The rate of convergence in law of the maximum of an exponential sample

Summary  We derive a uniform rate of convergence of (1– n^-1x)ⁿ to e^-x(x < 0). It provides a uniform rate of convergence for the distribution of the largest order statistic in a sample from an exponential distribution to the "double exponential" extreme value distribution. It likewise provides a rate of convergence for the distribution of the smallest order statistic from a uniform distribution.     

Exploring the relation between the r* approximation and the Edgeworth expansion

In this paper we study the relation between the r* saddlepoint approximation and the Edgeworth expansion when quite general assumptions for the statistic under consideration are fulfilled. We will show that the two term Edgeworth expansion approximates the r* formula up to an O(n ^?3/2) remainder; this provides a new way of looking at the order of the error of the r* approximation. This finding will be used to inspect the close connection between the r* formula and the Edgeworth B adjustment introduced in Phillips (Biometrika 65:91–98, 1978). We will show that, whenever an Edgeworth expansion exists, this adjustment approximates both the distribution function of the statistic and the r* formula to the same order degree as the Edgeworth expansion. Some numerical examples for the sample mean and U-statistics are given in order to shed light on the theoretical discussion.     

PERFORMANCE OF MURPHY'S TEST FOR TWO OUTLIERS IN A LINEAR MODEL

Abstract. In JOSHI and LALITHA (1986) a test for two outliers in the same direction in a linear model is discussed. Here the performance of this statistic is studied. For this, the exact non–null density function of the random variables involved in defining the statistic is obtained. Then a measure of performance is defined and it is applied to the case of a random sample from a normal distribution, as in this case the above said statistic reduces to the well known Murphy's test statistic. These values are then compared with the power values obtained by HAWKINS (1978).     

A linear approximation to the power function of a test

In this paper we obtain a linear approximation to the power function of a test that is very accurate for small sample sizes. This is especially useful for robust tests where not many power functions are available. The approximation is based on the von Mises expansion of the tail probability functional and on the Tail Area Influence Function (TAIF). The goals of the paper are, first to extend the definition of the TAIF to the case of non identically distributed random variables, defining the Partial Tail Area Influence Functions and the Vectorial Tail Area Influence Function; second, to obtain exact expressions for computing these new influence functions; and, finally, to find accurate approximations to the power function, that can be used in the case of non identically distributed random variables. We include some examples of the application of this linear approximation to tests that involve the Huber statistic and also saddlepoint tests, so proving that the approximations apply not only to simple problems but also to complex ones.     

浅谈似然函数的两种不同定义

本文通过两个具体例题表明当连续型总体可能的取值范围不是(-∞,+∞)时,利用第一种似然函数的定义解决点估计问题,学生不仅能够很容易地掌握最大似然估计法,同时对样本来自总体且估计离不开样本这一统计思想加深了理解。     

Some remarks concerning the quotient of sample median and sample range for a sample of size 2n+1 from a normal distribution

Consider an ordered sample ₍₁₎, ₍₂₎,…, _(2n+1) of size 2 n +1 from the normal distribution with parameters μ and . We then have with probability one
₍₁₎ < ₍₂₎ < … < (2 n +1).
The random variable
_n =_(n+1)/_(2n+1)-(1)
that can be described as the quotient of the sample median and the sample range, provides us with an estimate for μ/, that is easy to calculate. To calculate the distribution of h _n is quite a different matter***. The distribution function of h₁, and the density of h₂ are given in section 1. Our results seem hardly promising for general h_n. In section 2 it is shown that h_n is asymptotically normal.
In the sequel we suppose μ= 0 and = 1, i.e. we consider only the "central" distribution. Note that h_n can be used as a test statistic replacing Student's t. In that case the central h_n is all that is needed.     

Nonparametric fixed effects model for panel data with locally stationary regressors

We develop methods for inference in nonparametric time-varying fixed effects panel data models that allow for locally stationary regressors and for the time series length $T$ and cross-section size $N$ both being large. We first develop a pooled nonparametric profile least squares dummy variable approach to estimate the nonparametric function, and establish the optimal convergence rate and asymptotic normality of the resultant estimator. We then propose a test statistic to check whether the bivariate nonparametric function is time-varying or the time effect is separable, and derive the asymptotic distribution of the proposed test statistic. We present several simulated examples and two real data analyses to illustrate the finite sample performance of the proposed methods.     

