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.     

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.     

Asymptotic properties of Bayes risk for one-sided tests

Consider a given sequence {T_n} of estimators for a real-valued parameter θ. This paper studies asymptotic properties of restricted Bayes tests of the following form: reject H₀:θ ≤ θ₀ in favour of the alternative θ > θ₀ if T_n ≤ C_n, where the critical point C_n is determined to minimize among all tests of this form the expected probability of error with respect to the prior distribution. Such tests may or may not be fully Bayes tests, and so are called T_n-Bayes. Under fairly broad conditions it is shown that and the T_n-Bayes risk where a_n is the order of the standard error of T_n, - is the prior density, and μ is the median of F, the limit distribution of (T_n – θ)/a_nb(θ). Several examples are given.     

On multivariate variable-kernel density estimates for time series

Let X₁ be a strictly stationary multiple time series with values in R^d and with a common density f. Let X₁,.,.,X_n, be n consecutive observations of X₁. Let k = k_n, be a sequence of positive integers, and let H_ni be the distance from X_i to its kth nearest neighbour among X_j, j i. The multivariate variable-kernel estimate f_n, of f is defined by where K is a given density. The complete convergence of f_n, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.     

Estimation After Selection Under Reflected Normal Loss Function

Let X ₁ and X ₂ be two independent random variables from normal populations Π₁, Π₂ with different unknown location parameters θ₁ and θ₂, respectively and common known scale parameter σ. Let X ₍₂₎ = max (X ₁, X ₂) and X ₍₁₎ = min (X ₁, X ₂). We consider the problem of estimating the location parameter θ _M (or θ _J ) of the selected population under the reflected normal loss function. We obtain minimax estimators of θ _M and θ _J . Also, we provide sufficient conditions for the inadmissibility of invariant estimators of θ _M and θ _J .     

A note on some negative dependence notions

A random vector X = (X ₁,…,X _n ) is negatively associated if and only if for every pair of partitions X ₁ = (X _π(1),…,X _π(k)), X ₂ = (X _π(k+1),…,X _π(n)) of X , P( X ₁ ? A, X ₂ ? B) ≤ P( X ₁ ? A)P( X ₂ ? B) whenever A and B are open upper sets and π is any permutation of {1,…,n}. In this paper, we develop some of concepts of negative dependence, which are weaker than negative association but stronger than negative orthant dependence by requiring the above inequality to hold only for some upper sets A and B and applying the arguments in Shaked.     

Estimation of the Scale Parameter of the Selected Gamma Population Under the Entropy Loss Function

Let X ₁, X ₂,…, X _k be k (≥2) independent random variables from gamma populations Π₁, Π₂,…, Π _k with common known shape parameter α and unknown scale parameter θ _i , i = 1,2,…,k, respectively. Let X _(i) denotes the ith order statistics of X ₁,X ₂,…,X _k . Suppose the population corresponding to largest X _(k) (or the smallest X ₍₁₎) observation is selected. We consider the problem of estimating the scale parameter θ _M (or θ _J ) of the selected population under the entropy loss function. For k ≥ 2, we obtain the Unique Minimum Risk Unbiased (UMRU) estimator of θ _M (and θ _J ). For k = 2, we derive the class of all linear admissible estimators of the form cX ₍₂₎ (and cX ₍₁₎) and show that the UMRU estimator of θ _M is inadmissible. The results are extended to some subclass of exponential family.     

Asymptotic results for empirical and partial-sum processes: A review

Two processes of importance in statistics and probability are the empirical and partial-sum processes. Based on d-dimensional data X₁, … X_a the empirical measure is defined for any A ⊂ R^d by the sample proportion of observations in A. When normalized, F_n yields the empirical process W_n: = n^1/2 (F_n - F), where F denotes the “true” probability measure. To define partial-sum processes, one needs data that are assigned to specified locations (in contrast to the above, where specified unit masses are assigned to random locations). A suitable context for many applications is that of data attached to points of a lattice, say {X_j:j ϵ J^d} where J = {1, 2,…}, for which the partial sums are defined for any A ⊂ R^d by Thus S(A) is the sum of the data contained in A. When normalized, S yields the partial-sum process. This paper provides an overview of asymptotic results for empirical and partial-sum processes, including strong laws and central limit theorems, together with some indications of their inferential implications.     

Correlation estimation in Downton's bivariate exponential distribution using incomplete samples

Suppose (X, Y) has a Downton's bivariate exponential distribution with correlation ρ. For a random sample of size n from (X, Y), let X _r:n be the rth X-order statistic and Y _[r:n] be its concomitant. We investigate estimators of ρ when all the parameters are unknown and the available data is an incomplete bivariate sample made up of (i) all the Y-values and the ranks of associated X-values, i.e. (i, Y _[i:n]), 1≤i≤n, and (ii) a Type II right-censored bivariate sample consisting of (X _i:n , Y _[i:n]), 1≤i≤r<n. In both setups, we use simulation to examine the bias and mean square errors of several estimators of ρ and obtain their estimated relative efficiencies. The preferred estimator under (i) is a function of the sample correlation of (Y _i:n , Y _[i:n]) values, and under (ii), a method of moments estimator involving the regression function is preferred.     

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.     

On characterizing distributions via regression of functions of generalized order statistics

Let X(1,n,m₁,k),X(2,n,m₂,k),…,X(n,n,m,k) be n generalized order statistics from a continuous distribution F which is strictly increasing over (a,b),−∞≤a<b≤∞, the support of F. Let g be an absolutely continuous and monotonically increasing function in (a,b) with finite g(a⁺),g(b⁻) and E(g(X)). Then for some positive integer s,1<s≤n, we give characterization of distributions by means of

     

COMPARISON OF THREE NON-PARAMETRIC TESTS FOR BIVARIATE INTERCHANGEABILITY1

Given a random sample(X₁, Y₁), …,(X_n, Y_n) from a bivariate (BV) absolutely continuous c.d.f. H (x, y), we consider rank tests for the null hypothesis of interchangeability H₀: H(x, y). Three linear rank test statistics, Wilcoxon (W_N), sum of squared ranks (SSR_N) and Savage (S_N), are described in Section 1. In Section 2, asymptotic relative efficiency (ARE) comparisons of the three types of tests are made for Morgenstern (Plackett, 1965) and Moran (1969)BV alternatives with marginal distributions satisfying G(x) = F(x/θ) for some θ≠ 1. Both gamma and lognormal marginal distributions are used.     

Exponential Family Techniques for the Lognormal Left Tail

Let X be lognormal(μ,σ²) with density f(x); let θ > 0 and define . We study properties of the exponentially tilted density (Esscher transform) f_θ(x) = e^?θxf(x)/L(θ), in particular its moments, its asymptotic form as θ→∞ and asymptotics for the saddlepoint θ(x) determined by . The asymptotic formulas involve the Lambert W function. The established relations are used to provide two different numerical methods for evaluating the left tail probability of the sum of lognormals S_n=X₁+?+X_n: a saddlepoint approximation and an exponential tilting importance sampling estimator. For the latter, we demonstrate logarithmic efficiency. Numerical examples for the cdf F_n(x) and the pdf f_n(x) of S_n are given in a range of values of σ²,n and x motivated by portfolio value‐at‐risk calculations.     

Moments of Order Statistics from Weibull Distribution in the Presence of Multiple Outliers

In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X₁, X₂, …, X_{n ? p} are independent with density function f(x) while the remaining, p observations (outliers) X_{n ? p + 1}, …, X_n are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.