On moments of the supremum of normed weighted averages

Let {X, X_n; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (a_nk; 1 ≤ k ≤ n, n ≥ 1); a_nk, ? R and sup_{n, k} |a_n,k| < ∞}. Set S_n( A ) = ∑ⁿ_k=1a_{n, k}X_k for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{sup_{n ≥ 1}, |S_n( A )|/n^1/r}^p < ∞ or E{sup_{n ≥ 2} |S_n( A )|/2n log n)^1/2}^p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).     

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.     

Estimation of Conditional Ranks and Tests of Exogeneity in Nonparametric Nonseparable Models

Consider a nonparametric nonseparable regression model Y = ?(Z, U), where ?(Z, U) is strictly increasing in U and U ～ U[0, 1]. We suppose that there exists an instrument W that is independent of U. The observable random variables are Y, Z, and W, all one-dimensional. We construct test statistics for the hypothesis that Z is exogenous, that is, that U is independent of Z. The test statistics are based on the observation that Z is exogenous if and only if V = F_Y|Z(Y|Z) is independent of W, and hence they do not require the estimation of the function ?. The asymptotic properties of the proposed tests are proved, and a bootstrap approximation of the critical values of the tests is shown to be consistent and to work for finite samples via simulations. An empirical example using the U.K. Family Expenditure Survey is also given. As a byproduct of our results we obtain the asymptotic properties of a kernel estimator of the distribution of V, which equals U when Z is exogenous. We show that this estimator converges to the uniform distribution at faster rate than the parametric n^{? 1/2}-rate.     

The linear combination,product and ratio of Laplace random variables

The distributions of linear combinations, products and ratios of random variables arise in many areas of engineering. In this paper, the exact distributions of the linear combination α X+β Y, the product |X Y| and the ratio |X/Y| are derived when X and Y are independent Laplace random variables. The Laplace distribution, being the oldest model for continuous data, has been one of the most popular models for measurement errors in engineering.     

L'effet de l'erreur d'arrondi sur l'estimateur d'une densité par la méthode du noyau

In a model for rounded data suppose that the random sample X₁,.,.,X_n,. i.i.d., is transformed into an observed random sample X_1Δ,.,.,X_nΔ, where X_iΔ = 2vΔ if X_i, ∈ (2vΔ - Δ, 2vΔ + Δ), for i = 1,.,.,n. We show that the precision Δ of the observations has an important effect on the shape of the kernel density estimator, and we identify important points for the graphical display of this estimator. We examine the IMSE criteria to find the optimal window under the rounded-data model.     

The uniform convergence of the nadaraya-watson regression function estimate

If (X₁,Y₁), …, (X_n,Y_n) is a sequence of independent identically distributed R_d × R-valued random vectors then Nadaraya (1964) and Watson (1964) proposed to estimate the regression function m(x) = ? {Y₁|X₁ = x{ by where K is a known density and {h_n} is a sequence of positive numbers satisfying certain properties. In this paper a variety of conditions are given for the strong convergence to 0 of ess_Xsup|m_n (X)-m(X)| (here X is independent of the data and distributed as X₁). The theorems are valid for all distributions of X₁ and for all sequences {h_n} satisfying h_n → 0 and nh/log n→0.     

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|.     

Estimation of the location of a 0-type or ∞-type singularity by Poisson observations

We consider an inhomogeneous Poisson process X on [0, T]. The intensity function of X is supposed to be strictly positive and smooth on [0, T] except at the point θ, in which it has either a 0-type singularity (tends to 0 like |x| ^p , p∈(0, 1)), or an ∞-type singularity (tends to ∞ like |x| ^p , p∈(?1, 0)). We suppose that we know the shape of the intensity function, but not the location of the singularity. We consider the problem of estimation of this location (shift) parameter θ based on n observations of the process X. We study the Bayesian estimators and, in the case p>0, the maximum-likelihood estimator. We show that these estimators are consistent, their rate of convergence is n ^1/(p+1), they have different limit distributions, and the Bayesian estimators are asymptotically efficient.     

