of the systematic part $\mathbf{X} \varvec{\beta }$ , which is a natural consequence of relaxing the assumption that $\mathbf{X}$ and $\mathbf{V}$ are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of $\mathbf{X} \varvec{\beta }$ , as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.     

On Bahadur representation for sample quantiles under α-mixing sequence

In this paper, by relaxing the mixing coefficients to α(n) = O(n ^?β), β > 3, we investigate the Bahadur representation of sample quantiles under α-mixing sequence and obtain the rate as ${O(n^{-\frac{1}{2}}(\log\log n\cdot\log n)^{\frac{1}{2}})}$ . Meanwhile, for any δ > 0, by strengthening the mixing coefficients to α(n) = O(n ^?β ), ${\beta > \max\{3+\frac{5}{1+\delta},1+\frac{2}{\delta}\}}$ , we have the rate as ${O(n^{-\frac{3}{4}+\frac{\delta}{4(2+\delta)}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$ . Specifically, if ${\delta=\frac{\sqrt{41}-5}{4}}$ and ${\beta > \frac{\sqrt{41}+7}{2}}$ , then the rate is presented as ${O(n^{-\frac{\sqrt{41}+5}{16}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$ .     

Covariance matrix of the bias-corrected maximum likelihood estimator in generalized linear models

For the first time, we obtain a general formula for the \(n^{-2}\) asymptotic covariance matrix of the bias-corrected maximum likelihood estimators of the linear parameters in generalized linear models, where \(n\) is the sample size. The usefulness of the formula is illustrated in order to obtain a better estimate of the covariance of the maximum likelihood estimators and to construct better Wald statistics. Simulation studies and an application support our theoretical results.     

Rate-optimal tests for jumps in diffusion processes

Suppose one has a sample of high-frequency intraday discrete observations of a continuous time random process, such as foreign exchange rates and stock prices, and wants to test for the presence of jumps in the process. We show that the power of any test of this hypothesis depends on the frequency of observation. In particular, if the process is observed at intervals of length $1/n$ 1 / n and the instantaneous volatility of the process is given by $ \sigma _{t}$ σ t , we show that at best one can detect jumps of height no smaller than $\sigma _{t}\sqrt{2\log (n)/n}$ σ t 2 log ( n ) / n . We present a new test which achieves this rate for diffusion-type processes, and examine its finite-sample properties using simulations.     

On exact and optimal single-sampling plans by variables

We deal with sampling by variables with two-way protection in the case of a $N\>(\mu ,\sigma ^2)$ distributed characteristic with unknown $\sigma $ . The LR sampling plan proposed by Lieberman and Resnikoff (JASA 50: 457 ${-}$ 516, 1955) and the BSK sampling plan proposed by Bruhn-Suhr and Krumbholz (Stat. Papers 31: 195–207, 1990) are based on the UMVU and the plug-in estimator, respectively. For given $p_1$ (AQL), $p_2$ (RQL) and $\alpha ,\beta $ (type I and II errors) we present an algorithm allowing to determine the optimal LR and BSK plans having minimal sample size among all plans satisfying the corresponding two-point condition on the OC. An R (R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/ 2012) package, ExLiebeRes‘ (Krumbholz and Steuer ExLiebeRes: calculating exact LR- and BSK-plans, R-package version 0.9.9. http://exlieberes.r-forge.r-project.org 2012) implementing that algorithm is provided to the public.     

Structured orthogonal families of one and two strata prime basis factorial models

The models in structured families correspond to the treatments of a fixed effects base design \(\pi \) . Then the action of factors in \(\pi \) , on the fixed effects parameters of the models, is studied. Analyzing such a families enables the study of the action of nesting factors on the effects and interactions of nested factors. When \(\pi \) has an orthogonal structure, the family of models is said to be orthogonal. The models in the family can have one, two or more strata. Models with more than one stratum are obtained through nesting of one stratum models. A general treatment of the case in which the base design has orthogonal structure is presented and a special emphasis is given to the families of prime basis factorials models. These last models are, as it is well known, widely used in fertilization trials.     

Empirical likelihood for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data

In this paper, the empirical likelihood inferences for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data are investigated. We construct the empirical log-likelihood ratio function for the fixed-effects parameters and the mean parameters of random-effects. The empirical log-likelihood ratio at the true parameters is proven to be asymptotically $\chi ^2_{q+r}$ , where $q$ and $r$ are dimensions of the fixed and random effects respectively, and the corresponding confidence regions for them are then constructed. We also obtain the maximum empirical likelihood estimator of the parameters of interest, and prove it is the asymptotically normal under some suitable conditions. A simulation study and a real data application are undertaken to assess the finite sample performance of the proposed method.     

