Designs for approximately linear regression: Maximizing the minimum coverage probability of confidence ellipsoids

The classical D-optimality principle in regression design may be motivated by a desire to maximize the coverage probability of a fixed-volume confidence ellipsoid on the regression parameters. When the fitted model is exactly correct, this amounts to minimizing the determinant of the covariance matrix of the estimators. We consider an analogue of this problem, under the approximately linear model E[y|x] = θ^Tz(x) + f(x). The nonlinear disturbance f(x) is essentially unknown, and the experimenter fits only to the linear part of the response. The resulting bias affects the coverage probability of the confidence ellipsoid on θ. We study the construction of designs which maximize the minimum coverage probability as f varies over a certain class. Explicit designs are given in the case that the fitted response surface is a plane.     

On Non-Normal Invariance Principles for Multi-Response Permutation Procedures

A non-normal invariance principle is established for a restricted class of univariate multi-response permutation procedures whose distance measure is the square of Euclidean distance. For observations from a distribution with finite second moment, the test statistic is found asymptotically to have a centered chi-squared distribution. Spectral expansions are used to determine the asymptotic distribution for more general distance measures d, and it is shown that if d(x, y) = |x — y|^u, u? 2, the asymptotic distribution is not invariant (i.e. it is dependent on the distribution of the observations).     

Rejoinder to 'Ahmed, M.S. (1998). A note on regression-type estimators using multiple auxiliary information.'

In the estimators t ₃ , t ₄ , t ₅ of Mukerjee, Rao & Vijayan (1987), b _{y x} and b _{y z} are partial regression coefficients of y on x and z , respectively, based on the smaller sample. With the above interpretation of b _{y x} and b _{y z} in t ₃ , t ₄ , t ₅ , all the calculations in Mukerjee at al. (1987) are correct. In this connection, we also wish to make it explicit that b _{x z} in t ₅ is an ordinary and not a partial regression coefficient. The 'corrected' MSEs of t ₃ , t ₄ , t ₅ , as given in Ahmed (1998 Section 3) are computed assuming that our b _{y x} and b _{y z} are ordinary and not partial regression coefficients. Indeed, we had no intention of giving estimators using the corresponding ordinary regression coefficients which would lead to estimators inferior to those given by Kiregyera (1984). We accept responsibility for any notational confusion created by us and express regret to readers who have been confused by our notation. Finally, in consideration of the above, it may be noted that Tripathi & Ahmed's (1995) estimator t ₀ , quoted also in Ahmed (1998), is no better than t ₅ of Mukerjee at al. (1987).     

Estimation and testing stationarity for double-autoregressive models

Summary.  The paper considers the double-autoregressive model y _t  =  φ y _{t −1}+ ɛ _t with ɛ _t  =     . Consistency and asymptotic normality of the estimated parameters are proved under the condition E  ln | φ  +√ α η _t |<0, which includes the cases with | φ |=1 or | φ |>1 as well as     . It is well known that all kinds of estimators of φ in these cases are not normal when ɛ _t are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given.     

Estimating Mixing Densities in Exponential Family Models for Discrete Variables

This paper is concerned with estimating a mixing density g using a random sample from the mixture distribution f(x)=∫f x | θ)g(θ)dθ where f(· | θ) is a known discrete exponen tial family of density functions. Recently two techniques for estimating g have been proposed. The first uses Fourier analysis and the method of kernels and the second uses orthogonal polynomials. It is known that the first technique is capable of yielding estimators that achieve (or almost achieve) the minimax convergence rate. We show that this is true for the technique based on orthogonal polynomials as well. The practical implementation of these estimators is also addressed. Computer experiments indicate that the kernel estimators give somewhat disappoint ing finite sample results. However, the orthogonal polynomial estimators appear to do much better. To improve on the finite sample performance of the orthogonal polynomial estimators, a way of estimating the optimal truncation parameter is proposed. The resultant estimators retain the convergence rates of the previous estimators and a Monte Carlo finite sample study reveals that they perform well relative to the ones based on the optimal truncation parameter.     

