Conditional estimation of average on the basis of weighting data

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples: % MathType!End!2!1! and % MathType!End!2!1!.Let % MathType!End!2!1! and % MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination % MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived. The variance of the statistic % MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.     

Sequential experimental design and response optimisation

We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy _k . This may correspond to the case of a regression model, where one observesy _k =f(θ,x _k )+ε _k , with ε _k some random error, or to the Bernoulli case wherey _k ∈{0, 1}, with Pr[y _k =1|θ,x _k |=f(θ,x _k ). Special attention is given to sequences given by , with an estimated value of θ obtained from (x₁, y₁),...,(x _k ,y _k ) andd _k (x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon, where one wants to maximize ∑ _i=1 ^N w _i f(θ, x _i ) with {w _i } a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method for a binary response problem.     

Estimation of a normal mean relative to balanced loss functions

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function, % MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean % MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form % MathType!End!2!1! under % MathType!End!2!1!. We also consider the weighted balanced loss function, % MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ .     

Estimation bayesienne des effets dans le modele a effets aleatoires de classification double avec interaction

The problem of estimating the effects in a balanced two-way classification with interaction \documentclass{article}\pagestyle{empty}\begin{document}$i = 1, \ldots ,I;j = 1, \ldots ,J;k = 1, \ldots ,K$\end{document} using a random effect model is considered from a Bayesian view point. Posterior distributions of r_i, c_j and t_ij are obtained under the assumptions that r_i, c_j, t_ij and e_ijk are all independently drawn from normal distributions with zero meansand variances \documentclass{article}\pagestyle{empty}\begin{document}$\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document} respectively. A non informative reference prior is adopted for \documentclass{article}\pagestyle{empty}\begin{document}$\mu ,\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document}. Various features of thisposterior distribution are obtained. The same features of the psoterior distribution for a fixed effect model are also obtained. A numerical example is given.     

Biased estimation of a variance

Summary Let , whereX _i are i.i.d. random variables with a finite variance σ² and is the usual estimate of the mean ofX _i. We consider the problem of finding optimal α with respect to the minimization of the expected value of |S ²(σ)−σ²|^k for variousk and with respect to Pitman's nearness criterion. For the Gaussian case analytical results are obtained and for some non-Gaussian cases we present Monte Carlo results regarding Pitman's criteron. This research was supported by Science Fund of Serbia, grant number 04M03, through Mathematical Institute, Belgrade.     

Comparison of location parameters of two exponential distributions when scale parameters are different and unknown

Letx _i(1)≤x _i(2)≤…≤x _i(r_i) be the right-censored samples of sizesn _i from theith exponential distributions $\sigma _i^{ - 1} exp\{ - (x - \mu _i )\sigma _i^{ - 1} \} ,i = 1,2$ where μ_i and σ_i are the unknown location and scale parameters respectively. This paper deals with the posteriori distribution of the difference between the two location parameters, namely μ₂-μ₁, which may be represented in the form $\mu _2 - \mu _1 \mathop = \limits^\mathcal{D} x_{2(1)} - x_{1(1)} + F_1 \sin \theta - F_2 \cos \theta $ where $\mathop = \limits^\mathcal{D} $ stands for equal in distribution,F _i stands for the central F-variable with [2,2(r _i?1)] degrees of freedom and $\tan \theta = \frac{{n_2 s_{x1} }}{{n_1 s_{x2} }}, s_{x1} = (r_1 - 1)^{ - 1} \left\{ {\sum\limits_{j = 1}^{r_i - 1} {(n_i - j)(x_{i(j + 1)} - x_{i(j)} )} } \right\}$ The paper also derives the distribution of the statisticV=F ₁ sin σ?F ₂ cos σ and tables of critical values of theV-statistic are provided for the 5% level of significance and selected degrees of freedom.     

Efficiency comparison of M-estimates for scale at t-distributions

Using Fisher's information fort-distributions, the absolute asymptotic efficiency of some M-estimates for scale with known location parameter is calculated and graphically illustrated. The compared estimators are the standard deviationS _*, the mean absolute deviation, called mean deviationD _*, the median absolute deviation, called MAD_*, and some M-estimates for scale, one, which is very robust, and another one with high asymptotic efficiency fort-distributions close to the normal. The last one is considered with monotone (in the positive field) and with very late redescending χ-function too. Also the , an alternative and generalized excess measure defined as the double relative asymptotic variance of the underlying scale estimator in the previous paper, is calculated fort-distributions and graphically illustrated, because there is the relation that the higher the asymptotic efficiency of is, the lower is the corresponding .     