Consistent deconvolution in density estimation

Suppose we have n observations from X = Y + Z, where Z is a noise component with known distribution, and Y has an unknown density f. When the characteristic function of Z is nonzero almost everywhere, we show that it is possible to construct a density estimate f_n such that for all f, Iim_n| |=0.     

THE CONVERGENCE RATE OF SEQUENTIAL FIXED-WIDTH CONFIDENCE INTERVALS FOR REGULAR FUNCTIONALS

Let X₁X₂,.be i.i.d. random variables and let U_n= (n r)^-1S?^(n,r) h (X_i1,., X_ir,) be a U-statistic with EU_n= v, v unknown. Assume that g(X₁) =E[h(X₁,.,X_r) - v |X₁]has a strictly positive variance s?². Further, let a be such that φ(a) - φ(-a) =α for fixed α, 0 < α < 1, where φ is the standard normal d.f., and let S²_n be the Jackknife estimator of n Var U_n. Consider the stopping times N(d)= min {n: S²_n: + n^-1≤²a^-2},d > 0, and a confidence interval for v of length 2d,of the form I_n,d= [U_n,-d, Un + d]. We assume that Var U_n is unknown, and hence, no fixed sample size method is available for finding a confidence interval for v of prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, let I_N(d),d be a sequence of sequential confidence intervals for v. The asymptotic consistency of this procedure, i.e. lim_{d → 0}P(v ∈ I_N(d),d)=α follows from Sproule (1969). In this paper, the rate at which |P(v ∈ I_N(d),d) converges to α is investigated. We obtain that |P(v ∈ I_N(d),d) - α| = 0 (d^{1/2-(1+k)/2(1+m)}), d → 0, where K = max {0,4 - m}, under the condition that E|h(X₁, X_r)|^m < ∞m > 2. This improves and extends recent results of Ghosh & DasGupta (1980) and Mukhopadhyay (1981).     

Uniform asymptotic linearity of a process based on a signed rank statistic

Consider the process with, cf. (1.2) on page 265 in B1, X₁, …, X_N a sample from a distribution F and, for i = 1, …, N, R${}_{\text{|x}{}_{\text{1}}\text{, - q}{}_{\text{1}}\text{ø|}}$ , the rank of |X₁ - q₁ø| among |X₁ - q₁ø|, …, |X_N - q_Nø|. It is shown that, under certain regularity conditions on F and on the constants p_i and q_i, T^ø_N(t) is asymptotically approximately a linear function of ø uniformly in t and in ø for |ø| ≤ C. The special case where the p_i and the q_i, are independent of i is considered.     

Extremes of nonexchangeability

Summary For identically distributed random variables X and Y with joint distribution function H, we show that the supremum of |H(x,y)-H(y,x)| is 1/3. Using copulas, we define a measure of nonexchangeability, and study maximally nonexchangeable random variables and copulas. In particular, we show that maximally nonexchangeable random variables are negatively correlated in the sense of Spearman's rho. An erratum to this article is available at .     

Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X₁, …, X_N a random sample from a distribution F, let the process S^δ_N(t) be defined as where K²_N = σ^N_i=1(c_i ? c?)² and R x_i, + Δd, is the rank of X_i + Δd_i, among X₁ + Δd₁, …, X_N + Δd_N. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c_i and d_i, S^Δ_N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.     

A Bayesian analysis for the Wilcoxon signed-rank statistic

A Bayesian analysis is provided for the Wilcoxon signed-rank statistic (T⁺). The Bayesian analysis is based on a sign-bias parameter φ on the (0, 1) interval. For the case of a uniform prior probability distribution for φ and for small sample sizes (i.e., 6 ? n ? 25), values for the statistic T⁺ are computed that enable probabilistic statements about φ. For larger sample sizes, approximations are provided for the asymptotic likelihood function P(T⁺|φ) as well as for the posterior distribution P(φ|T⁺). Power analyses are examined both for properly specified Gaussian sampling and for misspecified non Gaussian models. The new Bayesian metric has high power efficiency in the range of 0.9–1 relative to a standard t test when there is Gaussian sampling. But if the sampling is from an unknown and misspecified distribution, then the new statistic still has high power; in some cases, the power can be higher than the t test (especially for probability mixtures and heavy-tailed distributions). The new Bayesian analysis is thus a useful and robust method for applications where the usual parametric assumptions are questionable. These properties further enable a way to do a generic Bayesian analysis for many non Gaussian distributions that currently lack a formal Bayesian model.     