CERTAIN BOUNDS CONCERNING CHARACTERISTIC FUNCTIONS1

The largest value of the constant c for which holds over the class of random variables X with non-zero mean and finite second moment, is c=π. Let the random variable (r.v.) X with distribution function F(·) have non-zero mean and finite second moment. In studying a certain random walk problem (Daley, 1976) we sought a bound on the characteristic function of the form for some positive constant c. Of course the inequality is non-trivial only provided that . This note establishes that the best possible constant c =π. The wider relevance of the result is we believe that it underlines the use of trigonometric inequalities in bounding the (modulus of a) c.f. (see e.g. the truncation inequalities in §12.4 of Loève (1963)). In the present case the bound thus obtained is the best possible bound, and is better than the bound (2) |1-?(θ)| ≥ |θEX|-θ²EX²\2 obtained by applying the triangular inequality to the relation which follows from a two-fold integration by parts in the defining equation (*). The treatment of the counter-example furnished below may also be of interest. To prove (1) with c=π, recall that sin u > u(1-u/π) (all real u), so Since |E sinθX|-|E sin(-θX)|, the modulus sign required in (1) can be inserted into (4). Observe that since sin u > u for u < 0, it is possible to strengthen (4) to (denoting max(0,x) by x₊) To show that c=π is the best possible constant in (1), assume without loss of generality that EX > 0, and take θ > 0. Then (1) is equivalent to (6) c < θEX²/{EX-|1-?(θ)|/θ} for all θ > 0 and all r.v.s. X with EX > 0 and EX². Consider the r.v. where 0 < x < 1 and 0 < γ < ∞. Then EX=1, EX²=1+γx², From (4) it follows that |1-?(θ)| > 0 for 0 < |θ| <π|EX|/EX² but in fact this positivity holds for 0 < |θ| < 2π|EX|/EX² because by trigonometry and the Cauchy-Schwartz inequality, |1-?(θ)| > |Re(1-?(θ))| = |E(1-cosθX)| = 2|E sin²θX/2| (10) >2(E sinθX/2)² (11) >(|θEX|-θ²EX²/2π)²/2 > 0, the inequality at (11) holding provided that |θEX|-θ²EX²/2π > 0, i.e., that 0 < |θ| < 2π|EX|/EX². The random variable X at (7) with x= 1 shows that the range of positivity of |1-?(θ)| cannot in general be extended. If X is a non-negative r.v. with finite positive mean, then the identity shows that (1-?(θ))/iθEX is the c.f. of a non-negative random variable, and hence (13) |1-?(θ)| < |θEX| (all θ). This argument fans if pr{X < 0}pr{X> 0} > 0, but as a sharper alternative to (14) |1-?(θ)| < |θE|X||, we note (cf. (2) and (3)) first that (15) |1-?(θ)| < |θEX| +θ²EX²/2. For a bound that is more precise for |θ| close to 0, |1-?(θ)|²= (Re(1-?(θ)))²+ (Im?(θ))² <(θ²EX²/2)²+(|θEX| +θ²EX²_-/π)², so (16) |1-?(θ)| <(|θEX| +θ²EX²_-/π) + |θ|³(EX²)²/8|EX|.     

Improved minimax estimation of a normal precision matrix

Let S_{p × p} have a Wishart distribution with parameter matrix Σ and n degrees of freedom. We consider here the problem of estimating the precision matrix Σ^?1 under the loss functions L₁(σ) tr (σ) - log |σ| and L₂(σ) = tr (σ). James-Stein-type estimators have been derived for an arbitrary p. We also obtain an orthogonal invariant and a diagonal invariant minimax estimator under both loss functions. A Monte-Carlo simulation study indicates that the risk improvement of the orthogonal invariant estimators over the James-Stein type estimators, the Haff (1979) estimator, and the “testimator” given by Sinha and Ghosh (1987) is substantial.     

Admissible linear estimators of the multivariate normal mean without extra information

Suppose y is normally distributed with mean IRⁿ and covariance σ²V, where σ²>0 and V>0 is known. The n. s. conditions that a linear estimator Ay+a of μ be admissible in the class of all estimators of μ which depend only on y are derived. In particular, the usual estimator δ₀(y)=y is admissible in this class. The results are applied to the normal linear model and the admissibilities of many well-known linear estimators are demonstrated.     

Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

Empirical bayes estimation of functionals of unknown probability measures

The empirical Dayes approach to one and two sal-npie problcrns has beeir considered by Korwar and Hollander (1976), Holiander and Korwar (1976) and Phadia and Susarla (1979). In this article we essen- tially generalize their empirical Bayes results by replacing the inlicaro-functions of. the sets (?∞,x) and {X≦Y} by arbitrary mea5, irable functions h(x) and h(x,y). More speclfically, the ernpiricaion yes estimation of esrimabie paramerers of degree one ani KG,I;ti kliown probability measure Pon (R,R) is considered. The asymptotic optimality of the these estimators, obtaining the exact risk expressions, is established. Also the results of Dalal and Phad (1983) we extended to the estimation of an estimable parametric function of an unknow probability measure P on (R² , B²)     

Nonparametric empirical bayes estimation of certain funtionals

For estimating functionals of the form ∫∫φ(x,y)dF(x) dF(y), nonparametric empirical Bayes estimators are developed which are competitors of the classical U-statistics. Asymptotic optimality of the proposed estimators is proved     

