Curve registration

Functional data analysis involves the extension of familiar statistical procedures such as principal components analysis, linear modelling, and canonical correlation analysis to data where the raw observation x_i is a function. An essential preliminary to a functional data analysis is often the registration or alignment of salient curve features by suitable monotone transformations h_i of the argument t , so that the actual analyses are carried out on the values x_i { h_i ( t )}. This is referred to as dynamic time warping in the engineering literature. In effect, this conceptualizes variation among functions as being composed of two aspects: horizontal and vertical, or domain and range. A nonparametric function estimation technique is described for identifying the smooth monotone transformations h_i , and is illustrated by data analyses. A second-order linear stochastic differential equation is proposed to model these components of variation.     

A characterization of the inverse autocorrelation function

The inverse autocorrelation function of a weakly stationary stochastic process X_t at lag h, γⁱ _h, is shown to equal the negative of the partial correlation between random variables X_t and X_t+h after elimination of the influence of random variables X_k, k≠t5,t+h.     

On generalized exponential transformations for proportions

ABSTRACT

Transformation of the response is a popular method to meet the usual assumptions of statistical methods based on linear models such as ANOVA and t-test. In this paper, we introduce new families of transformations for proportions or percentage data. Most of the transformations for proportions require 0 < x < 1 (where x denotes the proportion), which is often not the case in real data. The proposed families of transformations allow x = 0 and x = 1. We study the properties of the proposed transformations, as well as the performance in achieving normality and homoscedasticity. We analyze three real data sets to empirically show how the new transformation performs in meeting the usual assumptions. A simulation study is also performed to study the behavior of new families of transformations.     

On hypothesis tests in misspecified change-point problems for a Poisson process

Consider an inhomogeneous Poisson process X on [0, T] whose unk-nown intensity function “switches” from a lower function g_* to an upper function h_* at some unknown point ?_* that has to be identified. We consider two known continuous functions g and h such that g_*(t) ? g(t) < h(t) ? h_*(t) for 0 ? t ? T. We describe the behavior of the generalized likelihood ratio and Wald’s tests constructed on the basis of a misspecified model in the asymptotics of large samples. The power functions are studied under local alternatives and compared numerically with help of simulations. We also show the following robustness result: the Type I error rate is preserved even though a misspecified model is used to construct tests.     

On the Failure Probability of Used Coherent Systems

In analyzing the lifetime properties of a coherent system, the concept of “signature” is a useful tool. Let T be the lifetime of a coherent system having n iid components. The signature of the system is a probability vector s=(s₁, s₂, …, s_n), such that s_i=P(T=X_i:n), where, X_i:n, i=1, 2, …, n denote the ordered lifetimes of the components. In this note, we assume that the system is working at time t>0. We consider the conditional signature of the system as a vector in which the ith element is defined as p_i(t)=P(T=X_i:n|T>t) and investigate its properties as a function of time.     

Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

Evaluation of different statistical methods using SAS software: an in silico approach for analysis of real-time PCR data

Real-time polymerase chain reaction (PCR) is reliable quantitative technique in gene expression studies. The statistical analysis of real-time PCR data is quite crucial for results analysis and explanation. The statistical procedures of analyzing real-time PCR data try to determine the slope of regression line and calculate the reaction efficiency. Applications of mathematical functions have been used to calculate the target gene relative to the reference gene(s). Moreover, these statistical techniques compare C_t (threshold cycle) numbers between control and treatments group. There are many different procedures in SAS for real-time PCR data evaluation. In this study, the efficiency of calibrated model and delta delta C_t model have been statistically tested and explained. Several methods were tested to compare control with treatment means of C_t. The methods tested included t-test (parametric test), Wilcoxon test (non-parametric test) and multiple regression. Results showed that applied methods led to similar results and no significant difference was observed between results of gene expression measurement by the relative method.     

Performance characteristics of the adjusted r2 algorithm for determining the start of the terminal disposition phase and comparison with a simple r2 algorithm and a visual inspection method

The adjusted r² algorithm is a popular automated method for selecting the start time of the terminal disposition phase (t_z) when conducting a noncompartmental pharmacokinetic data analysis. Using simulated data, the performance of the algorithm was assessed in relation to the ratio of the slopes of the preterminal and terminal disposition phases, the point of intercept of the terminal disposition phase with the preterminal disposition phase, the length of the terminal disposition phase captured in the concentration‐time profile, the number of data points present in the terminal disposition phase, and the level of variability in concentration measurement. The adjusted r² algorithm was unable to identify t_z accurately when there were more than three data points present in a profile's terminal disposition phase. The terminal disposition phase rate constant (λ_z) calculated based on the value of t_z selected by the algorithm had a positive bias in all simulation data conditions. Tolerable levels of bias (median bias less than 5%) were achieved under conditions of low measurement variability. When measurement variability was high, tolerable levels of bias were attained only when the terminal phase time span was 4 multiples of t_1/2 or longer. A comparison of the performance of the adjusted r² algorithm, a simple r² algorithm, and t_z selection by visual inspection was conducted using a subset of the simulation data. In the comparison, the simple r² algorithm performed as well as the adjusted r² algorithm and the visual inspection method outperformed both algorithms. Recommendations concerning the use of the various t_z selection methods are presented.     