Limiting behavior of randomly weighted averages of symmetric heavy-tailed random variables

In this paper we consider a sequence of independent continuous symmetric random variables X₁, X₂, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages S_n = R⁽ⁿ⁾₁X₁ + ??? + R⁽ⁿ⁾_nX_n, where the random weights R⁽ⁿ⁾₁, …, R_n⁽ⁿ⁾ which are independent of X₁, X₂, …, X_n, are the cuts of (0, 1) by the n ? 1 order statistics from a uniform distribution. Indeed we prove that c_nS_n converges in distribution to a symmetric α-stable random variable with c_n = n^{1 ? 1/α}/Γ^1/α(α + 1).     

Some Estimators of Finite Population Variance of Stratified Sample Mean

We consider a continuous-time branching random walk on Z ^d , where the particles are born and die at a single lattice point (the source of branching). The underlying random walk is assumed to be symmetric. Moreover, corresponding transition rates of the random walk have heavy tails. As a result, the variance of the jumps is infinite, and a random walk may be transient even on low-dimensional lattices (d = 1, 2). Conditions of transience for a random walk on Z ^d and limit theorems for the numbers of particles both at an arbitrary point of the lattice and on the entire lattice are obtained.     

SOME LAW OF THE ITERATED LOGARITHM TYPE RESULTS FOR THE EMPIRICAL PROCESS

We consider Z^±_n= sup_{0< t ≤ 1/2}2 U^±_n (t)/(t(1- t))^1/2, where + and -denote the positive and negative parts respectively of the sample paths of the empirical process U_n. U^±_n and U_n are seen to behave rather differently, which is tied to the asymmetry of the binomial distribution, or to the asymmetry of the distribution of small order statistics. Csáki (1975) showed that log Z^±_n/log₂n is the appropriate normalization for a law of the iterated logarithm (LIL) for Z^±_n we show that Z^-_n/(2 log₂n)^1/2 is the appropriate normalization for Z^-_n. Csörgö & Révész (1975) posed the question: if we replace the sup over (0,1/2) above, by -the sup over [a_n, 1-a_n] where a_n→0, how fast can a_n→0 and still have |Z_n|/(2 log₂n)^1/2 maintain a finite lim sup a.s.? This question is answered herein. The techniques developed are then used in Section 4 to give an interesting new proof of the upper class half of a result of Chung (1949) for |U_n(t)|. The proofs draw heavily on James (1975); two basic inequalities of that paper are strengthened to their potential, and are felt to be of independent interest.     

Wavelet Estimation of a Density in a GARCH-type Model

We consider the GARCH-type model: S = σ² Z, where σ² and Z are independent random variables. The density of σ² is unknown whereas the one of Z is known. We want to estimate the density of σ² from n observations of S under some dependence assumption (the exponentially strongly mixing dependence). Adopting the wavelet methodology, we construct a nonadaptive estimator based on projections and an adaptive estimator based on the hard thresholding rule. Taking the mean integrated squared error over Besov balls, we prove that the adaptive one attains a sharp rate of convergence.     

Kshirsagar-Type Lower Bounds for Mean Squared Error of Prediction

Let Y be an observable random vector and Z be an unobserved random variable with joint density f(y, z | θ), where θ is an unknown parameter vector. Considering the problem of predicting Z based on Y, we derive Kshirsagar type lower bounds for the mean squared error of any predictor of Z. These bounds do not require the regularity conditions of Bhattacharyya bounds and hence are more widely applicable. Moreover, the new bounds are shown to be sharper than the corresponding Bhattacharyya bounds. The conditions for attaining the new lower bounds are useful for easy derivation of best unbiased predictors, which we illustrate with some examples.