Admissibility of generalized Bayes estimators in an one parameter nonregular family

Consider an estimation problem under squared error loss in an one parameter nonregular family of distributions with the lower endpoint of the support depending on an unknown parameter. Using Karlin's ([3]) method, sufficient conditions are given for generalized Bayes estimators to be admissible for estimating an arbitrary nonnegative, differentiable, monotone parametric function. The results are then applied to the case when both endpoints of the support of the distribution depend on the parameter . Finally, some examples are subsequently given.Research supported by a grant from Hanyang University, 1989.     

Relative Efficiency of the Fuzzy p-Value Approach to Hypothesis Testing

In missing data problems, it is often the case that there is a natural test statistic for testing a statistical hypothesis had all the data been observed. A fuzzy  p -value approach to hypothesis testing has recently been proposed which is implemented by imputing the missing values in the "complete data" test statistic by values simulated from the conditional null distribution given the observed data. We argue that imputing data in this way will inevitably lead to loss in power. For the case of scalar parameter, we show that the asymptotic efficiency of the score test based on the imputed "complete data" relative to the score test based on the observed data is given by the ratio of the observed data information to the complete data information. Three examples involving probit regression, normal random effects model, and unidentified paired data are used for illustration. For testing linkage disequilibrium based on pooled genotype data, simulation results show that the imputed Neyman Pearson and Fisher exact tests are less powerful than a Wald-type test based on the observed data maximum likelihood estimator. In conclusion, we caution against the routine use of the fuzzy  p -value approach in latent variable or missing data problems and suggest some viable alternatives.     

Exceptional Convergence of the Sample Variance

The case is investigated when Hoeffding's one sample U–statistic theorem for the sample variance S² is not applicable. It is shown that this occurs only when the parent distribution is the two–point distribution with jumps of equal magnitude. For this exceptional case the standardised S² is shown to converge in distribution to (1 – V )√2, where V has chi–square distribution with one degree of freedom.     

A bootstrap-assisted spectral test of white noise under unknown dependence

To test for the white noise null hypothesis, we study the Cramér-von Mises test statistic that is based on the sample spectral distribution function. Since the critical values of the test statistic are difficult to obtain, we propose a blockwise wild bootstrap procedure to approximate its asymptotic null distribution. Using a Hilbert space approach, we establish the weak convergence of the difference between the sample spectral distribution function and the true spectral distribution function, as well as the consistency of bootstrap approximation under mild assumptions. Finite sample results from a simulation study and an empirical data analysis are also reported.     

On order statistics when the sample size has a binomial distribution

R aghunandanan and P atil [1] derived the density function of the i-th order statistic from a sample with random size. For the case that the size has a bionmial distribution, a simpler derivation is given below.     

Approximating the distribution of Greenwood's statistic

Statistics based on uniform spacings are often used in goodness-of-fit problems. In this paper special attention is paid to the distribution of Greenwood's statistic. Although its asymptotic distribution is normal, the normal approximation is extremely bad, even for large sample sizes. It is shown that the Edge-worth expansion yields a considerably better approximation for the distribution of this statistic. Furthermore, an overview is given of the higher order asympto-tics for the sum of functions of uniform spacings, of which Greenwood's statistic is a special case.     

Jointness of growth determinants: Reply to comments by Rodney Strachan,Eduardo Ley and Mark F.J. Steel

This comment contrasts the jointness statistics proposed by Doppelhofer and Weeks ( 2009 ) with alternatives proposed by Strachan ( 2009 ) and Ley and Steel ( 2007 ). In contrast to the alternatives, our jointness statistic constitutes a formal test for dependence over the joint posterior distribution. The essential difference between our proposed measure and the alternatives is that our jointness measures uses the entire joint posterior distribution. We discuss differences in jointness, as well as inclusion and exclusion margins of the joint posterior distribution, and the impact on economic significance using the numerical examples given by Strachan ( 2009 ) and Ley and Steel ( 2009 ). Copyright © 2009 John Wiley & Sons, Ltd.     

