The asymptotic distribution of the sum of weighted squared residuals in binary choice models

It is claimed by some authors that the distribution of the sum of weighted squared residuals, used as a goodness of fit measure in binary choice models, behaves for large n as a x²_n– k–1 distribution. This claim seems to be based on a false analogy with the well–known Pearson x² statistic for frequency tables with a fixed number of cells and cell sizes tending to infinity. We derive the asymptotic (normal) distribution and show that the approximation by the x² distribution in general will not be valid. A new x² test is proposed based on the asymptotic normality of the measure.     

Scheduling identical jobs on uniform parallel machines

We address the problem of scheduling n identical jobs on m uniform parallel machines to optimize scheduling criteria that are nondecreasing in the job completion times. It is well known that this can be formulated as a linear assignment problem, and subsequently solved in O ( n ³) time. We give a more concise formulation for minsum criteria, and show that general minmax criteria can be minimized in O ( n ²) time. We present faster algorithms, requiring only O ( n + m log m ) time for minimizing makespan and total completion time, O ( n log n ) time for minimizing total weighted completion time, maximum lateness, total tardiness and the weighted number of tardy jobs, and O ( n log² n ) time for maximum weighted tardiness. In the case of release dates, we propose an O ( n log n ) algorithm for minimizing makespan, and an O ( mn ^2m+1) time dynamic programming algorithm for minimizing total completion time.     

An improved empirical Bayes test for positive exponential families

We exhibit an empirical Bayes test δ^* _n for a decision problem using a linear error loss in a class of positive exponential families. This empirical Bayes test δ^* _n possesses the asymptotic optimality, and its associated regret converges to zero with rate n ⁻¹(ln n )⁶ This rate of convergence improves the previous results in the literature in the sense that a faster rate of convergence is achieved under much weaker conditions. Examples are presented to illustrate the performance of the empirical Bayes test δ^* _n     

PSEUDO-R2 MEASURES FOR SOME COMMON LIMITED DEPENDENT VARIABLE MODELS

Abstract. A large number of different Pseudo- R ² measures for some common limited dependent variable models are surveyed. Measures include those based solely on the maximized likelihoods with and without the restriction that slope coefficients are zero, those which require further calculations based on parameter estimates of the coefficients and variances and those that are based solely on whether the qualitative predictions of the model are correct or not. The theme of the survey is that while there is no obvious criterion for choosing which Pseudo- R ² to use, if the estimation is in the context of an underlying latent dependent variable model, a case can be made for basing the choice on the strength of the numerical relationship to the OLS- R ² in the latent dependent variable. As such an OLS- R ² can be known in a Monte Carlo simulation, we summarize Monte Carlo results for some important latent dependent variable models (binary probit, ordinal probit and Tobit) and find that a Pseudo- R ² measure due to McKelvey and Zavoina scores consistently well under our criterion. We also very briefly discuss Pseudo- R ² measures for count data, for duration models and for prediction-realization tables.     

Skew probability curves with negative powers of time and related to random walks in series

Summary The well known probability distribution of first arrival times of a particle undergoing random walk or Brownian movement in one dimension is extended to allow for steps in series each in a different medium. Previously this led to considering a certain distribution defined by its cumulants, which form a simple series generalising that for the known distribution. This is illustrated by the particular case of two first passages in series. Approximations to the probability (density) curves are found, each of which consists of a sharp peak followed by a long tail where the ordinates are very nearly proportional to t-^W, W ≥ 3/2. A generalisation can yield smaller W down to ca. 0.1. It is concluded that this explains why negative powers of time are found in so many physiological clearance curves of all kinds. Numerical tables are based on distributions with very smal step times, and give ones built up from the sum of a varying number of steps. The parameters 01 the triangle formed by the inflection tangents are given in order to describe the peaks.     

A LIMIT THEOREM FOR BERNOULLI RV'S AND FELLER'S SHOE PROBLEM

Consider n sets of objects, each set consisting of m distinct types (for instance n place settings each made up of m distinct dishes and silverware pieces.) s items are drawn at random from the mn items. The distribution of the number of complete sets (each consisting of all m items) in the sample of s is asymptotically Poisson distributed with parameter (a /m )^m if s = an ^1–1 and n →∞. This fact can be interpreted in terms of a certain limit theorem for a sequence of i.i.d Bernoulli rv's.     

An efficient variant of the product and ratio estimators

Abstract  This article presents a variant of the usual ratio and product methods of estimation, with the intention 10 improve their efficiency. The first order large sample approximations to the bias and the mean square error of the proposed estimator are obtained and compared with those of the well-known methods (simple expansion, ratio, product, difference and linear regression methods). For a special case, the accuracy of the first order approximation (terms up to the order n-¹ ) is examined by including terms upto the order n^-2 . With suitable choice of a design parameter, the proposed estimator turns out to be superior to the three methods mentioned first. The relation to the other two methods is examined; if the design parameter can be chosen near to the optimal value, the proposed method is seen to be approximately as efficient as the linear regression estimator. Finally some extensions are indicated.     

