The formal posterior of a standard flat prior in MANOVA is incoherent

Summary A standard improper prior for the parameters of a MANOVA model is shown to yield an inference that is incoherent in the sense of Heath and Sudderth. The proof of incoherence is based on the fact that the formal Bayes estimate, sayδ ₀ , of the covariance matrix based on the improper prior and a certain bounded loss function is uniformly inadmissible in that there is another estimatorδ _l and an ɛ>0 such that the risk functions satisfyR(δ _l ,Σ)⩽R δ ₀ ,Σ)−ε for all values of the covariance matrix Σ. The estimatorδ _I is formal Bayes for an alternative improper prior which leads to a coherent inference. Research supported by National Science Foundation grants DMS-89-22607 (for Eaton) and DMS-9123358 (for Sudderth).     

Simultaneous marginal survival estimators when doubly censored data is present

A doubly censoring scheme occurs when the lifetimes T being measured, from a well-known time origin, are exactly observed within a window [L, R] of observational time and are otherwise censored either from above (right-censored observations) or below (left-censored observations). Sample data consists on the pairs (U, δ) where U = min{R, max{T, L}} and δ indicates whether T is exactly observed (δ = 0), right-censored (δ = 1) or left-censored (δ = −1). We are interested in the estimation of the marginal behaviour of the three random variables T, L and R based on the observed pairs (U, δ). We propose new nonparametric simultaneous marginal estimators [^(S)]_T, [^(S)]_L{\hat S_{T}, \hat S_{L}} and [^(S)]_R{\hat S_{R}} for the survival functions of T, L and R, respectively, by means of an inverse-probability-of-censoring approach. The proposed estimators [^(S)]_T, [^(S)]_L{\hat S_{T}, \hat S_{L}} and [^(S)]_R{\hat S_{R}} are not computationally intensive, generalize the empirical survival estimator and reduce to the Kaplan-Meier estimator in the absence of left-censored data. Furthermore, [^(S)]_T{\hat S_{T}} is equivalent to a self-consistent estimator, is uniformly strongly consistent and asymptotically normal. The method is illustrated with data from a cohort of drug users recruited in a detoxification program in Badalona (Spain). For these data we estimate the survival function for the elapsed time from starting IV-drugs to AIDS diagnosis, as well as the potential follow-up time. A simulation study is discussed to assess the performance of the three survival estimators for moderate sample sizes and different censoring levels.     

A study of generalized skew-normal distribution

Following the paper by Genton and Loperfido [Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math. 57 (2005), pp. 389–401], we say that Z has a generalized skew-normal distribution, if its probability density function (p.d.f.) is given by f(z)=2φ _p (z; ξ, Ω)π (z?ξ), z∈? ^p , where φ _p (·; ξ, Ω) is the p-dimensional normal p.d.f. with location vector ξ and scale matrix Ω, ξ∈? ^p , Ω>0, and π is a skewing function from ? ^p to ?, that is 0≤π (z)≤1 and π (?z)=1?π (z), ? z∈? ^p . First the distribution of linear transformations of Z are studied, and some moments of Z and its quadratic forms are derived. Next we obtain the joint moment-generating functions (m.g.f.’s) of linear and quadratic forms of Z and then investigate conditions for their independence. Finally explicit forms for the above distributions, m.g.f.’s and moments are derived when π (z)=κ (α′z), where α∈? ^p and κ is the normal, Laplace, logistic or uniform distribution function.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

Dispersive ordering—Some applications and examples

A basic concept for comparing spread among probability distributions is that of dispersive ordering. Let X and Y be two random variables with distribution functions F and G, respectively. Let F ⁻¹ and G ⁻¹ be their right continuous inverses (quantile functions). We say that Y is less dispersed than X (Y≤ _disp X) if G ⁻¹(β)−G ⁻¹(α)≤F ⁻¹(β)−F ⁻¹(α), for all 0<α≤β<1. This means that the difference between any two quantiles of G is smaller than the difference between the corresponding quantiles of F. A consequence of Y≤ _disp X is that |Y ₁−Y ₂| is stochastically smaller than |X ₁−X ₂| and this in turn implies var(Y)≤var(X) as well as E[|Y ₁−Y ₂|]≤E[|X ₁−X ₂|], where X ₁, X ₂ (Y ₁, Y ₂) are two independent copies of X(Y). In this review paper, we give several examples and applications of dispersive ordering in statistics. Examples include those related to order statistics, spacings, convolution of non-identically distributed random variables and epoch times of non-homogeneous Poisson processes. This work was supported in part by KOSEF through Statistical Research Center for Complex Systems at Seoul National University. Subhash Kochar is thankful to Dr. B. Khaledi for many helpful discussions.     

