Estimation of P[Y < X] for generalized exponential distribution

This paper deals with the estimation of P[Y < X] when X and Y are two independent generalized exponential distributions with different shape parameters but having the same scale parameters. The maximum likelihood estimator and its asymptotic distribution is obtained. The asymptotic distribution is used to construct an asymptotic confidence interval of P[Y < X]. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator and Bayes estimator of P[Y < X] are obtained. Different confidence intervals are proposed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a simulated data set has also been presented for illustrative purposes.Part of the work was supported by a grant from the Natural Sciences and Engineering Research Council     

Design and estimation problems when estimating a regression coefficient from survey data

Summary The values of a variablex are assumed known for all elements in a finite population. Between this variable and another variableY, whose values are registered in a sample survey, there is the usual linear regression relationship. This paper considers problems of design and of estimation of the regression coefficienta and the interceptb. The followingGodambe type theorem is proved: There exists no minimum variance unbiased linear estimator ofa andb. We also derive that the usual estimators ofa andb have minimum variance if attention is restricted to the class of linear estimators unbiased in any given sample.     

Model‐Assisted Regression Estimators for Longitudinal Data with Nonignorable Dropout

Estimation with longitudinal Y having nonignorable dropout is considered when the joint distribution of Y and covariate X is nonparametric and the dropout propensity conditional on (Y,X) is parametric. We apply the generalised method of moments to estimate the parameters in the nonignorable dropout propensity based on estimating equations constructed using an instrument Z, which is part of X related to Y but unrelated to the dropout propensity conditioned on Y and other covariates. Population means and other parameters in the nonparametric distribution of Y can be estimated based on inverse propensity weighting with estimated propensity. To improve efficiency, we derive a model‐assisted regression estimator making use of information provided by the covariates and previously observed Y‐values in the longitudinal setting. The model‐assisted regression estimator is protected from model misspecification and is asymptotically normal and more efficient when the working models are correct and some other conditions are satisfied. The finite‐sample performance of the estimators is studied through simulation, and an application to the HIV‐CD4 data set is also presented as illustration.     

Biases,variances and covariances of raking ratio estimators for marginal and cell totals and averages of observed characteristics

Summary On the basis of a simple random sample from a population, on which a cross-classification is defined with known marginal frequenciesN _i. andN _.j , one wishes to estimate the cell frequenciesN _ij , as well as cell totalsY _ij , marginal totalsY _i. andY _.j , and the grand totalY for characteristics measured on the units. Various authors have discussed so-called raking ratio estimators, which are built up from the estimated cell values by addition. They have given the bias and variance of this estimator ofY. This paper derives biases, variances and covariances for the corresponding estimators of the cell and marginal totals and of the corresponding marginal averages.     

Tendency towards normality of linear combinations of random variables

IfX andY are two random variables with the same means and variances, thenX is said to be nearer normal thanY if the absolute values of its cumulants are smaller than the corresponding cumulants ofY. Using this definition, it is shown that a linear combination of a finite number of independent identically distributed random variables is always nearer normal than its constituents, but that this is not necessarily true if not-identically distributed or not-independent variables are used. Some consequences of the results are reached for the testing of normality of time series and for the assumptions frequently made by social scientists about the distribution of their data.     

Semiparametric Estimator for Binary‐outcome Sample Selection: Prejudice Matters in Election

a semiparametric estimator for binary‐outcome sample‐selection models is proposed that imposes only single index assumptions on the selection and outcome equations without specifying the error term distribution. I adopt the idea in Lewbel (2000) using a ‘special regressor’ to transform the binary response Y so that the transformed Y becomes linear in the latent index, which then makes it possible to remove the selection correction term by differencing the transformed Y equation. There are various versions of the estimator, which perform differently trading off bias and variance. A simulation study is conducted, and then I apply the estimators to US presidential election data in 2008 and 2012 to assess the impact of racial prejudice on the elections, as a black candidate was involved for the first time ever in the US history.     

