Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

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.     

A remark on estimating the mean of a normal distribution with known coefficient of variation

Let X₁, X₂, …, X_n be iid N(μ, aμ²) (a>0) random variables with an unknown mean μ>0 and known coefficient of variation (CV) √a. The estimation of μ is revisited and it is shown that a modified version of an unbiased estimator of μ [cf. Khan RA. A note on estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1968;63:1039–1041] is more efficient. A certain linear minimum mean square estimator of Gleser and Healy [Estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1976;71:977–981] is also modified and improved. These improved estimators are being compared with the maximum likelihood estimator under squared-error loss function. Based on asymptotic consideration, a large sample confidence interval is also mentioned.     

Reduction of bias and skewness with applications to second order accuracy

Suppose [^(q)]{\widehat{\theta}} is an estimator of θ in \mathbbR{\mathbb{R}} that satisfies the central limit theorem. In general, inferences on θ are based on the central limit approximation. These have error O(n ^−1/2), where n is the sample size. Many unsuccessful attempts have been made at finding transformations which reduce this error to O(n ⁻¹). The variance stabilizing transformation fails to achieve this. We give alternative transformations that have bias O(n ⁻²), and skewness O(n ⁻³). Examples include the binomial, Poisson, chi-square and hypergeometric distributions.     

Bayesian and Robust Bayesian analysis under a general class of balanced loss functions

For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form L_{r, w, d0}(q, d)=wr(d₀, d)+ (1-w) r(q, d){L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}, as well as the weighted version q(q) L_{r, w, d0}(q, d){q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}, where ρ(θ, δ) is an arbitrary loss function, δ ₀ is a chosen a priori “target” estimator of q, w ? [0,1){\theta, \omega \in[0,1)}, and q(·) is a positive weight function. we develop Bayesian estimators under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω > 0 by relating such estimators to Bayesian solutions under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections between optimal actions derived under balanced and unbalanced losses.     

Variances of UMVU Estimators for Means and Variances After Using a Normalizing Transformation

Suppose that the function f is of recursive type and the random variable X is normally distributed with mean μ and variance α². We set C = f(x). Neyman & Scott (1960) and Hoyle (1968) gave the UMVU estimators for the mean E(C) and for the variance Var(C) from independent and identically distributed random variables X₁,…, X_n(n ≧ 2) having a normal distribution with mean μ and variance σ², respectively. Shimizu & Iwase (1981) gave the variance of the UMVU estimator for E(C). In this paper, the variance of the UMVU estimator for Var(C) is given.     

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.     

Regression analysis using order statistics

In this article, we study the joint distribution of X and two linear combinations of order statistics, a ^T Y ₍₂₎ and b ^T Y ₍₂₎, where a = (a ₁, a ₂) ^T and b = (b ₁, b ₂) ^T are arbitrary vectors in R ² and Y ₍₂₎ = (Y ₍₁₎, Y ₍₂₎) ^T is a vector of ordered statistics obtained from (Y ₁, Y ₂) ^T when (X, Y ₁, Y ₂) ^T follows a trivariate normal distribution with a positive definite covariance matrix. We show that this distribution belongs to the skew-normal family and hence our work is a generalization of Olkin and Viana (J Am Stat Assoc 90:1373–1379, 1995) and Loperfido (Test 17:370–380, 2008).     

A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model

Let [^(\varveck)]{\widehat{\varvec{\kappa}}} and [^(\varveck)]_r{\widehat{\varvec{\kappa}}_r} denote the best linear unbiased estimators of a given vector of parametric functions \varveck = \varvecKb{\varvec{\kappa} = \varvec{K\beta}} in the general linear models M = {\varvecy, \varvecX\varvecb, s²\varvecV}{{\mathcal M} = \{\varvec{y},\, \varvec{X\varvec{\beta}},\, \sigma^2\varvec{V}\}} and M_r = {\varvecy, \varvecX\varvecb | \varvecR \varvecb = \varvecr, s²\varvecV}{{\mathcal M}_r = \{\varvec{y},\, \varvec{X}\varvec{\beta} \mid \varvec{R} \varvec{\beta} = \varvec{r},\, \sigma^2\varvec{V}\}}, respectively. A bound for the Euclidean distance between [^(\varveck)]{\widehat{\varvec{\kappa}}} and [^(\varveck)]_r{\widehat{\varvec{\kappa}}_r} is expressed by the spectral distance between the dispersion matrices of the two estimators, and the difference between sums of squared errors evaluated in the model M{{\mathcal M}} and sub-restricted model M_r^*{{\mathcal M}_r^*} containing an essential part of the restrictions \varvecR\varvecb = \varvecr{\varvec{R}\varvec{\beta} = \varvec{r}} with respect to estimating \varveck{\varvec{\kappa}}.     

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.     

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 .     

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.     

Simulation study of new estimators combining the SUR ridge regression and the restricted least squares methodologies

In this paper, we propose two SUR type estimators based on combining the SUR ridge regression and the restricted least squares methods. In the sequel these estimators are designated as the restricted ridge Liu estimator and the restricted ridge HK estimator (see Liu in Commun Statist Thoery Methods 22(2):393–402, 1993; Sarkar in Commun Statist A 21:1987–2000, 1992). The study has been made using Monte Carlo techniques, (1,000 replications), under certain conditions where a number of factors that may effect their performance have been varied. The performance of the proposed and some of the existing estimators are evaluated by means of the TMSE and the PR criteria. Our results indicate that the proposed SUR restricted ridge estimators based on K _SUR, K _Sratio, K _Mratio and [(K)\ddot]{\ddot{K}} produced smaller TMSE and/or PR values than the remaining estimators. In contrast with other ridge estimators, components of [(K)\ddot]{\ddot{K}} are defined in terms of the eigenvalues of X^*￠ X^*{X^{{\ast^{\prime}}} X^{\rm \ast}} and all lie in the open interval (0, 1).     

