Sequential Classification Based on Regression

Let us consider m general populations π₁, …,π_m. Each object belonging to these populations is represented by (p ± 1) characteristics x₁, x₂,…,x_p,y. A certain object, which is an element of one of the m general populations π₁,…,π_m has to be classified into the correct population. It will be assumed that knowledge of the value of the characteristic y would permit its correct classification, but that the observation of this characteristic is expensive, difficult or dangerous, as e.g. in medical applications. y is correlated with a set of p characteristics x₁,x₂,…,x_p, which are observed sequentially. The classification procedure is based on the division of the space of the observed value of characteristics x₁,x₂,…,x_p into nonintersecting areas determined so as to minimize the value of BAYES' risk given by equation (3).     

The Estimation of a Multivariate Fitness Function from Several Samples Taken from a Population

The situation is considered where the multivariate distribution of certain variables X₁, X₂, …, X_p is changing with time in a population because natural selection related to the X's is taking place. It is assumed that random samples taken from the population at times t₁, t₂, …, t_s are available and it is desirable to estimate the fitness function w_t(x₁, x₂,…,x_p) which shows how the number of individuals with X_i = x_i, i = 1, 2, …, p at time t is related to the number of individuals with the same X values at time zero. Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating w_t. The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case there seems to have been very little selection on the population dispersion but considerable selection on means.     

A Further Problem of Interval Estimation of an Unknown Regressor in Model I of Linear Regression

For the model y = α + βx + ? (model I) of linear regression we dealt with in KUHNERT and HORN (1980) the determination of a confidence interval for that x₀ where the expectation Ey reaches a given value y₀. Here we start with realizations of random variables y (i = 1,…, m) being independent of x which are given in addition to the realizations of-y. Now y₀ denotes the unknown value of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \sum \limits_{i = 1}^m $\end{document} ciEy and x₀ the x-value where the expectation Ey reaches that value y₀. For this x₀ we give a confidence interval. Applications stem from dose response assays.     

An Unbalanced Nested Model with Random Effects Having Unequal Variances

Consider the model Y_ijk=μ + a_i + b_ij + e_ijk (i=1, 2,…, t; j=1,2,…, B_i; k=1,2…,n_ij), where μ is a constant and a¹,b_ij and e_ijk are distributed independently and normally with zero means and variances σ²_ad_ij and σ², respectively, where it is assumed that the d_i's and d_ij's are known. In this paper procedures for estimating the variance components (σ², σ²_a and σ²_b) and for testing the hypothesis σ²_b = 0 and σ²_a = 0 are presented. In the last section the mixed model y_ijk, where x_ijkkm are known constants and β_m's are unknown fixed effects (m = 1, 2,…,p), is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the β_m's.     

An Unbalanced Two-way Model With Random Effects Having Unequal Variances

Consider the model y_ijk=u ± a_i ± b_i ± c_ij ± e_ijk i=1, 2,…, t; j=1, 2,…b; k=1, 2,…,n_ij where μ is a constant and a_i, b_i, c_ij are distributed independently and normally with zero means and variances Δ₂ Δ2/bd_ij and δ₂ respectively. It is assumed that d_i's, and d_ij's are known (positive) constants (for all i and j). In this paper procedures for estimating the variance components (Δ₂, Δ²_b and Δ²_a) and for testing the hypothesis H_oc:Δ²_c/Δ² = y₃ and H_oa:Δ²_b/Δ² = y₄ (where y₂, y₃, and y₄, are specified constants) are presented. A generalization for the mixed model case is discussed in the last section.     