RELATIONS BETWEEN ERGODICITY AND MEAN DRIFT FOR MARKOV CHAINS1

If {X_n} is an irreducible aperiodic Markov chain on a state apace denote the mean one step change of position, or “drift”, of {X_n} at j. The main result of this note is to show that, when |µ(j)| is bounded, {X_n} admits a stationary distribution {π_j}if and only if for some N > 0 and some state i, lim inf ∑when this holds, the limit infimum is in fact . Many of the known sufficient or necessary criteria for the existence of a stationary distribution can then be derived easily from this and related results.     

Generating random numbers of prescribed distribution using physical sources

When constructing uniform random numbers in [0, 1] from the output of a physical device, usually n independent and unbiased bits B _j are extracted and combined into the machine number . In order to reduce the number of data used to build one real number, we observe that for independent and exponentially distributed random variables X _n (which arise for example as waiting times between two consecutive impulses of a Geiger counter) the variable U _n : = X _{2n – 1}/(X _{2n – 1} + X _2n ) is uniform in [0, 1]. In the practical application X _n can only be measured up to a given precision (in terms of the expectation of the X _n ); it is shown that the distribution function obtained by calculating U _n from these measurements differs from the uniform by less than /2.We compare this deviation with the error resulting from the use of biased bits B _j with P {B _j = 1{ = (where ] – [) in the construction of Y above. The influence of a bias is given by the estimate that in the p-total variation norm Q ^TV _p = ( |Q()| ^p )^1/p (p 1) we have P _Y – P ⁰ _Y ^TV _p (c _n · )^1/p with c _n p for n . For the distribution function F _Y – F ⁰ _Y 2(1 – 2^–n )|| holds.     

A probablistic analysis for an optimal screening problem

Summary We consider a lotL formed byN apparently similar unitsW ₁,…,W _N, where each of theW _i may come from one of two different populationsP ₁ andP ₂;T ₁,…,T _N denote the corresponding lifetimes. The units fromP _i undergo a failure of kindi and their survival function isS _i (t). We assume that the failure rate function are known and that the units fromP ₁ are ?substandard?: λ ₁ (t)≥λ ₂ (t), ∀t≥0. We want to putW ₁,…,W _N under a pre-operational test (burn-in test) in order to eliminate at least a great part of the substandard units and we face the problem of obtaining a rule for stopping the test under the assumption that, with the failure of a unit, it is possible to recognize the population from which the unit comes. Such a problem will be formalized as an optimal stopping problem for a suitably defined Markov process. Our study shall evidentiate some fundamental aspects of the problem and the role of the prior distribution of the (random) numberM ₀ of those units inL coming fromP ₁ (substandard). The latter distribution has a great influence on the form of the solution. This research was supported by the C.N.R. Project ?Statistica Bayesiana e Simulazione in Affidalità e Modellistica Biologica?.     

Comparing several exponential populations with more than one control

Suppose there are k ₁ (k ₁ ≥ 1) test treatments that we wish to compare with k ₂ (k ₂ ≥ 1) control treatments. Assume that the observations from the ith test treatment and the jth control treatment follow a two-parameter exponential distribution and , where θ is a common scale parameter and and are the location parameters of the ith test and the jth control treatment, respectively, i = 1, . . . ,k ₁; j = 1, . . . ,k ₂. In this paper, simultaneous one-sided and two-sided confidence intervals are proposed for all k ₁ k ₂ differences between the test treatment location and control treatment location parameters, namely , and the required critical points are provided. Discussions of multiple comparisons of all test treatments with the best control treatment and an optimal sample size allocation are given. Finally, it is shown that the critical points obtained can be used to construct simultaneous confidence intervals for Pareto distribution location parameters.     

Spectral analysis with replicated time series

A doubly stochastic process {x(b,t);b?B,t?Z} is considered, with (B,β,P_β) being a probability space so that for each b, {X(b,t);t ? Z} is a stationary process with an absolutely continuous spectral distribution. The population spectrum is defined as f(ω) = E_B[Q(b,ω)] with Q(b,ω) being the spectral density function of X(b,t). The aim of this paper is to estimate f(ω) by means of a random sample b₁,…,b_r from (B,β,P_β). For each b₁? B, the processes X(b₁,t) are observed at the same times t=1,…,N. Thus, r time series (x(b₁,t)} are available in order to estimate f(ω). A model for each individual periodogram, which involves f(ω), is formulated. It has been proven that a certain family of linear stationary processes follows the above model In this context, a kernel estimator is proposed in order to estimate f(ω). The bias, variance and asymptotic distribution of this estimator are investigated under certain conditions.     

