Closed forms for asymptotic bias and variance in autoregressive models with unit roots

AbstractFor a first-order autoregressive AR(1) model with zero initial value, xt = αxt−1 + εt, we provide closed-form analytical expressions for the asymptotic bias and variance of the maximum likelihood (ML) estimator α = ∑1n xtxt−1∑1n−1 xt2 when ¦α¦ = 1. For the bias, numerical accuracy of up to six significant digits is achieved for sample sizes n > 100

Estimation problems associated with the Weibull distribution

Series in descending powers of the sample size are developed for the moments of the coefficient of variation v* for the Weibull distribution F(t) = 1 -exp(-(t/b)/sup c/). A similar series for the moments of the estimator c* of the shape parameter c are derived from these. Comparisons are made with basic asymptotic assessments for the means and variances. From the first four moments, approximations are given to the distribution of v* and c*. In addition, an almost unbiased estimator of c is given when a sample is provided with the value of v*. Comments are given on the validity of the asymptotically normal assessments of the distributions

Asymptotic series and Stieltjes continued fractions for a gamma function ratio

AbstractAn analysis is given for the expansion (60 terms) of a gamma function ratio discussed by Stieltjes and others. A Stieltjes continued fraction is derived, affording lower and upper bound (but lacking a rigorous proof), along with continued fraction for the odd and even series

Continued fractions and the polygamma functions

AbstractIn a previous study we have shown that the polygamma functions (derivatives of the logarithm of the gamma function) relate to Stieltjes transforms in the square of the argument. These transforms in turn may be converted to Stieltjes continued fractions; in the background is a determined Stieltjes moment problem.In the present study we use the Hamburger form of the Stieltjes integral to produce a set of real monotonic increasing and monotonic decreasing approximants to each of the real and imaginary parts of a polygamma function when the argument is complex. The approximants involve rational fractions which appear to be new.Special attention is given to ln Γ(z) and the psi function

Exact Moments for Autor1egressive and Random walk Models for a Zero or Stationary Initial Value

For a first-order autoregressive AR(1) model with zero initial value, x i = ax i−1 ,_, + e i, we provide the bias, mean squared error, skewness, and kurtosis of the maximum likelihood estimator â. Brownian motion approximations by Phillips (1977, Econometrica 45, 463–485; 1978, Biometrika 65, 91–98; 1987, Econometrica 55, 277–301), Phillips and Perron (1988, Biometrika 75, 335–346), and Perron (1991, Econometric Theory 7, 236–252; 1991, Econometrica 59, 211–236), among others, yield an elegant unified theory but do not yield convenient formulas for calibration of skewness and kurtosis. In addition to the usual stationary case |α|

Replenishment-Depletion Urn in Equilibrium

Bernard's urn, beta integral transforms, finite difference calculus, generating functions, hypergeometric distributions, hypergeometric functions, moments, replenishment-depletion urn,

Moment series for moment estimators of the parameters of a Weibull density

Taylor series for the first four moments of the coefficients of variation in sampling from a 2-parameter Weibull density are given: they are taken as far as the coefficient of n/sup -24/. From these a four moment approximating distribution is set up using summatory techniques on the series. The shape parameter is treated in a similar way, but here the moment equations are no longer explicit estimators, and terms only as far as those in n/sup -12/ are given. The validity of assessed moments and percentiles of the approximating distributions is studied. Consideration is also given to properties of the moment estimator for 1/c