Large-sample quantile estimation in pareto laws

A large-sample method of estimation for the parameters of Pareto laws is investigatedo The estimates are derived by using a small subset of k sample quantiles out of the original observations. The optimum spacing of the k quantiles is also examined. A Monte Carlo study compares this method with the method of moments and that of maximum likelihood for a selected set of parameter values and sample sizes.     

Small sample distribution of the likelihood ratio test in the random effects model

In this paper we examine the small sample distribution of the likelihood ratio test in the random effects model which is often recommended for meta-analyses. We find that this distribution depends strongly on the true value of the heterogeneity parameter (between-study variance) of the model, and that the correct p-value may be quite different from its large sample approximation. We recommend that the dependence of the heterogeneity parameter be examined for the data at hand and suggest a (simulation) method for this. Our setup allows for explanatory variables on the study level (meta-regression) and we discuss other possible applications, too. Two data sets are analyzed and two simulation studies are performed for illustration.     

A p-value for testing the equivalence of the variances of a bivariate normal distribution

A p-value is developed for testing the equivalence of the variances of a bivariate normal distribution. The unknown correlation coefficient is a nuisance parameter in the problem. If the correlation is known, the proposed p-value provides an exact test. For large samples, the p-value can be computed by replacing the unknown correlation by the sample correlation, and the resulting test is quite satisfactory. For small samples, it is proposed to compute the p-value by replacing the unknown correlation by a scalar multiple of the sample correlation. However, a single scalar is not satisfactory, and it is proposed to use different scalars depending on the magnitude of the sample correlation coefficient. In order to implement this approach, tables are obtained providing sub-intervals for the sample correlation coefficient, and the scalars to be used if the sample correlation coefficient belongs to a particular sub-interval. Once such tables are available, the proposed p-value is quite easy to compute since it has an explicit analytic expression. Numerical results on the type I error probability and power are reported on the performance of such a test, and the proposed p-value test is also compared to another test based on a rejection region. The results are illustrated with two examples: an example dealing with the comparability of two measuring devices, and an example dealing with the assessment of bioequivalence.     

Large sample theory of the Langevin distribution

The Langevin (or von Mises-Fisher) distribution of random vector x on the unit sphere ω_q in $\text{R}$^q has a density proportional to exp κμ'x where μ'x is the scalar product of x with the unit modal vector μ and κ?0 is a concentration parameter. This paper studies estimation and tests for a wide variety of situations when the sample sizes are large. Geometrically simple test statistics are given for many sample problems even when the populations may have unequal concentration parameters.     

Least significant spacing for ‘one versus the rest’ normal populations

Consider sample means from k(≥2) normal populations where the variances and sample sizes are equal. The problem is to find the ‘least significant difference’ or ‘spacing’ (LSS) between the two largest means, so that if an observed spacing is larger we have confidence 1 - α that the population with largest sample mean also has the largest population mean.

When the variance is known it is shown that the maximum LSS occurs when k = 2, provided a < .2723. In other words, for any value of k we may use the usual (one-tailed) least significant difference to demonstrate that one population has a population mean greater than (or equal to) the rest.

When the variance is estimated bounds are obtained for the confidence which indicate that this last result is approximately correct.     

Semi-empirical pseudo-likelihood for estimating equations in the presence of missing responses

Consider a semiparametric model which parameterizes only the conditional distribution of Y given X, f(y|x,β), and allows the marginal distribution of X to be completely arbitrary. Under the semiparametric model, we develop semi-empirical pseudo-likelihood inference with estimating equation in the presence of missing responses. We define semi-empirical likelihood pseudo-score estimates for both the model parameter and the parameter in the estimating equation simultaneously. Also, we develop semi-empirical pseudo-likelihood ratio inference for them, respectively. A simulation was conducted to evaluate the finite sample properties of the proposed estimators and semi-empirical pseudo-likelihood approach.     