Efficient Estimation of a Distribution Function under Quadrant Dependence

LetX₁,X₂, ..., be real-valued random variables forming a strictly stationary sequence, and satisfying the basic requirement of being either pairwise positively quadrant dependent or pairwise negatively quadrant dependent. LetF^ be the marginal distribution function of theX_ips, which is estimated by the empirical distribution functionF_n and also by a smooth kernel-type estimateF_n, by means of the segmentX₁, ...,X_n. These estimates are compared on the basis of their mean squared errors (MSE). The main results of this paper are the following. Under certain regularity conditions, the optimal bandwidth (in the MSE sense) is determined, and is found to be the same as that in the independent identically distributed case. It is also shown thatn MSE(F_n(t)) andnMSE (F^_n(t)) tend to the same constant, asn→∞ so that one can not discriminate be tween the two estimates on the basis of the MSE. Next, ifi(n) = min {k∈{1, 2, ...}; MSE (F_k(t)) ≤ MSE (F_n(t))}, then it is proved thati(n)/n tends to 1, asn→∞. Thus, once again, one can not choose one estimate over the other in terms of their asymptotic relative efficiency. If, however, the squared bias ofF^_n(t) tends to 0 sufficiently fast, or equivalently, the bandwidthh_n satisfies the requirement thatnh³_n→ 0, asn→∞, it is shown that, for a suitable choice of the kernel, (i(n) ?n)/(nh_n) tends to a positive number, asn→∞ It follows that the deficiency ofF_n(t) with respect toF^_n(t),i(n) ?n, is substantial, and, actually, tends to ∞, asn→∞. In terms of deficiency, the smooth estimateF^_n(t) is preferable to the empirical distribution functionF_n(t)     

An Estimation Problem with Poisson Processes

For n independent Poisson processes such that the i th process has intensity function lM_i(t) =δ_iρ(t; α) we consider estimation of p(t; α) =∫_oρ:(u; α) du. Two procedures are developed, one using exact arrival times, the other using categorical arrival times. Two instances where p(t; α) =p(α t) are investigated further. An example applying the methodology to the active life of a judicial opinion is described.     

ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS

Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (q_ij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., q_ij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability p_ij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p_0N(t) (p^(m)_(N0)(t)), 0 ≤ m ≤N, where f^(m)(t) denotes the mth derivative of a function f(t) with f⁽⁰⁾(t) =f(t). If X(t) is a birth-death process, then p_ij(t) is represented as a linear combination of p_0N^(m)(t), 0 ≤m≤N - |i-j|.     

Berry–Esseen type bounds in heteroscedastic errors-in-variables model

ABSTRACT

Consider the heteroscedastic partially linear errors-in-variables (EV) model y_i = x_iβ + g(t_i) + ε_i, ξ_i = x_i + μ_i (1 ? i ? n), where ε_i = σ_ie_i are random errors with mean zero, σ²_i = f(u_i), (x_i, t_i, u_i) are non random design points, x_i are observed with measurement errors μ_i. When f( · ) is known, we derive the Berry–Esseen type bounds for estimators of β and g( · ) under {e_i,?1 ? i ? n} is a sequence of stationary α-mixing random variables, when f( · ) is unknown, the Berry–Esseen type bounds for estimators of β, g( · ), and f( · ) are discussed under independent errors.     

PIECEWISE POLYNOMIAL APPROXIMATIONS FOR HEAVY-TAILED DISTRIBUTIONS IN QUEUEING ANALYSIS

ABSTRACT

A basic difficulty in dealing with heavy-tailed distributions is that they may not have explicit Laplace transforms. This makes numerical methods that use the Laplace transform more challenging. This paper generalizes an existing method for approximating heavy-tailed distributions, for use in queueing analysis. The generalization involves fitting Chebyshev polynomials to a probability density function g(t) at specified points t ₁, t ₂, …, t _N . By choosing points t _i , which rapidly get far out in the tail, it is possible to capture the tail behavior with relatively few points, and to control the relative error in the approximation. We give numerical examples to evaluate the performance of the method in simple queueing problems.     

