Sample Quantiles Research Articles

On the Probability that a Sample Distribution Function Lies Below a Line Segment

The probability is determined that the sample distribution function (df) of a random sample of any size $n$, drawn from the uniform distribution, lies below a given line segment of any slope over some $(\alpha, \beta) \subset \lbrack 0, 1\rbrack$ (Section 1, Theorem 1.1 ff.). Probabilities of related events, also under conditioning, are derived. It is well known that results of this type are equivalent to similar ones for random samples from any continuous df. A catalogue of equivalent formulae is given, the various versions being advantageous on certain ranges of the parameters. These results rest upon and generalize a formula (Theorem 2.1 below) of Dempster (1959), Dwass (1959), and Pyke (1959) (his Lemma 1), which gives an explicit expression for any $n$ that the sample df lies below some (straight) line extended over the entire unit interval. Otherwise the proof uses familiar properties of order statistics, the whole argument being essentially a combinatorial one. The present result also generalizes, in particular, results by Wald and Wolfowitz (1939), by Birnbaum and Tingey (1951) (in both papers sample df below a line segment of slope 1 over [0,1]; especially the latter paper, which improves the first, is at the root of the approach of the present article as well as of the papers by Dempster, Dwass, Pyke), by Smirnov (1944 and 1961) (sample df below a line segment with slope 1 over (0,1) or $(\alpha, 1)$), by Chang (1955) (line segment of any nonnegative slope joining the origin with some point in the open unit square $I_2^0$), Csorgo (1965) (line segment of any slope may also end in the point (1, 1)), Birnbaum and Lientz (1969) (line segment of any nonnegative slope through the origin over any subinterval of the unit interval). Apparently the first author who determined explicitly the probability that the sample df lies below a line segment over an arbitrary subinterval $(\alpha, \beta)\subset\lbrack 0, 1\rbrack$ was Takacs (1964) (Theorem 3; see also Takacs (1967), pages 176-178). (The author is indebted to Professor J. Kiefer for reminding him of this reference.) However, he had the (not very crucial) restriction that the slope $\gamma$ of the line be $\geqq 1$. Moreover, his formula contains a double sum, whereas some of ours contain a single one. This fact proves to be of great advantage in the applications we have made so far. Theorem 1.2 below gives certain conditional probabilities that the sample df lies below a line segment over some subinterval of (0,1). These probability expressions (as any of those described above), for a suitable sequence of line segments depending on $n$ tend to the probability that the Brownian bridge (or conditioned Wiener) process lies below a line segment, as $n \rightarrow \infty$ (see [18], in particular pages 182-183). This follows from the Doob-Donsker theorem on weak convergence of probability measures (see also [13]). These asymptotic results provide a starting point different from and possibly simpler than that of Section 1 to compute approximate probabilities that the sample df lies below some curved line (for an application and some of the asymptotic formulae compare [7] and [17]). It is the objective of a later paper to determine the error of the approximation. Theorem 1.3 gives similar conditional probability expressions that the sample probability (a generalization of the sample df, defined below) for intervals of variable lengths lies below a line segment. It is these formulae that are needed in a new version of the proof of the Bahadur-Kiefer representation theorem for sample quantiles [1]. The formulae derived here are explicit (though involved) in contrast to recursive ones first considered in [12] and extended, more recently, e.g., in [9] and [25]. They have been used, e.g., to compute explicitly the probability that the sample df lies below a polygon (compare a forthcoming paper of the author). Another application is the derivation of asymptotic formulae paralleling and generalizing, e.g., those of Renyi and Csorgo ([4] and [10]). Implications for stochastic processes (such as those studied in [19] and for the theory of goodness of fit tests (compare [1]) are not considered. In [1] also the statistic $F_n(x) - F(x)$, divided by its standard deviation, has been proposed. Certain functionals of this statistic may be studied by the method of the present paper. On the other hand, the method is not immediately applicable to the study of probabilities that the sample df lies, e.g., between two lines (for a recent paper on this compare [16]). The vast existing literature on Kolmogorov-Smirnov type statistics (for recent surveys compare [20] and the appendix of [12a]) is not being surveyed for possible applications other than the few above-mentioned representative examples.

Linear Estimation of the Location and Scale Parameters of the Cauchy Distribution Based on Sample Quantiles

Abstract The asymptotically best linear estimate of the location or scale parameter, assuming one is known, based on a few, say k, order statistics selected from a large sample is considered. Through an analytic approach, some spacings (a spacing is a set of proportions 0 2.

