Steekproeven uit de halve cauchy verdeling

Summary  Let x₁…, x_n be a sample from a distribution with infinite expectation, then for n →∞ the sample average _n tends to +∞ with probability 1 (see [4]).
Sometimes _n contains high jumps due to large observations. In this paper we consider samples from the "absolute Cauchy" distribution. In practice, on may consider the logarithm of the observations as a sample from a normal distribution. So we found in our simulation. After rejecting the log-normality assumption, one will be tempted to regard the extreme observations as outliers. It is shown that the discarding of the outlying observations gives an underestimation of the expectation, variance and 99 percentile of the actual distribution.     

On Bayesian optimum allocation in some two phase sampling problems

Summary For estimating certain parametric functions, we consider the problem of allocatingN _i, the size of the sample from theith population,i=1,2,...,k, at the second phase of sampling of a two phase sampling procedure, given that we taken _i observations from the population at the first phase. We consider that the observations from theith population follow the exponential distribution with mean _i,i=1,2,...,k, and the functions to be estimated are (i) (d_i/_i) and (ii) (d_ii). When the total cost of sampling at the second phase is c _iN_i and is fixed, allocations using the Bayes approach are obtained so that the estimation is as precise as is possible.     

Distributions determined by conditioning on a single order statistic

LetX ₁,X ₂, …,X _n(n ? 2) be a random sample on a random variablex with a continuous distribution functionF which is strictly increasing over (a, b), ?∞ ?a <b ? ∞, the support ofF andX _1:n ?X _2:n ? … ?X _n:n the corresponding order statistics. Letg be a nonconstant continuous function over (a, b) with finiteg(a ⁺) andE {g(X)}. Then for some positive integers, 1 <s ?n $$E\left\{ {\frac{1}{{s - 1}}\sum\limits_{i - 1}^{s - 1} {g(X_{i:n} )|X_{s:n} } = x} \right\} = 1/2(g(x) + g(a^ + )), \forall x \in (a,b)$$ iffg is bounded, monotonic and \(F(x) = \frac{{g(x) - g(a^ + )}}{{g(b^ - ) - g(a^ + )}},\forall x \in (a,b)\) . This leads to characterization of several distribution functions. A general form of this result is also stated.     

Characterization of the exponential distribution function by Properties of the differenceX k+s∶n−X k∶n of order statistics

We considern independent and identically distributed random variables with common continuous distribution functionF concentrated on (0, ∞). LetX _1∶n≤X_2∶n...≤X_n∶n be the corresponding order statistics. Put $$d_s \left( x \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - P\left( {X_{s:n - k} \geqslant x} \right), x \geqslant 0,$$ and $$\delta _s \left( {x, \rho } \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - e^{ - \rho \left( {n - k} \right)x} ,\rho > 0,x \geqslant 0.$$ Fors=1 it is well known that each of the conditions d₁(x)=O ?x≥0 and δ₁ (x, p) = O ?x≥0 implies thatF is exponential; but the analytic tools in the proofs of these two statements are radically different. In contrast to this in the present paper we present a rather elementary method which permits us to derive the above conclusions for somes, 1≤n —k, using only asymptotic assumptions (either forx→0 orx→∞) ond _s(x) and δ₁ (x, p), respectively.     

On characterizing discrete distributions via conditional expectations of weak record values

Let (W _n ,n ≥ 0) denote the sequence of weak records from a distribution with support S = { α₀,α₁,...,α _N }. In this paper, we consider regression functions of the form ψ _n (x) = E(h(W _n ) |W _n+1 = x), where h(·) is some strictly increasing function. We show that a single function ψ _n (·) determines F uniquely up to F(α₀). Then we derive an inversion formula which enables us to obtain F from knowledge of ψ _n (·), ψ _n-1(·), h(·) and F(α₀).     

Risk-efficient nonparametric sequential estimators

Summary Letx ₁,x ₂,x ₃, ... be a sequence of independent identically distributed random variables andτ an estimable parameter of their distribution. We want to estimateτ by the correspondingU-statisticu _n with loss function (u _n −τ)² +cn. We derive a stopping time and prove its risk-efficiency in the sense of Starr (1966) without any assumption on the nature of the distribution function other than the existence of some moments. Research supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 72, at the Universit?t Bonn.     