Asymptotic distributions of φ-divergences of hypothetical and observed frequencies on refined partitions

For a wide class of goodness-of-fit statistics based on φ-divergences between hypothetical cell probabilities and observed relative frequencies, the asymptotic normality is established under the assumption n / m _n →γ∈(0,∞), where n denotes sample size and m _n the number of cells. Related problems of asymptotic distributions of φ-divergence errors, and of φ-divergence deviations of histogram estimators from their expected values, are considered too.     

De omvang van de steekproef bij een enkelvoudig steekproefsysteem

Summary  (Sample size for a single sampling scheme).
The operating characteristic of a sampling scheme may be specified by the producers 1 in 20 risk point ( p ₁), at which the probability of rejecting a batch is 0.05, and the consumers 1 in 20 risk point ( p ₂) at which the probability of accepting a batch of that quality is also 0.05.
A nomogram is given (fig. 2) to determine for single sampling schemes and for given values of p₁ and p ₂ the necessary sample size ( n ) and the allowable number of defectives in the sample ( c ).
The nomogram may reversedly be used to determine the producers and consumers 1 in 20 risk points for a given single sampling scheme.
The curves in this nomogram were computed from a table of percentage points of the χ² distribution. For v > 30 Wilson and Hilferty's approximation to the χ² distribution was used.     

Generalized piecewise linear histograms

We extend the concept of piecewise linear histogram introduced recently by Beirlant, Berlinet and Györfi. The disadvantage of that histogram is that in many models it takes on negative values with probability close to 1. We show that for a wide set of models, the extended class of estimates contains a bona fide density with probability tending to 1 as the sample size n increases to infinity. The mean integrated absolute error in the extended class of estimators decreases with the same rate n^–2/5 as in the original narrower class.     

On the generalized Poisson distribution

We use Euler's difference lemma to prove that, for θ > 0 and 0 ≤λ < 1, the function P _n defined on the non-negative integers by
P _n (θ, λ) = [θ(θ + n λ) ^{n −1}/ n !]e^{− n λ−θ}
defines a probability distribution, known as the Generalized Poisson Distribution.     

A note on the upper bound for the distance in total variation between the binomal and the Poisson distribution

Abstract  The total variation distance between the binomial B ( n, p ) distribution and the Poisson P ( np ) distribution is smaller than 2 ^1/2 p (1- p )^-1/2 according to V ERVAAT [4], [5]. We shall sharpen this inequality by using a result due to K EMPERMAN [1], C SISZÁR [2] and K ULLBACK [3].     

Uitbijternegerende schatters bij duplobepalingen

Outlyer-ignoring estimators for measurement in duplo.
By hypothesis a measurement u is the sum of two independent random variables, the normal random variable with expectation μ, and standard error σ, and a random error φ:

Basically two independent measurements u₁ and u₂ over u are to give the estimate x=1/2(u₁+ u₂) over μ.
However, to reduce the effect of the error φ on a final estimate of μ, one adds, according to a common practice, a third or even a fourth measurement u₃, u₄, in the case that the basic pair differs by more than a number A. For this extended set of measurements two outlyer-ignoring estimator y and z of μ are defined, and investigated against three specifications fo the error φ. Also an outlyer-ignoring estimate of σ is considered, and its application is illustrated by an example.     

First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties

Some properties of a first-order integer-valued autoregressive process (INAR)) are investigated. The approach begins with discussing the self-decomposability and unimodality of the 1-dimensional marginals of the process {X_n} generated according to the scheme X_n=α° X _n-i +e_n, where α° X _n-1 denotes a sum of X_{n - 1}, independent 0 - 1 random variables Y^(n-1), independent of X _n-1 with Pr -( y ^{(n - 1)}= 1) = 1 - Pr ( y ^(n-i)= 0) =α. The distribution of the innovation process ( e _n) is obtained when the marginal distribution of the process ( X _n) is geometric. Regression behavior of the INAR(1) process shows that the linear regression property in the backward direction is true only for the Poisson INAR(1) process.     