Linear Estimation of the Normal Distribution Standard Deviation

Recently in a series of articles ([1], [2], [3], [6], [7], [8]) the estimation of the standard deviation of the normal distribution based on the sample standard deviation, s, has been thoroughly explored-both for unbiased estimation ([1], [2], [3], [6]) and minimum mean square error estimation ([7], [8]). If one is willing to retain the assumption of normality then it is well known that many simple highly efficient linear estimates of the standard deviation also exist (for example in [4] Chapter 9). The usual drawbacks of these linear estimates are that they are appropriate either to small samples (e.g., the sample range) or large samples (e.g., estimates using a few sample quantiles) or else they do apply to all sample sizes (e.g., the BLUE estimates) but involve elaborate weighting schemes for the ordered observations-thus rendering them chiefly only of academic interest. In the following we examine one linear estimate first proposed by Downton [5]. It does not suffer the above mentioned drawbacks and further it compares extremely well with the estimates based on s. Let X1,n ? ? Xn,n represent the ordered observations from a random sample of size n drawn from a normal distribution. Downton's unbiased estimate of the distribution's standard deviation ois

Linear Estimation of the Location and Scale Parameters of the Cauchy Distribution Based on Sample Quantiles

Linear Estimation of the Location and Scale Parameters of the Cauchy Distribution Based on Sample Quantiles

Confidence intervals for the parameters of the logistic distribution

where a and b are location and scale parameters. The range of application of the logistic distribution as a probability model to describe random phenomenon covers such areas as psychosensory response systems, population growth, bioassay, life tests and physiochemical phenomena. Harter & Moore (1967) have given an extensive review of the problem of point estimation of the parameters in the logistic distribution using a censored sample. They evaluate the maximum likelihood estimators and the best linear unbiased estimators for the parameters, stating that the precision of the two is nearly the same for symmetric, or no, censoring, while the maximum likelihood estimators are better for strongly asymmetric censoring. Their evaluation is given for samples of size 10. They also present the asymptotic variances and covariances of the maximum likelihood estimators for 30 different censoring proportions. Gupta & Gnanadesikan (1966) have used the sample quantiles to estimate the parameters as proposed by Ogawa (1951). They indicate that Ogawa's estimator of a with b known is rather good while his estimator of b with a unknown is fair. They also evaluate the estimators, also linear in the order statistics, proposed by Jung (1955) and Blom (1958, p. 120, equation 10.3.12). When a and b are both unknown, they recommend that Blom's estimator of a and Jung's estimator of b be used, stating that these are nearly as efficient as the best linear unbiased estimators. Our concern is with inference based on the maximum likelihood estimators of a and b, which we shall denote by a and &. While there are examples, for example, the double exponential, in which the maximum likelihood estimators are not as good as the best linear unbiased estimators, it appears that they are always somewhat better in the case of the logistic distribution. For symmetric censoring, 2 is always unbiased and an unbiasing factor for 8 that depends only upon the sample size n, can be determined; this factor can be obtained by Monte Carlo methods. The task of finding the maximum likelihood estimators is easily accomplished by a brief computer program or with a bit of patience and a desk calculator, though the latter method