Some theoretical results concerning non parametric estimation by using a judgment poststratification sample

ABSTRACT

In this paper, some of the properties of non parametric estimation of the expectation of g(X) (any function of X), by using a judgment poststratification sample (JPS), have been discussed. A class of estimators (including the standard JPS estimator and a JPS estimator proposed by Frey and Feeman (2012 Frey, J., Feeman, T.G. (2012). An improved mean estimator for judgment post-stratification. Comput. Stat. Data Anal. 56(2):418–426.[Crossref], [Web of Science ®] , [Google Scholar], Comput. Stat. Data An.) is considered. The paper provides mean and variance of the members of this class, and examines their consistency and asymptotic distribution. Specifically, the results are for the estimation of population mean, population variance, and cumulative distribution function. We show that any estimators of the class may be less efficient than simple random sampling (SRS) estimator for small sample sizes. We prove that the relative efficiency of some estimators in the class with respect to balanced ranked set sampling (BRSS) estimator tends to 1 as the sample size goes to infinity. Furthermore, the standard JPS mean estimator, and Frey–Feeman JPS mean estimator are specifically studied and we show that two estimators have the same asymptotic distribution. For the standard JPS mean estimator, in perfect ranking situations, optimum values of H (the ranking class size), for different sample sizes, are determined non parametrically for populations that are not heavily skewed or thick tailed.     

On estimation of stress–strength parameter using record values from proportional hazard rate models

ABSTRACT

Based on record values, this article deals with inference for stress–strength reliability, R = P(X < Y), where the distributions of X and Y follow proportional hazard rate models but having different parameters. Maximum likelihood estimator, uniformly minimum variance unbiased estimator, Bayes estimator, and different confidence intervals for R are obtained. Numerical computations and simulation study are presented for illustrative purposes.     

Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions

For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  _θ P _θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.     

Characterization of the mixtures of Rayleigh distributions by conditional expectation of order statistics

The mixture of Rayleigh random variables X ₁and X ₂ are identified in terms of relations between the conditional expectation of ( X_2:2² -X_1:2²)^r{\left( {X_{2:2}^2 -X_{1:2}^2}\right)^{r}} given X _1:2 (or X_2:2^2k{X_{2:2}^{2k}} given X_1:2,"k ￡ r){X_{1:2},\forall k\leq r)} and hazard rate function of the distribution, where X _1:2 and X _2:2 denote the corresponding order statistics, r is a positive integer. In addition, we also mention some related theorems to characterize the mixtures of Rayleigh distributions. Finally, we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems) and a simulated example is used to illustrate our results.     

Optimal estimation for finite population mean in two-phase sampling

Using two-phase sampling scheme, we propose a general class of estimators for finite population mean. This class depends on the sample means and variances of two auxiliary variables. The minimum variance bound for any estimator in the class is provided (up to terms of ordern ⁻¹). It is also proved that there exists at least a chain regression type estimator which reaches this minimum. Finally, it is shown that other proposed estimators can reach the minimum variance bound, i.e. the optimal estimator is not unique.     

On MSE of EBLUP

We consider Best Linear Unbiased Predictors (BLUPs) and Empirical Best Linear Unbiased Predictors (EBLUPs) under the general mixed linear model. The BLUP was proposed by Henderson (Ann Math Stat 21:309–310, 1950). The formula of this BLUP includes unknown elements of the variance-covariance matrix of random variables. If the elements in the formula of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) are replaced by some type of estimators, we obtain the two-stage predictor called the EBLUP which is model-unbiased (Kackar and Harville in Commun Stat A 10:1249–1261, 1981). Kackar and Harville (J Am Stat Assoc 79:853–862, 1984) show an approximation of the mean square error (the MSE) of the predictor and propose an estimator of the MSE. The MSE and estimators of the MSE are also studied by Prasad and Rao (J Am Stat Assoc 85:163–171, 1990), Datta and Lahiri (Stat Sin 10:613–627, 2000) and Das et al. (Ann Stat 32(2):818–840, 2004). In the paper we consider the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976. Ża̧dło (On unbiasedness of some EBLU predictor. Physica-Verlag, Heidelberg, pp 2019–2026, 2004) shows that the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) may be treated as a generalisation of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) and proves model unbiasedness of the EBLUP based on the formula of the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) under some assumptions. In this paper we derive the formula of the approximate MSE of the EBLUP and its estimators. We prove that the approximation of the MSE is accurate to terms o(D ⁻¹) and that the estimator of the MSE is approximately unbiased in the sense that its bias is o(D ⁻¹) under some assumptions, where D is the number of domains. The proof is based on the results obtained by Datta and Lahiri (Stat Sin 10:613–627, 2000). Using our results we show some EBLUP based on the special case of the general linear model. We also present the formula of its MSE and estimators of its MSE and their performance in Monte Carlo simulation study.