ON THE DISTRIBUTIONS OF ORDER STATISTICS FOR A RANDOM SAMPLE SIZE*

The probability distribution of the i –th and j–th order statistics and of the range R of a sample of size n, taken from a population with probability density function f (x) have been obtained when the sample size n is a random variable N and has: (i) a generalized Poisson distribution; and (ii) a generalized negative bonimial distribution. Specific results are then obtained; (a) when f (x) is uniform over (0,1); and (b) when f(x) is exponential. All the results for N, being a Poisson, binomial and negative binomial rv follow as special cases.     

The conduct of the sample average when the first moment is infinite

Summary Many books about probability and statistics only mention the weak and the strong law of large numbers for samples from distributions with finite expectation. However, these laws also hold for distributions with infinite expectation and then the sample average has to go to infinity with increasing sample size. Being curious about the way in which this would happen, we simulated increasing samples (up to n= 40000) from three distributions with infinite expectation. The results were somewhat surprising at first sight, but understandable after some thought. Most statisticians, when asked, seem to expect a gradual increase of the average with the size of the sample. So did we. In general, however, this proves to be wrong and for different parent distributions different types of conduct appear from this experiment. The samples from the “absolute Cauchy”-distribution are most interesting from a practical point of view: the average takes a high jump from time to time and decreases in between. In practice it might well happen, that the observations causing the jumps would be discarded as outlying observations.     

Multivariate L1 mean

The center of a univariate data set {x ₁,…,x _n} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x ₁−μ|,…,|x _n−μ|). As the median and the mean are the minimizers of respectively the L ₁- and the L ₂-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x ₁,…,x _n} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L ₁-distances y′₁=(∥x ₁- μ∥₁,…,∥x _n- μ∥₁) and the vector of L ₂-distances y′₂=(∥x ₁- μ∥₂,…,∥x _n-μ∥₂). We define the L ₁-median and the L ₁-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₁; and then the L ₂-median and the L ₂-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₂. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L ₁-mean appears to be a new concept. Received January 1999     

Minimax estimation under convex loss when the parameter interval is bounded

In the present paper families of truncated distributions with a Lebesgue density forx=(x ₁,...,x _n ) ε ℝ ⁿ are considered, wheref ₀:ℝ → (0, ∞) is a known continuous function andC _n (ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator form a saddle point when the parameter interval is sufficiently small.     

A characterization of the exponential distribution by higher order gap

Summary SupposeX is a non-negative random variable with an absolutely continuous (with respect to Lebesgue measure) distribution functionF (x) and the corresponding probability density functionf(x). LetX ₁,X ₂,...,X _n be a random sample of sizen fromF andX _i,n is thei-th smallest order statistics. We define thej-th order gapg _i,j(n) asg _i,j(n)=X _i+j,n–X_i,n 1i<n, 1nn–i. In this paper a characterization of the exponential distribution is given by considering a distribution property ofg _i,j(n).     

A consistent test for multivariate normality based on the empirical characteristic function

LetX ₁,X ₂, …,X _n be independent identically distributed random vectors in IR ^d ,d ⩾ 1, with sample mean and sample covariance matrixS _n. We present a practicable and consistent test for the composite hypothesisH _d: the law ofX ₁ is a non-degenerate normal distribution, based on a weighted integral of the squared modulus of the difference between the empirical characteristic function of the residualsS _n ^−1/2 (X _j − ) and its pointwise limit exp (−1/2|t|²) underH _d. The limiting null distribution of the test statistic is obtained, and a table with critical values for various choices ofn andd based on extensive simulations is supplied.     

Copula Gaussian graphical models with penalized ascent Monte Carlo EM algorithm

Typical data that arise from surveys, experiments, and observational studies include continuous and discrete variables. In this article, we study the interdependence among a mixed (continuous, count, ordered categorical, and binary) set of variables via graphical models. We propose an ?₁‐penalized extended rank likelihood with an ascent Monte Carlo expectation maximization approach for the copula Gaussian graphical models and establish near conditional independence relations and zero elements of a precision matrix. In particular, we focus on high‐dimensional inference where the number of observations are in the same order or less than the number of variables under consideration. To illustrate how to infer networks for mixed variables through conditional independence, we consider two datasets: one in the area of sports and the other concerning breast cancer.     