Estimation of the Parameters of the Logistic Distribution by Sample Quantiles

Estimation of the Parameters of the Logistic Distribution by Sample Quantiles

The asymptotic joint distribution of an increasing number of sample quantiles

The asymptotic joint distribution of an increasing number of sample quantiles

Asymptotically Nearly Efficient Estimators of Multivariate Location Parameters

Common estimates of multivariate location parameters have the property that each component of the parameter is estimated using only the corresponding component of the observations. This is true of the sample mean, sample median and the vector of medians of averages (studied in [1]), as well as of the rank-order statistics often applied to testing for location. In some cases, particularly the multivariate normal, such estimators achieve asymptotic efficiency, but in general information is lost. This paper presents three methods of estimating multivariate location parameters which use more information than is available in the marginal distributions. These classes of estimators are asymptotically nearly efficient (ANE), in the sense that for every $\epsilon > 0$ there is an estimator in the class with asymptotic efficiency $> 1 - \epsilon$ (if efficiency is measured by a comparison of the asymptotic covariance matrix to the inverse of the information matrix). Our ANE estimators are motivated by those of Ogawa [6] for univariate location parameters. Ogawa obtained the asymptotically minimum-variance asymptotically unbiased estimator (ABLUE) for location or scale from a chosen set of sample quantiles. It was soon observed (Tischendorf [10]) that the reciprocal of the asymptotic variance of Ogawa's estimator (properly normalized) is essentially a Riemann sum for the information integral for the parameter being estimated. Thus under mild regularity conditions the ABLUE approaches asymptotic efficiency as larger sets of more closely spaced quantiles are chosen for use. Ogawa's estimators are therefore ANE for univariate location parameters. In the present paper we describe three classes of ANE estimators for multivariate location parameters. The first two consist of linear estimators, and represent multivariate generalizations of Ogawa's ANE class. Our three classes are as follows: (1) Choose a set of marginal sample quantiles in each direction from a continuous $r$-variate location parameter distribution. These quantiles generate a random partition of Euclidean $r$-space $R_r$, and for $r > 1$ the observed cell frequencies contain additional information. We obtain the ABLUE's in terms of the sample quantiles and the observed cell frequencies for $r = 2$ and show that they are ANE. (2) For all $r > 1$, linear ANE estimators are obtained by choosing a single sample quantile in each direction and partitioning $R_r$ by marking off fixed distances from these. The ABLUE's in terms of the $r$ chosen quantiles and the observed cell frequencies are ANE. These estimators have much simpler coefficients than do those of class (1). (3) Finally, ANE estimators can be obtained by exploiting analytic properties of RBAN estimators (Neyman [5]) for a sequence of multinomial problems related to the given location parameter family. These estimators are usually not expressed in closed form. Ogawa proceeded by applying least squares theory to the asymptotic distribution of his chosen set of sample quantiles. The ABLUE's of classes (1) and (2) are here derived by the same method, but establishing the joint asymptotic distribution of the marginal sample quantiles and the observed cell frequencies is non-trivial. Our method is to reduce the problem to one involving the multinomial distribution. A similar idea was used by Weiss [11] to obtain the joint asymptotic distribution of the quantiles alone, but the present problem requires more elaborate arguments. Section 2 contains a preliminary result for the multinomial distribution. The ANE classes (1) and (2) are discussed in Section 3, while Section 4 presents the third class. Estimators of all three classes will require use of a computer if the distribution function cannot be expressed in closed form, and may therefore be of limited practical usefulness. Section 5, however, contains an example for which estimators of classes (1) and (2) can be computed with relative ease. For this example, a bivariate logistic distribution, the performance of our estimators is compared with that of the sample mean and median and of Ogawa's univariate ANE estimators. Throughout, $K$ denotes a generic positive constant, $\mathscr{L}\{X\}$ is the probability law of the random variable $X$, and $\mathscr{L}\{X_n\} \rightarrow \mathscr{L}\{X\}$ designates convergence in law. $N(\mu, \Sigma)$ is the normal law with mean $\mu$ and covariance matrix $\Sigma$ (which may be $1 \times 1$).

Estimation of the parameters of the logistic distribution by sample quantiles

Estimation of the parameters of the logistic distribution by sample quantiles

Estimation of the parameters of the extreme value distribution by use of two or three order statistics

SUMMARY This paper discusses the use of order statistics in estimating the parameters of the extreme value distribution. The best linear unbiased estimator of the location and scale parameter is given, based on two or three order statistics, for samples up to size 20. For very large samples, the analogous estimators based on sample quantiles are obtained.

Testing the Mean and Standard Deviation of a Normal Distribution Using Quantiles

Assuming a normally distributed population and a large sample size, three tests of hypotheses are presented using sample quantiles. First, we test whether μ = μ1 and σ = σ1 or μ = μ2 and σ = σ2, using one optimum quantile. Then test statistics and acceptance regions are given for testing the mean and standard deviation independently. The efficiency of each test is given relative to the best test using all the sample values.

Asymptotic Normality of Sample Quantiles for $m$-Dependent Processes

The usual technique of deriving the asymptotic normality of a quantile of a sample in which the random variables are all independent and identically distributed [cf. Cramer (1946), pp. 367-369] fails to provide the same result for an $m$-dependent (and possibly non-stationary) process, where the successive observations are not independent and the (marginal) distributions are not necessarily all identical. For this reason, the derivation of the asymptotic normality is approached here indirectly. It is shown that under certain mild restrictions, the asymptotic almost sure equivalence of the standardized forms of a sample quantile and the empirical distribution function at the corresponding population quantile, studied by Bahadur (1966) [see also Kiefer (1967)] for a stationary independent process, extends to an $m$-dependent process, not necessarily stationary. Conclusions about the asymptotic normality of sample quantiles then follow by utilizing this equivalence in conjunction with the asymptotic normality of the empirical distribution function under suitable restrictions. For this purpose, the results of Hoeffding (1963) and Hoeffding and Robbins (1948) are extensively used. Useful applications of the derived results are also indicated.

