Bounds for the median of the negative binomial distribution

Summary Bounds are obtained for the median of the negative binomial distribution which are valid for all possible parameter values of the distribution when the median is defined as inf {x: P(X≤x)≥1/2}.     

Bounds for median and 50 percetage point of binomial and negative binomial distribution

Inequalities for comparing the binomial distribution function/operating characteristic function with value 0.5 have been established long ago. It is a trivial task to infer from these inequalities bounds for the 50 percentage point of the binomial operating characteristic function. However, it is also possible to use these inequalities to establish good bounds for the median of the binomial distribution function. By a relation of binomial distribution function/operating characteristic function and negative binomial distribution function/operating characteristic function we can get bounds for the respective parameters of negative binomial distribution, too.     

On the variance of the ratio estimator

Summary In an earlier paper [Rao 1966] an exact expression for the variance of the ratio estimator under theMidzuno-Sen sampling scheme is obtained and here we study some of the interesting properties of the coefficients involved in this expression which depend on the auxiliary information. Use of these coefficients is made of in finding out an exact expression for the Bias and Mean Square Error of the ratio estimator under Simple Random Sampling With-Out Replacement (SRSWOR) scheme.     

On the variance of the ratio estimator for midzuno-sen sampling scheme

Summary For the sampling scheme ofMidzuno [3] andSen [4], which provides unbiased ratio estimators an expression for the variance of the estimator does not seem to be available in literature. An expression for the same is derived in this note.     

Estimating the variance of the ratio estimator for theMidzuno-Sen sampling scheme

Summary The problem of estimating the variance of the ratio estimator for theMidzuno-Sen sampling scheme is further studied in this paper. Sufficient conditions are derived for which the suggested variance estimator is always positive definite.     

The combined inverse binomial sampling procedure

In the Combined Inverse Binomial Procedure (CIB), independent Bernoulli trials are performed one after another until for the first time either a fixed number of successes or a fixed number of failures are obtained. The sample size is the minimum of a pair of negative binomial variables. The distribution of the sample size and its expectation as well as the problems of estimation and testing of hypotheses in CIB are dealt with. It is seen that tests based on CIB procedures can lead to savings in sample size as compared to fixed sample size procedures in certain situations. It is also briefly indicated how the theory can be extended to the case of combined inverse multinomial procedures.     

Repeated substitution method: The ratio estimator for the population variance

This paper proposes a new unbiased estimator for the population variance in finite population sample surveys using auxiliary information. This estimator has a smaller mean squared error than the conventional unbiased estimator, the ratio estimator established by Isaki (1983) and it has the same precision than the regression estimator. Furthermore, it is a much more interesting estimator from the computation viewpoint.     

Bounds for the difference between median and mean of beta and negative binomial distributions

Summary We obtain bounds for the difference between the median and mean of the beta and negative binomial distributions using elementary methods.     

On the exact variance of an unbiased product type estimator and its comparison with the sample mean

We consider the usual product estimator adjusted for its bias, considered by Robson (1957). The exact variance of this estimator has been obtained through a direct method. Its superiority over the sample mean and the product estimator is demonstrated. The research of these authors has been partially supported by their respective NSERC grants A3661 and A3989.     

A simple ordered data estimator for inverse density weighted expectations

We consider estimation of means of functions that are scaled by an unknown density, or equivalently, integrals of conditional expectations. The “ordered data” estimator we provide is root n$n$ consistent, asymptotically normal, and is numerically extremely simple, involving little more than ordering the data and summing the results. No sample-size-dependent smoothing is required. A similarly simple estimator is provided for the limiting variance. The proofs include new limiting distribution results for functions of nearest-neighbor spacings. Potential applications include endogenous binary choice, willingness to pay, selection, and treatment models.     

The best quadratic estimator of the residual variance in regression analysis

De beste kwadratische schattingsfunctie van de storingsvariantie in regressie-analyse.
Dit artikel handelt over de schatting van de variantie σ² van de storingen in de regressieanalyse onder klassieke veronderstellingen: niet-stochastische waarden aangenomen door de verklarende variabelen en normaliteit, onafhankelijkheid en homoskedasticiteit van de storingen. Bekend is dat de schatting volgens maximale aannemelijkheid neerkomt op net bepalen van de kwadratensom van de volgens kleinste-kwadraten geschatte storingen en deling door T (het aantal waarne-mingen); voorts, dat de schatting die minimale variantie heeft binnen de klasse van schattingsfuncties die zuiver zijn en kwadratisch in de afhankelijke variabele (de beste zuivere kwadratische schattingsfunctie) gevonden wordt door genoemde kwadratensom te delen door T–A, waarbij λ het aantal te schatten coëfficiënten is [d.w.z. het aantal verklarende variabelen (+ 1 indien een constante term aanwezig is)]. Hier wordt aangetoond, dat de schattingsfunctie van σ² die een minimaal tweede moment heeft binnen de klasse van schattingsfuncties die kwadratisch zijn in de afhankelijke variabele (de beste kwadratische schattingsfunctie) gevonden wordt door de kwadratensom van de volgens kleinste kwadraten geschatte storingen te delen door T–Λ+ 2.     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

Locally minimum variance unbiased estimator in a discontinuous density function

The exact forms of the locally minimum variance unbiased estimators and their variances are given in the case of a discontinuous density function.     

Robust trend inference with series variance estimator and testing-optimal smoothing parameter

The paper develops a novel testing procedure for hypotheses on deterministic trends in a multivariate trend stationary model. The trends are estimated by the OLS estimator and the long run variance (LRV) matrix is estimated by a series type estimator with carefully selected basis functions. Regardless of whether the number of basis functions K is fixed or grows with the sample size, the Wald statistic converges to a standard distribution. It is shown that critical values from the fixed-K asymptotics are second-order correct under the large-K asymptotics. A new practical approach is proposed to select K that addresses the central concern of hypothesis testing: the selected smoothing parameter is testing-optimal in that it minimizes the type II error while controlling for the type I error. Simulations indicate that the new test is as accurate in size as the nonstandard test of Vogelsang and Franses (2005) and as powerful as the corresponding Wald test based on the large-K asymptotics. The new test therefore combines the advantages of the nonstandard test and the standard Wald test while avoiding their main disadvantages (power loss and size distortion, respectively).     

The effects of autocorrelation among errors on the consistency property of OLS variance estimator

In a linear regression model the ordinary least squares (OLS) variance estimator (S²) converges in probability to E(S²) even when the errors are autocorrelated. Of interest, however, is the rate of convergence. In this paper we shed some light on this question for the case of a linear trend model. In particular the relation between the rate of convergence and the correlation property of the errors is explored. It is shown that the retardation of the rate of convergence is not appreciable if the correlation is moderate, but it can be severe for extreme correlations.     

A generalization of an inequality on binomial coefficients

A generalization of an inequality on binomial coefficients solved by E. L. Johnson, D. Newman, K. Winston [1] is analyzed and at the worst the problem is reduced into the choice between two alternatives for the solutions. Supported by G.N.A.F.A. of the Consiglio Nazionale delle Ricerche.     

Some properties of the distribution of an operational ridge estimator

Metrika - The present paper investigates the properties of the sampling distribution of an operational general ridge estimator. Exact expression are given for the moments and their approximations...     

Variables sampling plans for the Poisson and the binomial

Variables sampling plans based upon continuous distributions are well known. The usual assumption is that a measurable characteristic associated with a product has a normal distribution, a case which has been treated extensively in the literature. Other continuous distributions, particularly the exponential, have also been used as models. In this paper we discuss variables sampling plans for situations in which the measurable characteristic has either a Poisson or a binomial distribution.