The distribution of the maximum of a first order autoregressive process: the continuous case

We give the cumulative distribution function of M _n , the maximum of a sequence of n observations from an autoregressive process of order 1. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. When the correlation is positive, $$P \left( M_n \leq x \right)\ =a_{n,x},$$ where $$a_{n,x}= \sum_{j=1}^\infty \beta_{jx}\ \nu_{jx}^{n} = O \left( \nu_{1x}^{n}\right),$$ where {?? _jx } are the eigenvalues of a non-symmetric Fredholm kernel, and ?? _1x is the eigenvalue of maximum magnitude. When the correlation is negative $$P \left( M_n \leq x \right)\ =a_{n,x} +a_{n-1,x}.$$ The weights ?? _jx depend on the jth left and right eigenfunctions of the kernel. These are given formally by left and right eigenvectors of an infinite Toeplitz matrix whose eigenvalues are just {?? _jx }. These results are large deviations expansions for extremes, since the maximum need not be standardized to have a limit. In fact, such a limit need not exist. The use of the derived expansion for P(M _n ?? x) is illustrated using both simulated and real data sets.     

Contributions to nonparametric generalized failure rate function estimation

For a specified distribution functionG with densityg, and unknown distribution functionF with densityf, the generalized failure rate function (x)=f(x)/gG ^–1 F(x) may be estimated by replacingf andF byf _n and , wheref _n is an empirical density function based on a sample of sizen from the distribution functionF, and . Under regularity conditions we show and, under additional restrictions whereC is a subset ofR and _n. Moreover, asymptotic normality is derived and the Berry-Esséen type bound is shown to be related to a theorem which concerns the sum of i.i.d. random variables. The order boundO(n^–1/2+c _n ^1/2 ) is established under mild conditions, wherec _n is a sequence of positive constants related tof _n and tending to 0 asn.Research was supported in part by the Army, Navy and Air Force under Office of Naval Research contract No. N00014-76-C-0608. AMS 1970 subject classifications. Primary 62G05. Secondary 60F15.     

A method to obtain new copulas from a given one

Given a strictly increasing continuous function φ from [0, 1] to [0, 1] and its pseudo-inverse φ^[−1], conditions that φ must satisfy for C_φ(x₁, . . . ,x_n)=φ^[−1](C(φ(x₁), . . . ,φ(x_n))) to be a copula for any copula C are studied. Some basic properties of the copulas obtained in this way are analyzed and several examples of generator functions φ that can be used to construct copulas C_φ are presented. In this manner, a method to obtain from a given copula C a variety of new copulas is provided. This method generalizes that used to construct Archimedean copulas in which the original copula C is the product copula, and it is related with mixtures     

Optimal risk rate in a functional inverse problem

In this note, we will consider the problem of recovering an unknown input function when the output function is observed in its entirety, blurred with functional error. An estimator is constructed whose risk converges at an optimal rate. In this functional model, convergence rates of order 1/n (n is the sample size) are possible, provided that the error distribution is sufficiently concentrated so as to compensate for the ill‐posedness of the inverse of the model operator.     

Slow convergence of the number of near-maxima for Burr XII distributions

A near-maximum is an observation which falls within a distance a of the maximum observation in an independent and identically distributed sample of size n. Subject to some conditions on the tail thickness of the population distribution, the number K _n (a) of near-maxima is known to converge in probability to one or infinity, or in distribution to a shifted geometric law. In this paper we show that for all Burr XII distributions K _n (a) converges almost surely to unity, but this convergence property may not become clear under certain cases even for very large n. We explore the reason of such slow convergence by studying a distributional continuity between Burr XII and Weibull distributions. We have also given a theoretical explanation of slow convergence of K _n (a) for the Burr XII distributions by showing that the rate of convergence in terms of P{K _n (a) > 1} tending to zero changes very little with the sample size n. Illustrations of the limiting behaviour K _n (a) for the Burr XII and the Weibull distributions are given by simulations and real data. The study also raises an important issue that although the Burr XII provides overall better fit to a given data set than the Weibull distribution, cautions should be taken for the extrapolation of the upper tail behaviour in the case of slow convergence.