Analysis of Extreme-Value Data by Sample Quantiles for Very Large Samples

This paper deals with estimating the location and the scale parameters of the extreme value distribution when the sample size is very large, using sample quantiles. An estimator is given for the location parameter when the scale parameter is known, based on one or more (up to 15) order statistics. Also given is an estimator for the scale parameter when the location parameter is known, built on two order statistics. An iterative procedure is utilized to estimate the parameters when both are unknown, using two order statistics. The problem of estimating the percentage point is also dealt with, and a comparison with Lieblein's method is included.

Analysis of Extreme-Value Data by Sample Quantiles for Very Large Samples

Analysis of Extreme-Value Data by Sample Quantiles for Very Large Samples

On the Uniform Convergences of the Distributions of Normed Sample Quantiles

On the Uniform Convergences of the Distributions of Normed Sample Quantiles

An Inequality for Expected Values of Sample Quantiles

An Inequality for Expected Values of Sample Quantiles

On Bahadur's Representation of Sample Quantiles

On Bahadur's Representation of Sample Quantiles

Estimation of the Parameters of the Exponential Distribution Based on Optimum Order Statistics in Censored Samples

In the theory of estimation it is well known that when all the observations in a sample are available, it is sometimes possible to find estimators that are the most efficient linear combinations of a given number of order statistics. In many practical situations, we encounter censored samples, that is, samples where values of some of the observations are not available, and it is desired to obtain linear estimators based on a few optimal order statistics from such a sample. The present study concerns the determination of the optimal set of order statistics for a given integer $k$ (where $k$ is much less than the number of observations in the censored sample), in estimating the parameters of the exponential distribution when the sample is censored. The study is based on the asymptotic theory and under Type II censoring scheme. The problem of estimation of the parameters of the exponential distribution in censored samples has been considered by Sarhan and Greenberg (1957). The choice of optimal set of order statistics for the scale parameter alone in a left censored sample has been studied numerically by Sarhan, Greenberg and Ogawa (1963). For the estimation of the parameters of exponential distribution based on optimal set of order statistics, we present in Section 2 the asymptotically best linear unbiased estimates (BLUE's) of the parameters based on $k$ sample quantiles of given orders when all the sample values are available and define the censored samples considered. In Sections 3-5, the detailed treatment for the determination of the $k$ optimum order statistics in singly and doubly censored sample is presented. In Section 6 some extremal properties of a related function are given. The results are always referred to in the text of Sections 2 to 5 to establish uniqueness of the optimal order statistics so determined. Further, for $k = 2(1)4$ and proportion of censoring the right $1 - \beta = .05(.05).40$, Table I has been prepared for the estimation of the scale parameter (assuming the location parameter known) furnishing the coefficients of the BLUE and the spacings corresponding to the optimum order statistics. For $k = 2(1)4$ and equal proportions of censoring on both sides from .05 to .25 at steps of .05, Table II has been prepared for the simultaneous estimation of the location and the scale parameters.

Systematic Statistics Used for Data Compression in Space Telemetry

Abstract The need for data compression, a consequence of the demands made on the telemetry system of a space vehicle, prompts consideration of the use of sample quantiles in estimating population parameters and obtaining tests of goodness of fit for large samples. In this paper optimal unbiased estimators of the mean and standard deviation are given using up to twenty quantiles when the parent population is normal. Moreover, the estimators are relatively insensitive to deviations from normality. A distribution-free goodness-of-fit test is presented based on the sum of the squares of four quantiles after an orthogonal transformation to independent normal deviates. If a frequency function is of the form f(x; p) = pf 1(x) + (1 - p) f 2(x), 0 < p < 1, where f 1 and f 2 are normal frequency functions, the distribution is likely to be bimodal. Another goodness-of-fit test is obtained using four quantiles, which is likely to have considerable power with a null hypothesis of normality and the alternative hypothesis of bimodality. The “data compression ratios” obtained with the use of a quantile system can be on the order of 100 to 1.

Systematic Statistics Used for Data Compression in Space Telemetry

Abstract The need for data compression, a consequence of the demands made on the telemetry system of a space vehicle, prompts consideration of the use of sample quantiles in estimating population parameters and obtaining tests of goodness of fit for large samples. In this paper optimal unbiased estimators of the mean and standard deviation are given using up to twenty quantiles when the parent population is normal. Moreover, the estimators are relatively insensitive to deviations from normality. A distribution-free goodness-of-fit test is presented based on the sum of the squares of four quantiles after an orthogonal transformation to independent normal deviates. If a frequency function is of the form f(x; p) = pf 1(x) + (1 - p) f 2(x), 0 < p < 1, where f 1 and f 2 are normal frequency functions, the distribution is likely to be bimodal. Another goodness-of-fit test is obtained using four quantiles, which is likely to have considerable power with a null hypothesis of normality and the alternative hypothes...