Conditional inference in finite population sampling under a calibration model

Several estimators, including the classical and the regression estimators of finite population mean, are compared, both theoretically and empirically, under a calibration model, where the dependent variable(y), and not the independent variable(x), can be observed for all units of the finite population. It is shown asymptotically that when conditioned on x, the bias of the classical estimator may be much smaller than that of the regression estimators; whereas when conditioned on y, the regression estimator may have much smaller conditional bias than the classical estimator. Since all the y's(not x's) can be observed, it seems appropriate to make comparison under the conditional distribution of each estimator with y fixed. In this case, the regression estimator has smaller variance, smaller conditional bias, and the conditional coverage probability closer to its nominal level     

A family of shrinkage estimators for the square of mean in normal distribution

This paper is devoted to the problem of estimating the square of population mean (μ²) in normal distribution when a prior estimate or guessed value σ₀ ² of the population variance σ² is available. We have suggested a family of shrinkage estimators , say, for μ² with its mean squared error formula. A condition is obtained in which the suggested estimator is more efficient than Srivastava et al’s (1980) estimator T_min. Numerical illustrations have been carried out to demonstrate the merits of the constructed estimator over T_min. It is observed that some of these estimators offer improvements over T_min particularly when the population is heterogeneous and σ² is in the vicinity of σ₀ ².     

Testing time series linearity via goodness-of-fit methods

We consider the problem of testing time series linearity. Existing time domain and spectral domain tests are discussed. A new approach relying on spectral domain properties of a time series under the null hypothesis of linearity is suggested. Under linearity, the normalized bispectral density function Z is a constant. Under the null hypothesis of linearity, properly constructed estimators of 2|Z|² have a non-central chi-squared distribution with two degrees of freedom and constant non-centrality parameter 2|Z|². If the null hypothesis is false, the non-centrality parameter is non-constant. This suggests goodness-of-fit tests might be effective in diagnosing non-linearity. Several approaches are introduced.     

Unbiased L1 and L∞ estimation

Sielken and Heartely 1973 have shown that the L₁ and L_∞ estimation problems may be formulated in such a way as to yield unbiased estimators of in the standard linear model y = Xβ + ε In this paper we will show that the L₁ estimation problem is closely related to the dual of the L_∞ estimation problem and vice versa. We will use this resu;t to obtain four fistiner lineat programming problems which yield unbiased L₁ and L_∞ estimators of β.     

Some unbiased estimators versus mean per unit and ratio estimators in finite population sample surveys

We present some unbiased estimators at the population mean in a finite population sample surveys with simple random sampling design where information on an auxiliary variance x positively correlated with the main variate y is available. Exact variance and unbiased estimate of the variance are computed for any sample size. These estimators are compared for their precision with the mean per unit and the ratio estimators. Modifications of the estimators are suggested to make them more precise than the mean per unit estimator or the ratio estimator regardless of the value of the population correlation coefficient between the variates x and y. Asymptotic distribution of our estimators and confidnece intervals for the population mean are also obtained.     

Estimators of Certain Functions of the Variance of a Normal Distribution

Estimators of σ^aand log σ which are functions of Σ(x?x)²/d are considered. Besides the usual sampling theory estimators, Bayesian point estimators which are the usual measures of location of the posterior distribution are given, and in each case an exact or asymptotic expression for the divisor d is stated.     

Nonparametric estimator for mean residual life and vitality function

In this paper, we study the estimation of the vitality function(v(x)=E(X|X>x) and mean residual life function(e(x)=E(X-x|X>x) from a sample ofX using the empirical estimator and kernel estimator. Under suitable conditions of regularity, the asymptotic normality of the kernel estimator is obtained. Partially supported by Consejeria de Cultura y Ed. (C.A.R.M.), under Grant PIB 95/90.     

On The Role of a Certain Eigenvalue in Estimating the Growth Rate of a Branching Process

In many situations, the data given on a p-type Galton-Watson process Z_n eP N^p will consist of the total generation sizes |Z_n| only. In that case, the maximum likelihood estimator ρ_ML of the growth rate ρ is not observable, and the asymptotic properties of the most obvious estimators of ρ based on the |Z_n|, as studied by Asmussen & Keiding (1978), show a crucial dependence on |ρ₁|,ρ₁ being a certain other eigenvalue of the offspring mean matrix. In fact, if |ρ₁|²≤ρ, then the speed of convergence compares badly with ρ_ML. In the present note, it is pointed out that recent results of Heyde (1981) on so-called Fibonacci branching processes provide further examples of this phenomenon, and an estimator with the same speed of convergence as ρ_ML and based on the |Z_n| alone is exhibited for the case p= 2, ρ₁²≥ρ.     

The Statistics of Simulating Chaos

When simulating a dynamical system, the computation is actually of a spatially discretized system, because finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretized simulations often have collapsing effects, to a fixed point or to short cycles. Statistical properties of these phenomena can be modelled with random mappings with an absorbing centre. The model gives results which are very much in line with computational experiments. The effects are discussed with special reference to the family of mappings f (x)=1-|1-2x|,x [0,1],1,<,,<,. Computer experiments show close agreement with predictions of the model.