Linear Combinations Of Order Statistics Research Articles

BOOTSTRAP APPROXIMATION OF DISTRIBUTIONS OF THE L-STATISTICS

We study consistency conditions of the bootstrap approximation for linear combinations of order statistics with coefficients determined by a weight function. Bibliography: 10 titles.

Characterization of Half-Normal Distribution Based on Linear Combinations of Order Statistics

In this paper two linear combinations of order statistics of two independent and identically distributed variables have been used to characterize the half-normal distribution.

Control charts based on linear combinations of order statistics

The last 20 years have seen an increasing emphasis on statistical process control as a practical approach to reducing variability in industrial applications. Control charts are used to detect problems such as outliers or excess variability in subgroup means that may have a special cause. We describe an approach to the computation of control limits for exponentially weighted moving average control charts where the usual statistics in classical charts are replaced by linear combinations of order statistics; in particular, the trimmed mean and Gini's mean difference instead of the mean and range, respectively. Control limits are derived, and simulated average run length experiments show the trimmed control charts to be less influenced by extreme observations than their classical counterparts, and lead to tighter control limits. An example is given that illustrates the benefits of the proposed charts. parameters; see, for example, Hunter (1986) and Montgomery (1996). On the other hand, EWMA charts have been shown to be more efficient than Shewharttype charts in detecting small shifts in the process mean; see, for example, Ng & Case (1989), Crowder (1989), Lucas & Saccucci (1990), Amin & Searcy (1991) and Wetherill & Brown (1991). In fact, the EWMA control chart has become popular for monitoring a process mean; see Hunter (1986) for a good discussion. More recently, EWMA charts have been developed for monitoring process variability;

Stability of order statistics under dependence

We precisely evaluate the upper and lower deviations of the expectation of every order statistic from an i.i.d. sample under arbitrary violations of the independence assumption, measured in scale units generated by various central absolute moments of the parent distribution of a single observation. We also determine the distributions for which the bounds are attained. The proof is based on combining the Moriguti monotone approximation of functions with the Holder inequality applied for proper integral representations of expected order statistics in the independent and dependent cases. The method allows us to derive analogous bounds for arbitrary linear combinations of order statistics.

On the Asymptotic Deficiency of a Test Based on a Linear Combination of Order Statistics

On the Asymptotic Deficiency of a Test Based on a Linear Combination of Order Statistics

Alternative empirical distributions based on weighted linear combinations of order statistics

A class of empirical distributions is introduced which are based on various weighted linear combinations of order statistics, and which have convergence properties the classical empirical distribution does not, or which stochastically or convexly dominate the classical empirical distribution

Lexicon-driven handwritten word recognition using optimal linear combinations of order statistics

In the standard segmentation-based approach to handwritten word recognition, individual character-class confidence scores are combined via averaging to estimate confidences in the hypothesized identities for a word. We describe a methodology for generating optimal linear combination of order statistics operators for combining character class confidence scores. Experimental results are provided on over 1000 word images.

Detection analysis of linearly combined order statistic CFAR algorithms in nonhomogeneous background environments

Automatic detection radars require some technique of adaptation against variations in the background clutter in order to control their false alarm rate. Recent interest has focused on order statistic based algorithms where they are quite robust against both interference and clutter edges. The primary purpose of this paper is to analyze the performance of a CFAR procedure that uses a linear combination of order statistics (LCOS) as a test statistic for its noise power level estimation. This processor is more general than the trimmed mean scheme. The LCOS-CFAR algorithm reduces to the cell-averaging (CA), order statistic (OS), censored cell averaging (CCA), censored mean level (CML) and trimmed mean (TM) CFAR algorithms for specific parameter values. The performance of the proposed detector is evaluated in the case where the reference sample set contains abrupt change in clutter power distribution amongst its elements and when the content of the sample set is contaminated with returns from the interferers. Besides providing a complete detection analysis for single pulse detection, this paper extends the single pulse analysis to the LCOS-CFAR processor with noncoherent integration of M-pulses when the background environments are nonhomogeneous. The primary and the secondary extraneous targets are assumed to be fluctuated in accordance with the Swerling II target model and closed-form expressions for the false alarm and detection probabilities are derived for the two cases.

Revisiting the estimation of the mean using order statistics

This paper deals with the problem of estimating expected values using Order Statistics of a small sample. Estimators using a linear combination of Order Statistics, are compared to usual linear estimators. It is shown that while the use of Order Statistics improves the estimation of white sequences, linear estimators are sometimes more effective for colored sequences. Thus, a new estimator based on both the variates and their Order Statistics is developed. It is shown that this new estimator performs at least as well as Order Statistics estimators, and at least as well as linear estimators. An adaptive scheme of these estimators is then given and applied to estimate a binary noisy signal.

On the trimmed mean and minimax‐variance L‐estimation in Kolmogorov neighbourhoods

AbstractWe consider the properties of the trimmed mean, as regards minimax‐variance L‐estimation of a location parameter in a Kolmogorov neighbourhood K() of the normal distribution:We first review some results on the search for an L‐minimax estimator in this neighbourhood, i.e. a linear combination of order statistics whose maximum variance in Kt() is a minimum in the class of L‐estimators. The natural candidate – the L‐estimate which is efficient for that member of Kt,() with minimum Fisher information – is known not to be a saddlepoint solution to the minimax problem. We show here that it is not a solution at all. We do this by showing that a smaller maximum variance is attained by an appropriately trimmed mean. We argue that this trimmed mean, as well as being computationally simple – much simpler than the efficient L‐estimate referred to above, and simpler than the minimax M‐ and R‐estimators – is at least “nearly” minimax.

