On Bayesian selection of the best normal population using theKullback–Leibler divergence measure

In this paper, we use the Bayesian approach to study the problem of selecting the best population among k different populations π₁, ..., π_k (k≥2) relative to some standard (or control) population π₀. Here, π₀ is considered to be the population with the desired characteristics. The best population is defined to be the one which is closest to the ideal population π₀ . The procedure uses the idea of minimizing the posterior expected value of the Kullback–Leibler (KL) divergence measure of π _i from π₀. The populations under consideration are assumed to be multivariate normal. An application to regression problems is also presented. Finally, a numerical example using real data set is provided to illustrate the implementation of the selection procedure.     

The Efficiency of Some Distribution-Free Tests

Laten T₁ en T₂ twee toetsen zijn voor dezelfde hypothese θ=θ₀betreffende de waarde van een parameter θ, Zij verder de onbetrouwbaarheidsdrempel van beide toetsen gelijk aan α en het onderscheidingsvermogen tegen de alternatieve hypothese θ=θ₁ geliik aan 1-β. Indien toets T₁ nu n₁ waarnemingen vergt en toets T₂n₂ waarnemingen, dan wordt de relatieve doeltreffendheid (Eng.: efficiency) van toets T₁ ten opzichte van toets T₂ (als toetsen voor θ=θ₀ tegen θ=θ₁ gegeven door: e = n₂/n₁. Indien men de waarde van θ₁ op een bepaalde wijze naar θ₀ laat convergeren bij toenemende n₁, is het in vele gevallen, door gebruik te maken van een stelling van α en β Deze limiet-waarde wordt de asymptotische relatieve doeltreffendheid (volgens Pitman) genoemd. In dit artikel wordt een overzicht gegeven van hetgeen bekend is over de asymptotische relatieve doeltreffendheid van een aantal verdelingsvrije toetsen ten opzichte van de corresponderende standaardtoetsen.
De conclusie van de schrijver is, dat men bij het gebruik van verdelingsvrije methoden met een hoge doeltreffendheid (bijv. de symmetrietoets en de twee-steek-proeven-toets van Wilcoxon, de toets van Kruskal voor k steekproeven en de methode van m rangschikkingen) slechts zeer weinig informatie kan verliezen en dat zelfs het gebruik van minder doeltreffende verdelingsvrije methoden gerechtvaardigd kan zijn.     

On a crossroad of resampling plans: bootstrapping elementary symmetric polynomials

We investigate the validity of the bootstrap method for the elementary symmetric polynomials S ^{( k )} _n =( ⁿ _k )⁻¹Σ_{1≤ i 1}< ... < i _k ≤ n X _{i 1} ... X _{i k} of i.i.d. random variables X ₁, ..., X _n . For both fixed and increasing order k , as n→∞ the cases where μ=E X ₁[moe2]0, the nondegenerate case, and where μ=E X ₁=0, the degenerate case, are considered.     

Selection from Logistic populations

Assume k ( k ≥ 2) independent populations π₁, π₂μ_k are given. The associated independent random variables X_i,( i = 1,2,… k ) are Logistically distributed with unknown means μ₁, μ₂, μ_k and equal variances. The goal is to select that population which has the largest mean. The procedure is to select that population which yielded the maximal sample value. Let μ₍₁₎≤μ₍₂₎≤…≤μ_(k) denote the ordered means. The probability of correct selection has been determined for the Least Favourable Configuration μ₍₁₎=μ₍₂₎==μ_{(k – 1)}=μ_(k)–δ where δ > 0. An exact formula for the probability of correct selection is given.     

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.     

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 Simulation Approach to Nonparametric Empirical Bayes Analysis

We deal with general mixture of hierarchical models of the form m(x) = f_øf(x |θ) g (θ)dθ , where g(θ) and m(x) are called mixing and mixed or compound densities respectively, and θ is called the mixing parameter. The usual statistical application of these models emerges when we have data x_i, i = 1,…,n with densities f(x_i|θ_i) for given θ_i, and the θ₁ are independent with common density g(θ) . For a certain well known class of densities f(x |θ) , we present a sample-based approach to reconstruct g(θ) . We first provide theoretical results and then we use, in an empirical Bayes spirit, the first four moments of the data to estimate the first four moments of g(θ) . By using sampling techniques we proceed in a fully Bayesian fashion to obtain any posterior summaries of interest. Simulations which investigate the operating characteristics of our proposed methodology are presented. We illustrate our approach using data from mixed Poisson and mixed exponential densities.     

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.     

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.     

A test for equality of two exponential distributions

Abstract Let X ₁., X _n1 and Y ₁., Y _n1, be two independent random samples from exponential populations. The statistical problem is to test whether or not two exponential populations are the same, based on the order statistics X _[1],. X _[r1] and Y [₁],. Y _[rs] where 1 r₁ n ₁ and 1 r₂ n ₂. A new test is given and an asymptotic optimum property of the test is proved.     

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.     

