On the rate of convergence of recursive kernel estimates of probability densities

Recursive estimates f_n^r(x)of the rth derivative f^r(x)(r=0,1)of the univariate probability density f(x) for strictly stationary processes {X_j,} are considered. The asymptotic variance-covariance of f_n^r(x)is established for stationary triangular arrays of random variables satisfying various asymptotic independence-uncorrelatedness conditions.     

Some characterizations of discrete distributions based on weak records

Let X ₁, X ₂,... be iid random variables (rv's) with the support on nonnegative integers and let (W _n , n≥0) denote the corresponding sequence of weak record values. We obtain new characterization of geometric and some other discrete distributions based on different forms of partial independence of rv's W _n and W _n+r —W _n for some fixed n≥0 and r≥1. We also prove that rv's W ₀ and W _n+1 —W _n have identical distribution if and only if (iff) the underlying distribution is geometric.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

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.     

Admissible tests for the mean with additional information

Let X be a po-normal random vector with unknown µ and unknown covariance matrix ∑ and let X be partitioned as X = (X^′ ₍₁₎, …, X^′ _(r))′ where X_(j)is a subvector of X with dimension p_jsuch that ∑^r _j=1Pj = P0. Some admissible tests are derived for testing H₀: μ = 0 versus H₁: μ ¦0 based on a sample drawn from the whole vector X of dimension p and r additional samples drawn from X₍₁₎, X₍₂₎, …, X(r) respectively, All (r+1) samples are assumed to be independent. The distribution of some of the tests' statistics involved are also derived.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

Distributions of patterns of two successes separated by a string of <Emphasis Type="Italic">k</Emphasis>-2 failures

Let Z ₁, Z ₂, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z _t  = 1) and q = Pr(Z _t  = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns E₁{\mathcal{E}_{1}}: two successes are separated by at most k−2 failures, E₂{\mathcal{E}_{2}}: two successes are separated by exactly k −2 failures, and E₃{\mathcal{E}_{3}} : two successes are separated by at least k − 2 failures. Denote by N_n,k⁽ⁱ⁾{ N_{n,k}^{(i)}} (respectively M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}) the number of occurrences of the pattern E_i{\mathcal{E}_{i}} , i = 1, 2, 3, in Z ₁, Z ₂, . . . , Z _n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} (resp. W_r,k⁽ⁱ⁾){W_{r,k}^{(i)})} be the waiting time for the r − th occurrence of the pattern E_i{\mathcal{E}_{i}}, i = 1, 2, 3, in Z ₁, Z ₂, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of N_n,k⁽ⁱ⁾{N_{n,k}^{(i)}}, M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}, T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} and W_r,k⁽ⁱ⁾{W_{r,k}^{(i)}} (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.     

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.     

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.     

A NEW APPROXIMATION FOR FISHER'S Z

As the sample size increases, the coefficient of skewness of the Fisher's transformation z= tanh^-1r, of the correlation coefficient decreases much more rapidly than the excess of its kurtosis. Hence, the distribution of standardized z can be approximated more accurately in terms of the t distribution with matching kurtosis than by the unit normal distribution. This t distribution can, in turn be subjected to Wallace's approximation resulting in a new normal approximation for the Fisher's z transform. This approximation, which can be used to estimate the probabilities, as well as the percentiles, compares favorably in both accuracy and simplicity, with the two best earlier approximations, namely, those due to Ruben (1966) and Kraemer (1974). Fisher (1921) suggested approximating distribution of the variance stabilizing transform z=(1/2) log ((1 +r)/(1r)) of the correlation coefficient r by the normal distribution with mean = (1/2) log ((1 + p)/(lp)) and variance =l/(n3). This approximation is generally recognized as being remarkably accurate when ||Gr| is moderate but not so accurate when ||Gr| is large, even when n is not small (David (1938)). Among various alternatives to Fisher's approximation, the normalizing transformation due to Ruben (1966) and a t approximation due to Kraemer (1973), are interesting on the grounds of novelty, accuracy and/or aesthetics. If r?= r/√ (1r²) and r?|Gr = |Gr/√(1|Gr²), then Ruben (1966) showed that (1) g_n (r,|Gr) ={(2n5)/2}^1/2r?r{(2n3)/2}^1/2r?|GR, {1 + (1/2)(r?r²+r?|Gr²)}^1/2 is approximately unit normal. Kraemer (1973) suggests approximating (2) t_n (r, |Gr) = (r|GR¹) √ (n2), √(11r²) √(1|Gr²) by a Student's t variable with (n2) degrees of freedom, where after considering various valid choices for |Gr¹ she recommends taking |Gr¹= |Gr*, the median of r given n and |Gr.     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