A Berry-Esseen Bound for Linear Combinations of Order Statistics

The rate of convergence of the distribution function of a linear combination of order statistics of n independent and identically distributed random variables with a common distribution function F to its normal limit is investigated. Under the assumptions some α 1, , α 2, β 2 > 0 and with some 0 ≤ κ < 4/3 and appropriate moment conditions a Berry-Esseen bound is given. If the coefficients are generated by a sequence of weight functions of a special structure, then the rate is shown to be . Finally, the result is applied for a statistic, which is widely used in auditing.

Berry-Esseen Bounds for Statistics of Weakly Dependent Samples

We prove Berry--Esseen bounds for a general class of asymptotically normal statistics which are functions of N weakly dependent random variables under easily verifiable conditions. In particular, we show, for some δ >0 , the validity of the bound O (N - 1/2log δ N) for U -statistics, studentized means, functions of sample means, functionals of empirical distribution functions and linear combinations of order statistics.

A formula for deficiency (L-andR-tests)

In this paper an analogue of the formulas [D. M. Chibisov,Teor. Veroyatn. Primen.,30, 269–288 (1985);Izv. Akad. Nauk UzSSR,6, 23–30 (1982)] for the difference between the power of a given asymptotically efficient test and that of the most powerful test is justified for one-sample L-and R-tests, i.e., tests based on linear combinations of order statistics and linear rank statistics. This formula directly yields the Hodges-Lehmann deficiency of corresponding tests. A general theorem is stated which is applied to L-and R-tests. The explicit expressions given by this formula for L- and R-tests are also presented. The expression related to R-tests agrees with the one obtained in [W. Albers, P. J. Bickel, and W. R. Van Zwet,Ann. Statist.,4, 108–156 (1976);6, 1170–1171 (1978)]. We present here a nontechnical (heuristic) proof of these results.

OS-CFAR thresholding in decentralized radar systems

In a decentralized detection scheme, several sensors perform a binary (hard) decision and send the resulting data to a fusion center for the final decision. If each local decision has a constant false alarm rate (CFAR), the final decision is ensured to be CFAR. We consider the case that each local decision is a threshold decision, and the threshold is proportional, through a suitable multiplier, to a linear combination of order statistics (OS) from a reference set (a generalization of the concept of OS thresholding). We address the following problem: given the fusion rule and the relevant system parameters, select each threshold multiplier and the coefficients of each linear combination so as to maximize the overall probability of detection for constrained probability of false alarm. By a Lagrangian maximization approach, we obtain a general solution to this problem and closed-form solutions for the AND and OR fusion logics. A performance assessment is carried on, showing a global superiority of the OR fusion rule in terms of detection probability (for operating conditions matching the design assumptions) and of robustness (when these do not match). We also investigate the effect of the hard quantization performed at the local sensors, by comparing the said performance to those achievable by the same fusion rule in the limiting case of no quantization.

Strong laws for 𝐿- and 𝑢-statistics

Strong laws of large numbers are given for L L -statistics (linear combinations of order statistics) and for U U -statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.

On the rate of convergence ofL- andR-statistics under alternatives

Asymptotic distributions of test statistics under alternatives are important from the point of view of their power properties. When the limiting distributions of test statistics are specified under the hypothesis in a certain sense, LeCam's third lemma ([4], Chapter 6) enables one to obtain their limiting distributions under close alternatives. In this paper we generalize LeCam's third lemma by using the rate of convergence in the case of asymptotically efficient test statistics. A general lemma is proved which is specified to linear combinations of order statistics (L-statistics) and linear rank statistics (R-statistics). Edgeworth-type asymptotic expansions for these statistics under alternatives are considered in [3].

Asymptotic expansions forL- andR-statistics under alternatives

In this paper we establish asymptotic expansions (a.e.) under alternatives for the distribution functions of sums of independent identically distributed random variables (i.i.d.r.v.'s.), linear combinations of order statistics, and one-sample rank statistics (L- and R-statistics). The general Lemma from [V. E. Bening,Bull. Moscow State Univ., Ser. 15, 2 36–44 (1994)] is applied to these statistics. Section 1 contains the statement of the theorem, in Sec. 2 the theorems is proved; its proof involves four auxiliary lemmas, also contained in Sec. 2. Finally Sec. 3 contains the proofs of these lemmas.

On analogues of the berry?esseen inequality for L-statistics

In this paper we present estimates for the rate of convergence of distributions of linear combinations of order statistics (L-statistics) to the normal law, having an optimal order under light moment assumptions regarding the initial distribution.

$L$-Statistics in Complex Survey Problems

We study linear combinations of order statistics ($L$-statistics) in survey problems with a stratified multistage sampling design. Two general types of $L$-statistics are considered: smooth $L$-statistics with weights generated by a smooth function and nonsmooth $L$-statistics (sample quantiles). The trimmed sample mean, the decile mean and variance, the sample Lorenz curve and the sample Gini's family parameters are examples of smooth $L$-statistics or functions of smooth $L$-statistics used in survey problems. It is shown that under weak conditions the smooth $L$-statistics are asymptotically normal and their asymptotic variances can be consistently estimated by jackknifing. For the sample quantiles, their asymptotic normality requires more conditions on the finite population distribution functions. Consistent estimators for the asymptotic variances of the sample quantiles are derived. Asymptotic validity of the Woodruff's confidence intervals for population quantiles is also proved.

Asymptotic distributions οf linear combinations of order statistics

We study the asymptotic distributions of linear combinations of order statistics (L-statistics) which can be expressed as differentiable statistical functionals and we obtain Berry-Esseen type bounds and the Edgeworth series for the distribution functions