Measuring robustness for weighted distributions: Bayesian perspective

There are many situations where the usual random sample from a population of interest is not available, due to the data having unequal probabilities of entering the sample. The method of weighted distributions models this ascertainment bias by adjusting the probabilities of actual occurrence of events to arrive at a specification of the probabilities of the events as observed and recorded. We consider two different classes of contaminated or mixture of weight functions, Γ _a ={w(x):w(x)=(1−ε)w ₀(x)+εq(x),q∈Q} and Γ _g ={w(x):w(x)=w ₀ ^1−ε (x)q ^ε(x),q∈Q} wherew ₀(x) is the elicited weighted function,Q is a class of positive functions and 0≤ε≤1 is a small number. Also, we study the local variation of ϕ-divergence over classes Γ _a and Γ _g . We devote on measuring robustness using divergence measures which is based on the Bayesian approach. Two examples will be studied.     

Bounds for Expectations of Differences of Generalized Order Statistics Based on General and Life Distributions

Let X _? ^(r), r ≥ 1, denote generalized order statistics based on an arbitrary distribution function F with finite pth absolute moment for some 1 ≤ p ≤ ∞. We present sharp upper bounds on E(X _? ^(s) ? X _? ^(r)), 1 ≤ r < s, for F being either general or life distribution. The bounds are expressed in various scale units generated by pth central absolute or raw moments of F, respectively. The distributions achieving the bounds are specified.     

The number of records within a random interval of the current record value

LetX ₁,X ₂, … be a sequence of i.i.d. random variables with some continuous distribution functionF. LetX(n) be then-th record value associated with this sequence and μ _n ⁻ , μ _n ⁺ be the variables that count the number of record values belonging to the random intervals(f−(X(n)), X(n)), (X(n), f+(X(n))), wheref−, f+ are two continuous functions satisfyingf−(x)<x, f+(x)>x. Properties of μ _n ⁻ , μ _n ⁺ are studied in the present paper. Some statistical applications connected with these variables are also provided.     

Sparse conformal predictors

Conformal predictors, introduced by Vovk et al. (Algorithmic Learning in a Random World, Springer, New York, 2005), serve to build prediction intervals by exploiting a notion of conformity of the new data point with previously observed data. We propose a novel method for constructing prediction intervals for the response variable in multivariate linear models. The main emphasis is on sparse linear models, where only few of the covariates have significant influence on the response variable even if the total number of covariates is very large. Our approach is based on combining the principle of conformal prediction with the ℓ ₁ penalized least squares estimator (LASSO). The resulting confidence set depends on a parameter ε>0 and has a coverage probability larger than or equal to 1−ε. The numerical experiments reported in the paper show that the length of the confidence set is small. Furthermore, as a by-product of the proposed approach, we provide a data-driven procedure for choosing the LASSO penalty. The selection power of the method is illustrated on simulated and real data.     

Bayes Estimation of Change Point in Non Standard Mixture of Left-Truncated Exponential With Unknown Proportions

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,…, X _n were observed from the mixture of a degenerate and left-truncated exponential (LTE) distribution, with reliability R _1τ at time τ and minimum life length η with unknown proportion p ₁ and θ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in reliability R _2τ at time τ and unknown proportion p ₂ and θ₂. This distribution occurs in many practical situations, for instance; life of a unit may have a LTE distribution but some of the units fail instantaneously. Apart from mixture distributions, the phenomenon of change point is also observed in several situations in life testing and reliability estimation problems. It may happen that at some point of time instability in the sequence of failure times is observed. The problem of study is: When and where this change has started occurring. This is called change point inference problem. The estimators of m, R ₁(t ₀), R ₂(t ₀), p ₁, and p ₂ are derived under asymmetric loss functions namely Linex loss & general entropy loss functions. Both the non informative and informative prior are considered. The effects of prior consideration on Bayes estimates of change point are also studied.     

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.     

Robustness of weighted L p–depth and L p–median

Summary: L ^p –norm weighted depth functions are introduced and the local and global robustness of these weighted L ^p –depth functions and their induced multivariate medians are investigated via influence function and finite sample breakdown point. To study the global robustness of depth functions, a notion of finite sample breakdown point is introduced. The weighted L ^p –depth functions turn out to have the same low breakdown point as some other popular depth functions. Their influence functions are also unbounded. On the other hand, the weighted L ^p –depth induced medians are globally robust with the highest possible breakdown point for any reasonable estimator. The weighted L ^p –medians are also locally robust with bounded influence functions for suitable weight functions. Unlike other existing depth functions and multivariate medians, the weighted L ^p depth and medians are easy to calculate in high dimensions. The price for this advantage is the lack of affine invariance and equivariance of the weighted L ^p depth and medians, respectively.*The author thanks the referees for their very insightful and constructive comments and suggestions which led to corrections and substantial improvements. Supported in part by NSF Grants DMS-0071976 and DMS-0134628.     