Simultaneous closeness among order statistics to population quantiles

We derive expressions for the probability that an individual order statistic is closest to the target parameter among the order statistics from a complete random sample. Results are given for random variables with bounded and complete support. We then apply these general results to location-scale parameter families of distributions with specific applications to estimation of percentiles. In this case, simultaneous-closeness probabilities depend upon the parameters through the value of p in the percentile and the sample size, n. Results are finally illustrated with the estimation of percentiles for normal and exponential distributions.     

A note on bias reduction of maximum likelihood estimates for the scalar skew t distribution

The skew t distribution is a flexible parametric family to fit data, because it includes parameters that let us regulate skewness and kurtosis. A problem with this distribution is that, for moderate sample sizes, the maximum likelihood estimator of the shape parameter is infinite with positive probability. In order to try to solve this problem, Sartori (2006) has proposed using a modified score function as an estimating equation for the shape parameter. In this note we prove that the resulting modified maximum likelihood estimator is always finite, considering the degrees of freedom as known and greater than or equal to 2.     

Improved variance bounds for unbiased estimation in inverse sampling

Sathe (1977) derived sharper variance bounds for inverse sample unbiased estimator of the negative binomial parameter p. In the present writing improved upper/lower variance bounds are achieved and the relative improvement is numerically illustrated.     

Confidence intervals for the scale parameter of exponential distribution based on Type II doubly censored samples

This paper deals with the problem of interval estimation of the scale parameter in the two-parameter exponential distribution subject to Type II double censoring. Base on a Type II doubly censored sample, we construct a class of interval estimators of the scale parameter which are better than the shortest length affine equivariant interval both in coverage probability and in length. The procedure can be repeated to make further improvement. The extension of the method leads to a smoothly improved confidence interval which improves the interval length with probability one. All improved intervals belong to the class of scale equivariant intervals.     

Sample size determination for the confidence interval of mean comparison adjusted by multiple covariates

The current method of determining sample size for confidence intervals does not accommodate multiple covariate adjustment. Under the normality assumption, the effect of multiple covariate adjustment on the standard error of the mean comparison is related to a Hotelling T ² statistic. Sample size can be calculated to obtain a desired probability of achieving a predetermined width in the confidence interval of the mean comparison with multiple covariate adjustment, given that the confidence interval includes the population parameter.     

On the behaviour of tests based on sample spacings for moderatesamples

We revisit the question about optimal performance of goodness-of-fit tests based on sample spacings. We reveal the importance of centering of the test-statistic and of the sample size when choosing a suitable test-statistic from a family of statistics based on power transformations of sample spacings. In particular, we find that a test-statistic based on empirical estimation of the Hellinger distance between hypothetical and data-supported distribution does possess some optimality properties for moderate sample sizes. These findings confirm earlier statements about the robust behaviour of the test-statistic based on the Hellinger distance and are in contrast to findings about the asymptotic (when sample size approaches infinity) of statistics such as Moran's and/or Greenwood's statistic. We include simulation results that support our findings.     

Change PP plot and continous sample quantile function

Functional inference recommends data analysis of a sample of n observations by functional and graphical representations of its probability models using various functions on 0 < u < 1, including the quantile function. This paper discusses: charge PP plots and a continuous version of the sample quantile function which use the mid-distinct value probability integral transform; comparison density functions; comparison interpretation of probability integral transform; maximum spacings method of one sample parameter estimation.     

An improved Cp criterion for spline smoothing

Spline smoothing is a popular technique for curve fitting, in which selection of the smoothing parameter is crucial. Many methods such as Mallows’ C_p, generalized maximum likelihood (GML), and the extended exponential (EE) criterion have been proposed to select this parameter. Although C_p is shown to be asymptotically optimal, it is usually outperformed by other selection criteria for small to moderate sample sizes due to its high variability. On the other hand, GML and EE are more stable than C_p, but they do not possess the same asymptotic optimality as C_p. Instead of selecting this smoothing parameter directly using C_p, we propose to select among a small class of selection criteria based on Stein's unbiased risk estimate (SURE). Due to the selection effect, the spline estimate obtained from a criterion in this class is nonlinear. Thus, the effective degrees of freedom in SURE contains an adjustment term in addition to the trace of the smoothing matrix, which cannot be ignored in small to moderate sample sizes. The resulting criterion, which we call adaptive C_p, is shown to have an analytic expression, and hence can be efficiently computed. Moreover, adaptive C_p is not only demonstrated to be superior and more stable than commonly used selection criteria in a simulation study, but also shown to possess the same asymptotic optimality as C_p.     

Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions

The skew normal model is a class of distributions that extends the Gaussian family by including a shape parameter. Despite its nice properties, this model presents some problems with the estimation of the shape parameter. In particular, for moderate sample sizes, the maximum likelihood estimator is infinite with positive probability. As a solution, we use a modified score function as an estimating equation for the shape parameter. It is proved that the resulting modified maximum likelihood estimator is always finite. For confidence intervals a quasi-likelihood approach is considered. When regression and scale parameters are present, the method is combined with maximum likelihood estimators for these parameters. Finally, also the skew t distribution is considered, which may be viewed as an extension of the skew normal. The same method is applied to this model, considering the degrees of freedom as known.     

A note on smoothing parameter selection for penalized spline smoothing

In nonparametric regression the smoothing parameter can be selected by minimizing a Mean Squared Error (MSE) based criterion. For spline smoothing one can also rewrite the smooth estimation as a Linear Mixed Model where the smoothing parameter appears as the a priori variance of spline basis coefficients. This allows to employ Maximum Likelihood (ML) theory to estimate the smoothing parameter as variance component. In this paper the relation between the two approaches is illuminated for penalized spline smoothing (P-spline) as suggested in Eilers and Marx Statist. Sci. 11(2) (1996) 89. Theoretical and empirical arguments are given showing that the ML approach is biased towards undersmoothing, i.e. it chooses a too complex model compared to the MSE. The result is in line with classical spline smoothing, even though the asymptotic arguments are different. This is because in P-spline smoothing a finite dimensional basis is employed while in classical spline smoothing the basis grows with the sample size.     

Empirical bayes rules for selecting good populations

A problem of selecting populations better than a control is considered. When the populations are uniformly distributed, empirical Bayes rules are derived for a linear loss function for both the known control parameter and the unknown control parameter cases. When the priors are assumed to have bounded supports, empirical Bayes rules for selecting good populations are derived for distributions with truncation parameters (i.e. the form of the pdf is f(x|θ)= p_i(x)c_i(θ)I_{(0, θ)}(x)). Monte Carlo studies are carried out which determine the minimum sample sizes needed to make the relative errors less than ε for given ε-values.     

A Note on the Fisher Information in Exponential Distribution

ABSTRACT

It is known that the Fisher information in any set of order statistics can be simplified to a sum of double integrals. In this article, we show that it can be further simplified to a sum of single integrals for the scale parameter of an exponential distribution. Moreover, we use the result and provide a simple method of obtaining the optimal spacing of order statistics.     

On the parameters of Zenga distribution

In 2010 Zenga introduced a new three-parameter model for distributions by size that can be used to represent income, wealth, financial and actuarial variables. This paper proposes a summary of its main properties, followed by a focus on the interpretation of the parameters in terms of inequality. The scale parameter μ is equal to the expectation, and it does not affect the inequality, while the two shape parameters α and θ are inverse and direct inequality indicators respectively. This result is obtained through stochastic orders based on inequality curves. A procedure to generate a random sample from Zenga distribution is also proposed. The second part of this article looks at the parameter estimation. Analytical solution of method of moments is obtained. This result is used as a starting point of numerical procedures to obtain maximum likelihood estimates both on ungrouped and grouped data. In the application, three empirical income distributions are considered and the aforementioned estimates are evaluated. A comparison with other well-known models is provided, by the evaluation of three goodness-of-fit indexes.