Efficiency of iterated WLS in the linear model with completely unknown heteroskedasticity

Iterated weighted least squares (IWLS) is investigated for estimating the regression coefficients in a linear model with symmetrically distributed errors. The variances of the errors are not specified; it is not assumed that they are unknown functions of the explanatory variables nor that they are given in some parametric way.
IWLS is carried out in a random number of steps, of which the first one is OLS. In each step the error variance at time t is estimated with a weighted sum of m squared residuals in the neighbourhood of t and the coefficients are estimated using WLS. Furthermore an estimate of the co-variance matrix is obtained. If this estimate is minimal in some way the iteration process is stopped.
Asymptotic properties of IWLS are derived for increasing sample size n . Some particular cases show that the asymptotic efficiency can be increased by allowing more than two steps. Even asymptotic efficiency with respect to WLS with the true error variances can be obtained if m is not fixed but tends to infinity with n and if the heteroskedasticity is smooth.     

The geometry of log-normal and related distributions and an application to tracer-dilution curves

Summary
Most of a typical log-normal curve lies very close to two straight lines, namely the tangents at its points of inflection. This holds good for nearly all the rising part of the curve and for most of the top half beyond the maximum.
When log_e (t – to) is normally distributed with standard deviation , whether the whole curve can be observed or not, , to, the mean, the mode and the area under the whole curve can be derived from the triangle formed by these two tangents and the base line. The triangle also provides a simple test of goodness-of-fit. The corresponding inflection triangles of two other skew distributions are investigated in outline namely a chi² (or gamma) distribution and one arising in Brownian movement or random walk in one dimension with drift. These are compared with the log-normal one. The normal distribution and its inflection triangle are given as a limiting case of the log-normal one and it is shown that the curve on the cover of Statistica Neerlandica cannot be a normal one.
The application is to the extrapolation problem in heart dye-dilution curves, which record the distribution of passage times of an indicator through the lungs and heart. The usual semi-log extrapolation for determining the area under the complete curve is discussed critically; it is shown how the inflection triangle provides a simpler procedure that is probably more accurate. The general formulae, an outline of the application and numerical tables for 0 < 0.8 are given in the main text; the mathematical derivations of all the basic formulae are in an appendix (sections 11 to 16).
The information needed for practical applications without studying the theory is contained in sections 3 to 6 and the associated tables and figures.     

Approximation for the cumulative and inverse gamma distribution

The gamma distribution function can be expressed in terms of the Normal distribution and density functions with sufficient accuracy for most practical purposes.
The distribution function for the density x^Λ-1e^-x/μ/μ^ΛΓ(A) on 0 -R(Λ){(1 + 1/1 2Λ) φ(z) + 11 -z/4Λ^1/2+2(z²+ 2)/45Λ] φ(z) /3 Λ^1/2} where φ(z)≅1/[1 +_e^{-2z(√2/π+z2 /28)}] and φ(z) = e^{-z2 /2}/√2π are the Normal distribution and density functions, y is the appropriate root of y-y²/6+y³/36-y⁴/270= In (x/Λμ), z= Λ^1/2 y, and R( Λ) is the remainder term in Stirling's approximation for In Γ(Λ).     

On residuals and their autocorrelations in fitted time series models

Summary When discrete autoregressive-moving average time series are fitted by least squares, both the residuals and their autocorrelations are for large n representable as singular linear transformations of the true errors (or white noise) and their autocomlations, respectively, and the matrices of these transformations arc both of the form I-X(X'X) ^-1X, where the rank of X is the number of parameters estimated. However, the large-sample properties of these two sets of statistics are fundamentally different, a phenomenon which is of considerable importance for the use of the residual autocorrelations in performing tests of fit of these models.     

Superpositie van twee frequentieverdelingen

Summary  (Superposition of two frequency distributions)
Notation:
n: number of observations
M: arithmetic mean
: standard deviation
μ_r: r^th moment coefficient
β₁: coefficient of skewness
β₂: coefficient of kurtosis.
The suffixes a and b apply to the component distributions. The suffix t applies to the resulting distributions.

The problem: Given the first r moments of two frequency distributions (to begin with μ₀). Find the first r moments of the distribution resulting from superposition of the two components ( r ≥ 5 ).
Formulae [1]. … [ 5 ] (§ 3 ) give the results in their most general form up to μ₄.
Some special cases are treated in § 4, and eight different cases of superposition of two normal distributions in § 5.
In § 6 some remarks are made about the reverse situation, i.e. the splitting into two normal components of a combined frequency distribution.     

Distance between sampling with and without replacement

Summary Two random samples of size n are taken from a set containing N objects of H types, first with and then without replacement. Let d be the absolute (L₁-)distance and I the K ullback -L eibler information distance between the distributions of the sample compositions without and with replacement. Sample composition is meant with respect to types; it does not matter whether order of sampling is included or not. A bound on I and d is derived, that depends only on n, N, H. The bound on I is not higher than 2 I. For fixed H we have d 0, I 0 as N if and only if n/N 0. Let W _r be the epoch at which for the r-th time an object of type I appears. Bounds on the distances between the joint distributions of W ₁., W _r without and with replacement are given.