On testing for independence against right tail increasing in bivariate models

A random variableY is right tail increasing (RTI) inX if the failure rate of the conditional distribution ofX givenY>y is uniformly smaller than that of the marginal distribution ofX for everyy0. This concept of positive dependence is not symmetric inX andY and is stronger than the notion of positive quadrant dependence. In this paper we consider the problem of testing for independence against the alternative thatY is RTI inX. We propose two distribution-free tests and obtain their limiting null distributions. The proposed tests are compared to Kendall's and Spearman's tests in terms of Pitman asymptotic relative efficiency. We have also conducted a Monte Carlo study to compare the powers of these tests.Research supported by an NSERC Canada operating grant at the University of Alberta.Part of this research was done while visiting the University of Alberta supported by the NSERC Canada grant of the first author.     

Sulla somma di due numeri aleatori egualmente distribuiti

In questo lavoro si studia il problema di ricerca della distribuzione di probabilità comune da assegnare a due numeri aleatori discreti che assumano i primin valori interi naturali in modo che la loro somma abbia moda di minima probabilità.Il problema è affrontato sia dal punto di vista teorico tramite gli strumenti della programmazione matematica, sia dal punto di vista numerico.

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LetX andY be two random numbers with the same distribution function; in this paper we consider the problem of finding a random numberX+Y having mode with minimal probability. In particular we have considered only the case ofX andY assuming the firstn integer values, so thatp (dimensionn) is the common distribution andq (dimension 2n–1) is the distribution ofX+Y; then the problem is to minimizem=maxq ₁.In the known literature it appears that theoretical results and numerical experience have brought to various conjectures not confermed. In this paper the problem is considered from the mathematical programming point of view. Several theoretical results are obtained even if the full solution of the problem is not reached. Anyway, such results, limiting the search range of a solution, suggested extended numerical testing, also for rather large values ofn, so that non trivial conclusions can be derived.

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Pervenuto il 22-1-82     

Uniform minimum variance unbiased estimators for bivariate families

Summary We consider uniform minimum variance unbiased (UMVU) estimation of an unbiased estimable function of distribution parameters for bivariate truncation (non-regular) parameter families. In particular, we derive the UMVU estimator of the probability thatY is less thanX.     

Strong uniform convergence for the estimator of the regression function under ϕ-mixing conditions

Suppose the observations (X _i,Y _i), i=1,…, n, are ϕ-mixing. The strong uniform convergence and convergence rate for the estimator of the regression function was studied by serveral authors, e.g. G. Collomb (1984), L. Gy?rfi et al. (1989). But the optimal convergence rates are not reached unless the Y _i are bounded or the E exp (a|Y _i|) are bounded for some a>0. Compared with the i.i.d. case the convergence of the Nadaraya-Watson estimator under ϕ-mixing variables needs strong moment conditions. In this paper we study the strong uniform convergence and convergence rate for the improved kernel estimator of the regression function which has been suggested by Cheng P. (1983). Compared with Theorem A in Y. P. Mack and B. Silverman (1982) or Theorem 3.3.1 in L. Gy?rfi et al. (1989), we prove the convergence for this kind of estimators under weaker moment conditions. The optimal convergence rate for the improved kernel estimator is attained under almost the same conditions of Theorem 3.3.2 in L. Gy?rfi et al. (1989). Received: September 1999     

Kernel estimation of a smooth distribution function based on censored data

The problem of estimating a smooth distribution functionF at a pointτ based on randomly right censored data is treated under certain smoothness conditions onF. The asymptotic performance of a certain class of kernel estimators is compared to that of the Kaplan-Meier estimator ofF(τ). It is shown that the relative deficiency of the Kaplan-Meier estimator ofF(τ) with respect to the appropriately chosen kernel type estimator tends to infinity as the sample sizen increases to infinity. Strong uniform consistency and the weak convergence of the normalized process are also proved. Research Surported in part by NIH grant 1R01GM28405.     

On bootstrapping the mode in the nonparametric regression model with random design

In the nonparametric regression model with random design and based on i.i.d. pairs of observations (X _i, Y _i), where the regression function m is given by m(x)=?(Y _i|X _i=x), estimation of the location θ (mode) of a unique maximum of m by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve m is considered. In order to obtain asymptotic confidence intervals for θ, the suitably normalized distribution of is bootstrapped in two ways: we present a paired bootstrap (PB) where resampling is done from the empirical distribution of the pairs of observations and a smoothed paired bootstrap (SPB) where the bootstrap variables are generated from a smooth bivariate density based on the pairs of observations. While the PB requires only relatively small computational effort when carried out in practice, it is shown to work only in the case of vanishing asymptotic bias, i.e. of “undersmoothing” when compared to optimal smoothing for mode estimation. On the other hand, the SPB, although causing more intricate computations, is able to capture the correct amount of bias if the pilot estimator for m oversmoothes. Received: May 2000     