Corrections and amendments

In the Competing risk analysis g_i(t) the failure rates h_i(t) and g_i(t) (as defined ) are, in general, not the same. However, of the survival times due to different causes are independent, then g_i (t) = h_i (t) this is a model studied by chiang (1961). In this note we have exhibited some counterexamples illustrating the relationships between g_i (t)= h_i (t) and the independence of survival times.     

General scores statistics on ranks in the analysis of unbalanced designs

The authors consider the situation of incomplete rankings in which n judges independently rank k_i ∈ {2, …, t} objects. They wish to test the null hypothesis that each judge picks the ranking at random from the space of k_i! permutations of the integers 1, …, k_i. The statistic considered is a generalization of the Friedman test in which the ranks assigned by each judge are replaced by real‐valued functions a(j, k_i), 1 ≤ j ≤ k_i ≤ t of the ranks. The authors define a measure of pairwise similarity between complete rankings based on such functions, and use averages of such similarities to construct measures of the level of concordance of the judges' rankings. In the complete ranking case, the resulting statistics coincide with those defined by Hájek & ?idák (1967, p. 118), and Sen (1968). These measures of similarity are extended to the situation of incomplete rankings. A statistic is derived in this more general situation and its properties are investigated.     

Precise large deviations for the difference of two sums of WUOD and non identically distributed random variables with dominatedly varying tails

In this article, we study large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j ? ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j ? ∑^N₂(t)_{j = 1}X_2j, where {X_1j, j ? 1} is a sequence of widely upper orthant dependent (WUOD) random variables with non identical distributions {F_1j(x), j ? 1}, {X_2j, j ? 1} is a sequence of independent identically distributed random variables, n₁(t) and n₂(t) are two positive integer-valued functions, and {N_i(t), t ? 0}²_{i = 1} with EN_i(t) = λ_i(t) are two counting processes independent of {X_ij, j ? 1}²_{i = 1}. Under several assumptions, some results of precise large deviations for non random difference and random difference are derived, and some corresponding results are extended.     

Quantification of symmetry for functional data with application to equine lameness classification

This paper presents a study on symmetry of repeated bi-phased data signals, in particular, on quantification of the deviation between the two parts of the signal. Three symmetry scores are defined using functional data techniques such as smoothing and registration. One score is related to the L ₂-distance between the two parts of the signal, whereas the other two are constructed to specifically measure differences in amplitude and phase. Moreover, symmetry scores based on functional principal component analysis (PCA) are examined. The scores are applied to acceleration signals from a study on equine gait. The scores turn out to be highly associated with lameness, and their applicability for lameness quantification and detection is investigated. Four classification approaches turn out to give similar results. The scores describing amplitude and phase variation turn out to outperform the PCA scores when it comes to the classification of lameness.     

OPTIMAL DESIGNS FOR FITTING A PROPORTIONAL HAZARDS REGRESSION MODEL TO DATA SUBJECT TO CENSORING

Suppose the probability model for failure time data, subject to censoring, is specified by the hazard function λ(t)exp(β_T x), where x is a vector of covariates. Analytical difficulties involved in finding the optimal design are avoided by assuming that λ is completely specified and by using D-optimality based on the information matrix for β Optimal designs are found to depend on β, but some results of practical consequence are obtained. It is found that censoring does not affect the choice of design appreciably when β^T x ≥ 0 for all points of the feasible region, but may have an appreciable effect when β_ix_i 0, for all i and all points in the feasible experimental region. The nature of the effect is discussed in detail for the cases of one and two parameters. It is argued that in practical biomedical situations the optimal design is almost always the same as for uncensored data.     

A note on the mean past and the mean residual life of a (n − k + 1)-out-of-n system under multi monitoring

In the study of the reliability of technical systems, k-out-of-n systems play an important role. In the present paper, we consider a (n − k + 1)-out-of-n system consisting of n identical components such that the lifetimes of components are independent and have a common distribution function F. It is assumed that the number of monitoring is l and the total number of failures of the components at time t _i is m _i , i = 1, . . . , l − 1. Also at time t _l (t ₁ < . . . < t _l ) the system have failed or the system is still working. Under these conditions, the mean past lifetime, the mean residual lifetime of system and their properties are investigated.     

Why is one choice different?

Let X_i be nonnegative independent random variables with finite expectations and . The value is what can be obtained by a “prophet”. A “mortal” on the other hand, may use k1 stopping rules t₁,…,t_k yielding a return E[max_i=1,…,kX_ti]. For nk the optimal return is where the supremum is over all stopping rules which stop by time n. The well known “prophet inequality” states that for all such X_i's and one choice and the constant “2” cannot be improved on for any n2. In contrast we show that for k=2 the best constant d satisfying for all such X_i's depends on n. On the way we obtain constants c_k such that .