EL inference for partially identified models: Large deviations optimality and bootstrap validity

This paper addresses the issue of optimal inference for parameters that are partially identified in models with moment inequalities. There currently exists a variety of inferential methods for use in this setting. However, the question of choosing optimally among contending procedures is unresolved. In this paper, I first consider a canonical large deviations criterion for optimality and show that inference based on the empirical likelihood ratio statistic is optimal. Second, I introduce a new empirical likelihood bootstrap that provides a valid resampling method for moment inequality models and overcomes the implementation challenges that arise as a result of non-pivotal limit distributions. Lastly, I analyze the finite sample properties of the proposed framework using Monte Carlo simulations. The simulation results are encouraging.     

Another look at the Tail Area Influence Function

A new way of computing the Tail Area Influence Function (TAIF) exactly is proposed and a new finite sample robustness measure, based on the TAIF, is introduced. The main properties of this robustness measure are also studied, for both finite and asymptotic sample sizes. Next, a very accurate approximation to the finite sample power function of a test is obtained; this is based on the TAIF plus an iterative procedure. The results are valid when there are no nuisance parameters.     

On the nearness of record values to order statistics from Pitman’s measure of closeness

In the two-sample prediction problem, record values from the present sample may be used as predictors of order statistics from a future sample. In this paper, we investigate the nearness of record statistics (upper and lower) to order statistics from a location-scale family of distributions in the sense of Pitman closeness and discuss the corresponding monotonicity properties. We then determine the closest record value to a specific order statistic from a future sample. Even though in general it depends on the parent distribution, exact and explicit expressions are derived for the required probabilities in the case of exponential and uniform distributions, and some computational results are presented as well. Finally, we consider the mean squared error criterion and examine the corresponding results in the exponential case.     

Conditions for invariance of the multivariate versions of grubbs’ test and bartlett’s test under a general dependency structure

Summary This paper generalizes a result by Stadje (1984) by deriving conditions for which a general dependency structure for multivariate observations, given in Pavur (1987), yields a positive definite covariance structure. This general dependency structure allows the sample covariance matrix to be distributed as a constant times a Wishart random matrix. It is then demonstrated that the maximum squared-radii test and a test for equal population covariance matrices have null distributions which remain unchanged when the new general dependency structure, rather than the usual independence structure, for the vector observations, is assumed. Moreover, under a general dependency structure for which the population covariance matrices are unequal, it is shown that the distribution of the test statistic for testing equal covariance matrices is identical to the distribution of the same test statistic when the population covariance matrices are equal and the observations are independent.     

PERFORMANCE OF STUDENTIZED RANGE STATISTIC FOR TWO OUTLIERS IN A LINEAR MODEL

Abstract. In JOSHI and LALITHA (1986) a statistic K is proposed for detecting two outliers present in the opposite directions in a linear model. This statistic reduces to the Studentized range statistic for the case of a simple random sample from a normal population. In this paper, the performance of this statistic is studied. For this an approximate non–null density of the random variables involved in defining the statistic is derived. An exact density is also obtained by using the methods of LALITHA and JOSHI (1986), where the results are proved for Murphy's statistic. A measure of performance is then defined and an example is also given for highlighting the usage of the above said statistic.     

Testing for the shape parameter of generalized extreme value distribution based on the L_q-likelihood ratio statistic

This paper studies the applications of extreme value theory on analysis for closing price data of the Dow-Jones industrial index and Danish fire insurance claims data. The generalized extreme value (GEV) distribution is considered in analyzing the real data, and the hypothesis testing problem for the shape parameter of GEV distribution is investigated based on a new test statistic—the $L_q$ -likelihood ratio ( $L_q$ R) statistic. The $L_q$ R statistic can be treated as a generalized form of the classical likelihood ratio (LR) statistic. We show that the asymptotic behavior of proposed statistic is characterized by the degree of distortion $q$ . For small and modest sample sizes, the $L_q$ R statistic is still available when $q$ is properly chosen. By simulation studies, the proposed statistic not only performs the asymptotic properties, but also outperforms the classical LR statistic as the sample sizes are modest or even small. Meanwhile, the test power based on the new statistic is also simulated by Monte Carlo methods. At last, the models are diagnosed by graphical methods.