Het verdelen van steekproeven over subpopulaties bij accountantscontroles

Summary “Stratificationprocedures for a typical auditing problem”. During the past ten years, much experience was gained in The Netherlands in using random sampling methods for typical auditing problems. Especially, a method suggested by VAN. HEERDEN [2] turned out to be very fruitful. In this method a register of entries is considered to be a population of T guilders, if all entries total up to T guilders. The sample size n₀ is determined in such a way that the probability β not to find any mistake in the sample, if a fraction p₀ or more of T is incorrect, is smaller than a preassigned value β₀. So n₀ should satisfy (l-p)ⁿ⁰≤β₀ for p≥p₀. A complication arises if it is not possible to postpone sampling until the whole population T is available. One then wants to take samples from a population which is growing up to T. Suppose one is going to take samples n_i from e.g. r subpopulations Using the minimax procedure, it is shown, that in this case one should choose the sizes n_i equal to (T_i/T)n₀. The minimax-value of the probability not to find any incorrect guilder in the r samples, taken together is equal to β₀.     

Time Series Analysis of Repeated Surveys: The State–space Approach

Cross sectional estimates from repeated surveys form a time series { y_t }. These estimates can be viewed as the sum y _t = Y _t + e _t of two processes, { Y _t }, the population process and { e _t }, the survey error process. Serial correlations in the latter series are usually present, mainly due to sample overlap. Other sources of data such as censuses, administrative records and demographic population counts are also available. The state–space modelling approach to the analysis of repeated surveys allows combining information from different sources, incorporating benchmarking constraints in a natural way. Results from these methods seem to compare favourably with those from X-11-ARIMA in filtering out survey errors.     

On divergence and convergence of sums of nonnegative random variables

Abstract  If X ₁, X ₂,… are exponentially distributed random variables thenσ^∞_{k= 1} X_k=∞ with probability 1 iff σ^∞_{k= 1} EX_k=∞. This result, which is basic for a criterion in the theory of Markov jump processes for ruling out explosions (infinitely many transitions within a finite time) is usually proved under the assumption of independence (see FREEDMAN (1971), p. 153–154 or BREI-MAN (1968), p. 337–338), but is shown in this note to hold without any assumption on the joint distribution. More generally, it is investigated when sums of nonnegative random variables with given marginal distributions converge or diverge whatever are their joint distributions.     

A central limit theorem for sums of correlated products

Consider a sequence of random points placed on the nonnegative integers with i.i.d. geometric (1/2) interpoint spacings y _i . Let x _i denote the numbers of points placed at integer i . We prove a central limit theorem for the partial sums of the sequence x ₀ y ₀, x ₁ y ₁, . . . The problem is connected with a question concerning different bootstrap procedures.     

On the asymptotically uniform distribution modulo 1 of extreme order statistics

Let (X_m)∞₁ be a sequence of independent and identically distributed random variables. We give sufficient conditions for the fractional part of rnax (X₁., X_n) to converge in distribution, as n ←∞ to a random variable with a uniform distribution on [0, 1).     

Substituut-variabelen in correlatieberekingen

The recently repeated assertion that in correlation analysis it makes little difference whether one variable (x₂) is used instead of another one (x₃), provided the coefficient of correlation (r₂₃) between x₂ and x₃ is high, is scrutinized.
To that purpose the ranges of coefficients of correlation with respect to the substitute variable are expressed in formula 3. Moreover, by way of example, extreme values of coefficients of simple correlation (r₁₃ and r₃₄), of multiple correlation (R_1.34 and R_3.14) and of regression (α₁₃ and α₁₄, α₃₁ and α₃₄) relating to the substitute variable, are calculated on the basis of empirical values of coefficients of simple correlation relating to the substituted and the remaining variables.
The outcome of those calculations are summarized in the tables 1 and 3, and in the graph.
Table 1 presents ranges of r₁₃ for given values of r₁₂ and r₂₃, table 3 shows extreme values of coefficients of single and multiple correlation and regression in case an additional variable x₄ is introduced and r₁₂, r₁₄, r₂₄ and r₂₃ are given. The graph shows an ellipse as the boundary of the inner closed domain of compatible values of r₁₃ and r₃₄.
Those results clearly indicate the need for caution in substituting one variable by another.     

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.     

Generalized densities of order statistics

Let X ₁, ... , X _n be independent identically distributed random variables with distribution F . We derive expressions for generalized joint 'densities' of order statistics of X ₁, ... , X _n , for arbitrary distributions F , in terms of Radon–Nikodym derivatives with respect to product measures based on F . We then give formulae for conditional distributions of order statistics and use them to derive results concerning Markov properties of order statistics, formulae for distributions of trimmed sums, and other useful representations. Our approach leads to simple and natural expressions which appear not to have been given before.     

A property of Stein's estimator for the mean of a multivariate normal distribution

A uniform bound on the risk (under squared error loss) of Stein's estimator Ψ₁ for the mean of the multivariate normal distribution is given. Using the bound, the asymptotic behaviour of the risk of Ψ₁ under a Bayesian assumption is obtained.