A note on LeCam’s bound for the distance between the Poisson binomial and the Poisson distribution

n possibly different success probabilities p ₁, p ₂, ..., p _n is frequently approximated by a Poisson distribution with parameter λ = p ₁ + p ₂ + ... + p _n . LeCam's bound p ² ₁ + p ² ₂ + ... + p _n ² for the total variation distance between both distributions is particularly useful provided the success probabilities are small. The paper presents an improved version of LeCam's bound if a generalized d-dimensional Poisson binomial distribution is to be approximated by a compound Poisson distribution. Received: May 10, 2000; revised version: January 15, 2001     

Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes

We derive approximating formulas for the mean and the variance of an autocorrelation estimator which are of practical use over the entire range of the autocorrelation coefficient ρ. The least-squares estimator ∑ ^{n −1} _{i =1}ε _i ε _{i +1} / ∑ ^{n −1} _{i =1}ε² _i is studied for a stationary AR(1) process with known mean. We use the second order Taylor expansion of a ratio, and employ the arithmetic-geometric series instead of replacing partial Cesàro sums. In case of the mean we derive Marriott and Pope's (1954) formula, with (n− 1)⁻¹ instead of (n)⁻¹, and an additional term α (n− 1)⁻². This new formula produces the expected decline to zero negative bias as ρ approaches unity. In case of the variance Bartlett's (1946) formula results, with (n− 1)⁻¹ instead of (n)⁻¹. The theoretical expressions are corroborated with a simulation experiment. A comparison shows that our formula for the mean is more accurate than the higher-order approximation of White (1961), for |ρ| > 0.88 and n≥ 20. In principal, the presented method can be used to derive approximating formulas for other estimators and processes. Received: November 30, 1999; revised version: July 3, 2000     

On confidence intervals for nonmonotone parametric functions and an application to the squared mean of the normal distribution

Consider the problem of obtaining a confidence interval for some function g(θ) of an unknown parameter θ, for which a (1-α)-confidence interval is given. If g(θ) is one-to-one the solution is immediate. However, if g is not one-to-one the problem is more complex and depends on the structure of g. In this note the situation where g is a nonmonotone convex function is considered. Based on some inequality, a confidence interval for g(θ) with confidence level at least 1-α is obtained from the given (1-α) confidence interval on θ. Such a result is then applied to the n(μ, σ ²) distribution with σ known. It is shown that the coverage probability of the resulting confidence interval, while being greater than 1-α, has in addition an upper bound which does not exceed Θ(3z_1−α/2)-α/2.     

Exchangeability,predictive sufficiency and Bayesian bootstrap

Summary Let {X _n } be a sequence of random variables conditionally independent and identically distributed given the random variable Θ. The aim of this paper is to show that in many interesting situations the conditional distribution of Θ, given (X ₁,…,X _n ), can be approximated by means of the bootstrap procedure proposed by Efron and applied to a statisticT _n (X ₁,…,X _n ) sufficient for predictive purposes. It will also be shown that, from the predictive point of view, this is consistent with the results obtained following a common Bayesian approach.     

Goodness-of-fit testing of a weak memoryless property of life distributions

A life distribution is said to have a weak memoryless property if its conditional probability of survival beyond a fixed time point is equal to its (unconditional) survival probability at that point. Goodness‐of‐fit testing of this notion is proposed in the current investigation, both when the fixed time point is known and when it is unknown but estimable from the data. The limiting behaviour of the proposed test statistic is obtained and the null variance is explicitly given. The empirical power of the test is evaluated for a commonly known alternative using Monte Carlo methods, showing that the test performs well. The case when the fixed time point t₀ equals a quantile of the distribution F gives a distribution‐free test procedure. The procedure works even if t₀ is unknown but is estimable.     

An analysis of quantile measures of kurtosis: center and tails

The consequences of substituting the denominator Q ₃(p)  −  Q ₁(p) by Q ₂  −  Q ₁(p) in Groeneveld’s class of quantile measures of kurtosis (γ ₂(p)) for symmetric distributions, are explored using the symmetric influence function. The relationship between the measure γ ₂(p) and the alternative class of kurtosis measures κ₂(p) is derived together with the relationship between their influence functions. The Laplace, Logistic, symmetric Two-sided Power, Tukey and Beta distributions are considered in the examples in order to discuss the results obtained pertaining to unimodal, heavy tailed, bounded domain and U-shaped distributions. The authors thank the referee for the careful review.     

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.