Zur Parameter- und Prozentpunktschätzung von Lebensdauerverteilungen bei kleinem Stichprobenumfang

Summary Some of the many methods developed for estimating parameters or percentage points of the Weibull distribution are compared. It is shown that the known estimation of the reciprocal shape parameter with the aid of a straight line in the extremal probability paper is rather biased for small sample sizes. To avoid the bias, correction factors are given, and the efficiency of the resulting unbiased estimator is calculated for sample sizesn=2, 3, …, 9. Results ofJ. Lieblein concerning the double exponential distribution are slightly modified in order to get best linear unbiased estimators for parameters and for the logarithms of percentage points of the Weibull distribution. Other methods are shortly discussed and a median-unbiased estimator for the shape parameter is derived.      

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

A note on reliability properties of k-record statistics

LetY _k,n denote the nth (upper) k-record value of an infinite sequence of independent, identically distributed random variables with common continuous distribution function F. We show that if the nth k-record valueY _k,n has an increasing failure rate (IFR), thenY _l,n (l<k) andY _k+1,n+1(n≤k+1) also have IFR distributions. On the other hand, ifY _k,n has a decreasing failure rate (DFR), thenY _l,n (1>k) has also a DFR distribution. We also present some results concerning log convexity and log concavity ofY _k,n .     

Estimation in the pareto distribution

The unique minimum variance unbiased (UMVU) estimate of the probability distribution function of the Pareto distribution is derived. It is shown that the distribution function and ther ^th moment associated with the UMVU estimate are also UMVU estimators. The p.d.f. and its estimator are compared graphically. An estimate of the 100p ^th percentile is given. It is seen that a function of this estimator has a chi-square distribution.     

Strong representation of the presmoothed quantile function estimator for censored data

We consider lifetime data subject to right random censorship. In this context, this paper deals with the topic of estimating the distribution function of the lifetime and the corresponding quantile function. As it has been shown that the classical Kaplan–Meier estimator of the distribution function can be improved by means of presmoothing ideas, we introduce a quantile function estimator via the presmoothed distribution function estimator studied by Cao et al. [Journal of Nonparametric statistics, Vol. 17 (2005) pp. 31–56.] The main result of this paper is an almost sure representation of this presmoothed estimator. As a consequence, its strong consistency and asymptotic normality are established. The performance of this new quantile estimator is analyzed in a simulation study and applied to a real data example.     

On the Lp error of the Grenander-type estimator in the Cox model

We consider the Cox regression model and study the asymptotic global behavior of the Grenander-type estimator for a monotone baseline hazard function. This model is not included in the general setting of Durot (2007). However, we show that a similar central limit theorem holds for L_p-error of the Grenander-type estimator. As an illustration of application of our main result, we propose a test procedure for a Weibull baseline distribution, based on the L_p-distance between the Grenander estimator and a parametric estimator of the baseline hazard. Simulation studies are performed to investigate the performance of this test.     

Convergence of a branching type recursion with non-stationary immigration

The asymptotic distribution of a branching type recursion with non-stationary immigration is investigated. The recursion is given by , where (X _l ) is a random sequence, (L _n ⁻¹⁽¹⁾ ) are iid copies ofL _n−1,K is a random number andK, (L _n ⁻¹⁽¹⁾ ), {(X _l ),Y _n } are independent. This recursion has been studied intensively in the literature in the case thatX _l ≥0,K is nonrandom andY _n =0 ∀n. Cramer, Rüschendorf (1996b) treat the above recursion without immigration with starting conditionL ₀=1, and easy to handle cases of the recursion with stationary immigration (i.e. the distribution ofY _n does not depend on the timen). In this paper a general limit theorem will be deduced under natural conditions including square-integrability of the involved random variables. The treatment of the recursion is based on a contraction method. The conditions of the limit theorem are built upon the knowledge of the first two moments ofL _n . In case of stationary immigration a detailed analysis of the first two moments ofL _n leads one to consider 15 different cases. These cases are illustrated graphically and provide a straight forward means to check the conditions and to determine the operator whose unique fixed point is the limit distribution of the normalizedL _n .