Likelihood as an index of fit

Summary In this paper likelihood is characterized as an index which measures how much a model fits a sample. Some properties required to an index of fit are introduced and discussed, while stressing how they describe aspects inner to idea of fit. Finally we prove that, if an index of fit is maximal when the model reaches the distribution of the sample, then such an index is an increasing continuous transform of , where thep _i's are the theoretical relative frequencies provided by the model and theq _i's are the actual relative frequencies of the sample.     

Quadratic subspaces and construction of Bayes invariant quadratic estimators of variance components in mixed linear models

Gnot et al. (J Statist Plann Inference 30(1):223–236, 1992) have presented the formulae for computing Bayes invariant quadratic estimators of variance components in normal mixed linear models of the form where the matrices V _i , 1 ≤ i ≤ k − 1, are symmetric and nonnegative definite and V _k is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV ₁ M,...,MV _k-1 M,M, where M is an orthogonal projector on the null space of X′. In the paper we discuss methods of construction of such a basis. We survey Malley’s algorithms for finding the smallest quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms. We also consider a class of matrices sharing some of the symmetries common to MV ₁ M,...,MV _k-1 M,M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving the efficiency of Malley’s algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices MV ₁ M,...,MV _k-1 M,M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods. Dedicated to the memory of Professor Stanisław Gnot.     

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.     

Nonparametric estimation of a multivariate multiple regression function

Let (X₁, X₂, Y₁, Y₂) be a four dimensional random variable having the joint probability density function f(x₁, x₂, y₁, y₂). In this paper we consider the problem of estimating the regression function \({{E[(_{Y_2 }^{Y_1 } )} \mathord{\left/ {\vphantom {{E[(_{Y_2 }^{Y_1 } )} {_{X_2 = X_2 }^{X_1 = X_1 } }}} \right. \kern-0em} {_{X_2 = X_2 }^{X_1 = X_1 } }}]\) on the basis of a random sample of size n. We have proved that under certain regularity conditions the kernel estimate of this regression function is uniformly strongly consistent. We have also shown that under certain conditions the estimate is asymptotically normally distributed.     

Bootstrap inference in local polynomial regression of time series

In this paper we consider the inferential aspect of the nonparametric estimation of a conditional function , where X _t,m represents the vector containing the m conditioning lagged values of the series. Here is an arbitrary measurable function. The local polynomial estimator of order p is used for the estimation of the function g, and of its partial derivatives up to a total order p. We consider α-mixing processes, and we propose the use of a particular resampling method, the local polynomial bootstrap, for the approximation of the sampling distribution of the estimator. After analyzing the consistency of the proposed method, we present a simulation study which gives evidence of its finite sample behaviour.     

Constructing an unbiased estimator of population mean in finite populations using auxiliary information

The objective of this paper is to construct an unbiased estimator (up to order 0(1/n)) of the population mean of the study variatey which is more efficient than the sample mean of the ‘n’ obsrvedy-values. In particular, the unbiased estimators are discussed for the cases of positive and negative correlations of the study variatey and the auxiliary variatex.     

Hypotheses testing about the drift parameter in linear stochastic differential equation driven by stable processes

In this paper, we consider the problem of hypotheses testing about the drift parameter \(\theta \) in the process \(\text {d}Y^{\delta }_{t} = \theta \dot{f}(t)Y^{\delta }_{t}\text {d}t + b(t)\text {d}L^{\delta }_{t}\) driven by symmetric \(\delta \)-stable Lévy process \(L^{\delta }_{t}\) with \(\dot{f}(t)\) being the derivative of a known increasing function f(t) and b(t) being known as well. We consider the hypotheses testing \(H_{0}: \theta \le 0\) and \(K_{0}: \theta =0\) against the alternatives \(H_{1}: \theta >0\) and \(K_{1}: \theta \ne 0\), respectively. For these hypotheses, we propose inverse methods, which are motivated by sequential approach, based on the first hitting time of the observed process (or its absolute value) to a pre-specified boundary or two boundaries until some given time. The applicability of these methods is illustrated. For the case \(Y^{\delta }_{0}=0\), we are able to calculate the values of boundaries and finite observed times more directly. We are able to show the consistencies of proposed tests for \(Y^{\delta }_{0}\ge 0\) with \(\delta \in (1,2]\) and for \(Y^{\delta }_{0}=0\) with \(\delta \in (0,2]\) under quite mild conditions.