Robust One-Way ANOVA under Possibly Non-Regular Conditions

Consider the one-way ANOVA problem of comparing the means m₁, m₂, …, m_c of c distributions F₁(x) = F(x — m₁), …, F_c(x) = F(x — m_c). Solutions are available based on (i) normal-theory procedures, (ii) linear rank statistics and (iii) M-estimators. The above model presupposes that F₁, F₂, …, F_c have equal variances (= homoscedasticity). However practising statisticans content that homoscedasticity is often violated in practice. Hence a more realistic problem to consider is F₁(x) = F((x — m₁)/σ₁), …, F_c(x) = F((x — m_c)/σ_c), where F is symmetric about the origin and σ₁, …, σ_c are unknown and possibly unequal (= heteroscedasticity). Now we have to compare m₁, m₂, …, m_c. At present, nonparametric tests of the equality of m₁, m₂, …, m_c are available. However, simultaneous tests for paired comparisons and contrasts and do not seem to be available. This paper begins by proposing a solution applicable to both the homoscedastic and the heteroscedastic situations, assuming F to be symmetric. Then the assumptions of symmetry and the identical shapes of F₁, …, F_c are progressively relaxed and solutions are proposed for these cases as well. The procedures are all based on either the 15% trimmed means or the sample medians, whose quantiles are estimated by means of the bootstrap. Monte Carlo studies show that these procedures tend to be superior to the Wilcoxon procedure and Dunnett's normal theory procedure. A rigorous justification of the bootstrap is also presented. The methodology is illustrated by a comparison of mean effects of cocaine administration in pregnant female Sprague-Dawley rats, where skewness and heteroscedascity are known to be present.     

Remarks upon Empirical Regression Belt

In the presented paper the method of the empirical regression belt is demonstrated. An empirical regression curve r(x), which is determined by the realizations (measured points) (x₁, y₁), i = 1,…., n of a continuous two-dimensional random variable (X, Y), is enclosed by a belt, the local width of which varies dependent on local frequency and variance of the measured points. This empirical regression belt yields certain information for evaluating the empirical regression curve, providing a useful basis for the biomathematical forming of a model. By giving three examples derived from morphometrics the authors discuss important qualities of the empirical regression belt.     

A confidence interval for the unknown value of the regressor with given expectation of the regressand in model I of linear regression

In the case of model I of linear regression there is derived a confidence interval for that x_o where the “true line” will reach a given value y_o. The interval can be given by the intersections between the line y = y_o and the hyperbolas providing pointwise confidence intervals of the expectations of y.     

Multivariate Probability in Terms of Marginal Probability and Correlation Coefficient

This paper is motivated by a practical problem relating to student performance in a number of subjects of equal standing. Its mathematical formulation is to find an approximation to a multivariate probability of the form Pr {X₁⩾a, X₂⩾a, …, X_N⩾a} for arbitrary a and N, in terms of p = Pr {X₁⩾a} and q = Corr (X_i, X_j), i≠j, where X_i, i = 1, …, N are exchangeable random variables with mean 0 and variance unity.     

Estimation and Verification of Hypotheses in Some Zyskind-Martin Models with Missing Values

Several theorems on estimation and verification of linear hypotheses in some Zyskind-Martin (ZM) models are given. The assumptions are as follows. Let y = Xβ + e or (y, Xβ, σ²V) be a fixed model where y is a vector of n observations, X is a known matrix nXp with rank r(X) = r ≦ p < n, where p is a number of coordinates of the unknown parameter vector β, e is a random vector of errors with covariance matrix σ²V, where σ² is unknown scalar parameter, V is a known non-negative definite matrix such that R(X) ? R(V). Symbol R(A) denotes a vector space generated by columns of matrix A. The expected value of y is Xβ. In this paper four following Zyskind-Martin (ZM) models are considered: ZMd, ZMa, ZMc and ZMqd (definitions in sec. 1) when vector y y₁ y₂ involves a vector y₁ of m missing values and a vector y₂ with (n — m) observed values. A special transformation of ZM model gives again ZM model (cf. theorem 2.1). Ten properties of actual (ZMa) and complete (ZMc) Zyskind-Martin models with missing values (cf. theorem 2.2) test functions F are given in (2.11)) are presented. The third propriety constitutes a generalization of R. A. Fisher's rule from standard model (y, Xβ, σ²I) to ZM model. Estimation of vector y₁ (cf. 3.3) of vector β (cf. th. 3.2) and of scalar σ² (cf. th. 3.4) in actual ZMa model and in diagonal quasi-ZM model (ZMqd) are presented. Relation between y? ₁ and β is given in theorem 3.1. The results of section 2 are illustrated by numerical example in section 4.     