Algorithms for the construction of optimizing distributions

We will consider the following problem.Maximise Φ(p)over P={p=(p₁,P₂,…,p_j):P_j≧0,∑p_j=1}". We require to calcute an optimizing distribution. Examples arise in optimal regression design,maximum likelihood estimation and stratified sazmpling problems. A class of multiplicative algorithms, indexed by functions which depend on the derivatives of Φ(·)is considered for solving this problem.Iterations are of the form:p_j ^(r+1)αp_j ^(r)f(x_j ^(r)), where x_{j (r)}=d_j ^(r) or F_j ^(r)and d_j ^(r)=?Φ/?p_j While F_j ^(r)=D_j ^(r)?∑p_i ^(r)d_i ^(r) (a directional derivative)at p=p^(r)f(·)satisfies some suitable properties and may depend on one or more free parameters. These iterations neatly submit to the constraints ofv the problem. Some results will be reported and extensions to problems dependin on two or more distributions and to problems with additional constraints will be considered.     

√n-CONSISTENT ESTIMATION IN A RANDOM COEFFICIENT AUTOREGRESSIVE MODEL

This paper deals with √n-consistent estimation of the parameter μ in the RCAR(l) model defined by the difference equation X_j=(μ+U_j)X_j-l+ej (jε Z), where {e_j: jε Z} and {U_j: jε Z} are two independent sets of i.i.d. random variables with zero means, positive finite variances and E[(μ+U₁)²] < 1. A class of asymptotically normal estimators of μ indexed by a family of bounded measurable functions is introduced. Then an estimator is constructed which is asymptotically equivalent to the best estimator in that class. This estimator, asymptotically equivalent to the quasi-maximum likelihood estimator derived in Nicholls & Quinn (1982), is much simpler to calculate and is asymptotically normal without the additional moment conditions those authors impose.     

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).     

On some joint distributions in fluctuation theory

For non-negative integral valued interchangeable random variables v₁, v₂,…,v_n, Takács (1967, 70) has derived the distributions of the statistics ?_n' ?¹_n' ?^(c)_n and ?^(-c)_n concerning the partial sums N_r = v₁ + v₂ + ··· + v_rr = 1,…,n. This paper deals with the joint distributions of some other statistics viz., (α^(c)_n, δ^(c)_n, Z_n), (β^(c)_n, Z_n) and (β^(-c)_n, Z_n) concerning the partial sums N_r = ε₁ + ··· + ε_rr = 1,2,…,n, of geometric random variables ε₁, ε₂,…,ε_n.     

Optimal component test plans for a parallel system based on Type-II censoring

Consider a parallel system with n independent components. Assume that the lifetime of the jth component follows an exponential distribution with a constant but unknown parameter λ_j, 1≤j≤n. We test r_j components of type-j for failure and compute the total time T_j of r_j failures for the jth component. Based on T=(T₁,T₂,…,T_n) and r=(r₁,r₂,…,r_n), we derive optimal reliability test plans which ensure the usual probability requirements on system reliability. Further, we solve the associated nonlinear integer programming problem by a simple enumeration of integers over the feasible range. An algorithm is developed to obtain integer solutions with minimum cost. Finally, some examples have been discussed for various levels of producer’s and consumer’s risk to illustrate the approach. Our optimal plans lead to considerable savings in costs over the available plans in the literature.     

Precise large deviations for the difference of two sums of END random variables with heavy tails

Assume that there are two types of insurance contracts in an insurance company, and the ith related claims are denoted by {X_ij, j ? 1}, i = 1, 2. In this article, the asymptotic behaviors of precise 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 are investigated, and under several assumptions, some corresponding asymptotic formulas are obtained.