Linear models that allow perfect estimation

The general Gauss–Markov model, Y = Xβ + e, E(e) = 0, Cov(e) = σ ² V, has been intensively studied and widely used. Most studies consider covariance matrices V that are nonsingular but we focus on the most difficult case wherein C(X), the column space of X, is not contained in C(V). This forces V to be singular. Under this condition there exist nontrivial linear functions of Q′Xβ that are known with probability 1 (perfectly) where ${C(Q)=C(V)^\perp}$ . To treat ${C(X) \not \subset C(V)}$ , much of the existing literature obtains estimates and tests by replacing V with a pseudo-covariance matrix T = V + XUX′ for some nonnegative definite U such that ${C(X) \subset C(T)}$ , see Christensen (Plane answers to complex questions: the theory of linear models, 2002, Chap. 10). We find it more intuitive to first eliminate what is known about Xβ and then to adjust X while keeping V unchanged. We show that we can decompose β into the sum of two orthogonal parts, β = β ₀ + β ₁, where β ₀ is known. We also show that the unknown component of X β is ${X\beta_1 \equiv \tilde{X} \gamma}$ , where ${C(\tilde{X})=C(X)\cap C(V)}$ . We replace the original model with ${Y-X\beta_0=\tilde{X}\gamma+e}$ , E(e) = 0, ${Cov(e)=\sigma^2V}$ and perform estimation and tests under this new model for which the simplifying assumption ${C(\tilde{X}) \subset C(V)}$ holds. This allows us to focus on the part of that parameters that are not known perfectly. We show that this method provides the usual estimates and tests.     

Order selection in finite mixtures of linear regressions

Finite mixture models can adequately model population heterogeneity when this heterogeneity arises from a finite number of relatively homogeneous clusters. An example of such a situation is market segmentation. Order selection in mixture models, i.e. selecting the correct number of components, however, is a problem which has not been satisfactorily resolved. Existing simulation results in the literature do not completely agree with each other. Moreover, it appears that the performance of different selection methods is affected by the type of model and the parameter values. Furthermore, most existing results are based on simulations where the true generating model is identical to one of the models in the candidate set. In order to partly fill this gap we carried out a (relatively) large simulation study for finite mixture models of normal linear regressions. We included several types of model (mis)specification to study the robustness of 18 order selection methods. Furthermore, we compared the performance of these selection methods based on unpenalized and penalized estimates of the model parameters. The results indicate that order selection based on penalized estimates greatly improves the success rates of all order selection methods. The most successful methods were \(MDL2\) , \(MRC\) , \(MRC_k\) , \(ICL\) – \(BIC\) , \(ICL\) , \(CAIC\) , \(BIC\) and \(CLC\) but not one method was consistently good or best for all types of model (mis)specification.     

Estimating common standard deviation of two normal populations with ordered means

Independent random samples are taken from two normal populations with means $\mu _1$ and $\mu _2$ and a common unknown variance $\sigma ^2.$ It is known that $\mu _1\le \mu _2.$ In this paper, estimation of the common standard deviation $\sigma $ is considered with respect to a scale invariant loss function. A general minimaxity result is proved and a class of minimax estimators is derived. An admissibility result is proved in this class. Further a class of equivariant estimators with respect to a subgroup of affine group is considered and dominating estimators in this class are obtained. The risk performance of some of these estimators is compared numerically.     

Estimation in a competing risks proportional hazards model under length-biased sampling with censoring

What population does the sample represent? The answer to this question is of crucial importance when estimating a survivor function in duration studies. As is well-known, in a stationary population, survival data obtained from a cross-sectional sample taken from the population at time $t_0$ represents not the target density $f(t)$ but its length-biased version proportional to $tf(t)$ , for $t>0$ . The problem of estimating survivor function from such length-biased samples becomes more complex, and interesting, in presence of competing risks and censoring. This paper lays out a sampling scheme related to a mixed Poisson process and develops nonparametric estimators of the survivor function of the target population assuming that the two independent competing risks have proportional hazards. Two cases are considered: with and without independent censoring before length biased sampling. In each case, the weak convergence of the process generated by the proposed estimator is proved. A well-known study of the duration in power for political leaders is used to illustrate our results. Finally, a simulation study is carried out in order to assess the finite sample behaviour of our estimators.