A Semiparametric Estimator of the Crossing Point in the Two‐Sample Linear Shift Function: Application to Crossing Lifetime Distributions

Let X and Y be two random variables with continuous distribution functions F and G. Consider two independent observations X₁, … , X_m from F and Y₁, … , Y_n from G. Moreover, suppose there exists a unique x* such that F(x) > G(x) for x < x* and F(x) < G(x) for x > x* or vice versa. A semiparametric model with a linear shift function (Doksum, 1974) that is equivalent to a location‐scale model (Hsieh, 1995) will be assumed and an empirical process approach (Hsieh, 1995) is used to estimate the parameters of the shift function. Then, the estimated shift function is set to zero, and the solution is defined to be an estimate of the crossing‐point x*. An approximate confidence band of the linear shift function at the crossing‐point x* is also presented, which is inverted to yield an approximate confidence interval for the crossing‐point. Finally, the lifetime of guinea pigs in days observed in a treatment‐control experiment in Bjerkedal (1960) is used to demonstrate our procedure for estimating the crossing‐point. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Confidence Intervals on Ratios of Variance Components for the Unbalanced Two Factor Nested Model

The model considered in this article is the two-factor nested unbalanced variance component model: for p = 1, 2, …, P; q = 1, 2, …, Q_p; and r = 1, 2, …, R_pq. The random variables Y_pqr are observable. The constant μ is an unknown parameter, and A_p, B_pq and C_pqr are (unobservable) normal and independently distributed random variables with zero means and finite variances σ²_A, σ²_B, and σ²_C, respectively. Approximate confidence intervals on ?_A and ?_B using unweighted means are derived, where The performance of these approximate confidence intervals are evaluated using computer simulation. The simulated results indicate that these proposed confidence intervals perform satisfactorily and can be used in applied problems.     

No-escape regions and oscillations in second order predator-prey recurrences

We are concerned with the second order recurrence x _n+1 = x _n f(x _n, y _n), y _n+1 = y _n g(x _n, y _n), where n N ₀, x ₀ > 0, y ₀ > 0, and the reproduction rates f and g simulate predator-prey interaction. Under conditions on the sign of f – 1 and g–1 we show the existence of a nontrivial no-escape region D, i.e. a compact set D {(x, y): x > 0, y > 0} which is invariant under the recurrence and has the property that every sequence enters D after finitely many steps. Under further conditions on the shape of the isoclines {f = 1} and {g = 1} and on the stationary points {f = 1} {g = 1} we are able to show the existence of sustained oscillations.This work has been supported by the Deutsche Forschungsgemeinschaft     

Estimation and Prediction of Generalized Growth Curve with Grouping Variances in AR(q) Dependence Structure

In this paper we consider maximum likelihood analysis of generalized growth curve model with the Box‐Cox transformation when the covariance matrix has AR(q) dependence structure with grouping variances. The covariance matrix under consideration is Σ = D _σ CD _σ where C is the correlation matrix with stationary autoregression process of order q, q < p and D _σ is a diagonal matrix with p elements divided into g(≤ p) groups, i.e., D _σ is a function of {σ₁, …, σ_g} and – 1 < ρ < 1 and σ_l, l = 1, …, g, are unknown. We consider both parameter estimation and prediction of future values. Results are illustrated with real and simulated data.     

The Point Evaluation Method for Approximating Function Means,Variances and Covariances

A perennial problem in statistics is the determination of biases, variances and covariances for functions of random variables X₁, X₂, …, X_n which themselves have a known distribution. A common approach is through equations based upon Taylor series approximations but a “point evaluation” method may sometimes be a useful alternative. This involves approximating the multivariate distribution of the X variables by the 2ⁿ points given by X₁=μ₁±₁, X₂ = μ₂ ±₂, …, X_n = = μ_n μ_n, where μ_i is the mean and σ_i the standard deviation of Xi, with appropriate point weights. An advantage over the Taylor series approach is that function derivatives do not have to be explicitely calculated. The point evaluation method is particularly useful in cases where the X variables are uncorrelated. Then the evaluation of the 2ⁿ points can be replaced by the evaluation of 2n points. The point evaluation method is illustrated with powers of a normally distributed variable, and with estimation of gene frequencies from ABO blood group frequencies.     

秃病蚤蒙冀亚种与长爪沙鼠密度的状态空间模型

 给出了一个描述秃病蚤蒙冀亚种Nosopsyllus laeviceps kuzenkovi与长爪沙鼠M eriones unguiculatus密度的状态空间模型,模型为x_t+1=-0.336x_t+0057y _t+0107z_t, y_t+1=0.586x_t-0369y_t-0274z_t, z_t+1=0.189x _t+0247y_t-0309z_t, 其中x_t,y_t和z_t分别为t月份的沙鼠密度, 巢秃病蚤 和体秃病蚤指数。结果表明,模型能够较好地描述野外秃病蚤蒙 冀亚种与长爪沙鼠间的关系,并发现沙鼠密度显著地影响巢秃病蚤指数。     

Optimal Fixing of the Bandwidth-Parameter for the Empirical Regression1,2

The estimator ?₀(x) of the regression r(x) = E (Y | × = x) from measured points (x_i, y_i), i = 1(1) n, of a continuous two-dimensional random variable (X, Y) with unknown continuous density function f(x, y) and with moments up to the second order can be made with the help of a density estimation f?₀(x, y) (see e.g. SCHMERLING and PEIL, 1980). Here f?₀(x, y) still contains free parameters (so-called band-width-parameters), the values of which have to be optimally fixed in the concrete case. This fixing can be done by using a modification of the maximum-likelihood principle including jackknife techniques. The parameter values can be also found from the estimators for r(x). Here the cross-validation principle can be applied. Some numerical aspects of these possibilities for optimally fixing the bandwidth-parameter are discussed by means of examples. If ?₀(x) is used as a smoothing operator for time series the optimal choice of the parameter values is dependent on the purpose of application of the smoothed time series. The fixing will then be done by considering the so-called filter-characteristic of ?C₀(x).     

A Note on the Mann-Whitney Statistic for Lehmann Alternatives

For the case of the LEHMANN alternatives the paper presents some new facts on the MANN -WHITNEY statistic and, in particular, its variance V(p, m, n), where p = P(x_i<y_i). Explicit formulas for U and V are used to prove, among other things, the following propositions: For any m, n, V is a one-hump function of p, and the hump always lies in the interval (1/2(3 - √5), 1/2(√5 - 1)). If no restrictions are imposed on p the boundaries of this interval are sharp. Given s = m + n, V(1/2, s/2,s/2) is maximal among all values V(p, m, n). The formulas allow, moreover, the improvement of the known bounds for the variance of p? = U/mn.     

Data for estimating eaten prey masses from Eurasian lynx<Emphasis Type="Italic">Lynx Lynx</Emphasis> scats in Central and East Europe

To derive a model which allows estimating eaten prey masses from lynxLynx lynx Linnaeus, 1758 scats, we fed 3 roe deerCapreolus capreolus, 2 wild boarsSus scrofa, 1 fallow deerDama dama, 1 mouflonOvis ammon musimon, 1 European hareLepus europaeus and 1 daily diet of miceMus musculus to two adult lynx. The percentage of prey use decreased with an increase in the offered body mass of the prey individual. Conversion factors from dry matter scat mass to fresh matter mass of eaten prey (y) increased linearly as the eaten mass of the prey individual (x) rose:y = 15.06 + 1.330x,R ² = 0.643,F _1,7 = 12.6,p < 0.01. For estimating eaten prey masses from lynx scats we recommend to use (1) this equation as well as (2) prey type-specific conversion factors and (3) prey type-specific quotients of the eaten prey mass per scat.