Logarithmic Ratio-Type Estimator of Population Coefficient of Variation

The estimation of population coefficient of variation is one of the challenging aspects in sampling survey techniques for the past decades and much effort has been employed to develop estimators to produce its efficient estimate. In this paper, we proposed logarithmic ratio type estimator for the estimating population coefficient of variation using logarithm transformation on the both population and sample variances of the auxiliary character. The expression for the mean squared error (MSE) of the proposed estimator has been derived using Taylor series first order approximation approach. Efficiency conditions of the proposed estimator over other estimators in the study has also been derived. The empirical study was conducted using two-sets of populations and the results showed that the proposed estimator is more efficient. This result implies that, the estimate of proposed estimator will be closer to the true parameter than the estimates of other estimators in the study.

Chi-Square and Student Bridge Distributions and the Behrens–Fisher Statistic

We prove that the Behrens–Fisher statistic follows a Student bridge distribution, the mixing coefficient of which depends on the two sample variances only through their ratio. To this end, it is first shown that a weighted sum of two independent normalized chi-square distributed random variables is chi-square bridge distributed, and secondly that the Behrens–Fisher statistic is based on such a variable and a standard normally distributed one that is independent of the former. In case of a known variance ratio, exact standard statistical testing and confidence estimation methods apply without the need for any additional approximations. In addition, a three pillar bridges explanation is given for the choice of degrees of freedom in Welch’s approximation to the exact distribution of the Behrens–Fisher statistic.

Contrasting GARCH Daily Variance Predictions Between Foreign Exchange Returns and Carry Trade Strategy Returns

GARCH does a better job predicting carry trade strategy returns than the foreign exchange returns. Using two time periods from 1998-2010 and then from 2010-2018 we find the daily variance to have dropped between period 1 and period 2 for both daily foreign exchange spot prices and daily carry trade strategy returns. Comparing daily spot price returns to daily carry trade strategy returns in each time period we find differences in the sample variances. Daily variance in the foreign exchange market has changed within the last twenty years. As one of the most liquid markets, the FX market has seen significant losses in both spot price returns and carry trade strategy returns. The current market place has historic low levels of volatility. Since the 2008 financial crash asset prices have steadily risen in levels reducing downside risk. In addition, foreign exchange rates have been stable. Given a decade’s worth of slow and steady growth coupled with stabilization has reduced the variance in returns. Popular predictive volatility model GARCH has a long-run variance component. The inclusion of such a parameter allows for the model to remember past financial crisis or “normal” times. Returns in the carry trade are marked by periods of severe downward risk and losses. The carry trade will often have long-term trends of small positive returns only to give back all of the returns in one downward move.

Exponential Autoregressive Parameters Estimation

The purpose of the research was to compare the least squares estimatate and the least absolute deviation estimate depending on the probability distribution of the renewal process of the autoregressive equation. To achieve this goal, the sequence of observations of the exponential autoregressive process was repeatedly reproduced using computer simulation, and the least squares estimate and the least absolute deviation estimate were calculated for each sequence. The resulting estimation sequences were used to calculate the sample variances of the least squares estimate and the least absolute deviation estimate. The best estimate was the one with the lowest sample variance. The quantitative measure for the estimates comparison was the sample relative efficiency of estimates, defined as the inverse ratio of their sample variances. Normal distribution, contaminated normal distribution, i.e. Tukey distribution, with different values of the proportion and intensity of contamination, logistic distribution, Laplace distribution and Student distribution with different degrees of freedom, in particular, with one degree of freedom, that is, Cauchy distribution, were used as models of probability distribution of the renewal process. For each probability distribution, asymptotic values of the sample relative efficiency were obtained with an unlimited increase in the sample size of the observations of the autoregressive process. Findings of research show that the least absolute deviation estimate is better than the least squares estimate for Laplace distribution and the contaminated normal distribution with sufficiently large levels of the proportion and intensity of contamination. In other cases, the least squares estimate is preferable.

Stratied Finite Empirical Bernstein Sampling

We derive a concentration inequality for the uncertainty in the mean computed by stratified random sampling, and provide an online sampling method based on this inequality.  Our concentration inequality is versatile and considers a range of factors including: the data ranges, weights, sizes of the strata, the number of samples taken, the estimated sample variances, and whether strata are sampled with or without replacement.  Sequentially choosing samples to minimize this inequality leads to a online method for choosing samples from a stratified population.  We evaluate and compare the effectiveness of our method against others for synthetic data sets, and also in approximating the Shapley value of cooperative games.  Results show that our method is competitive with the performance of Neyman sampling with perfect variance information, even without having prior information on strata variances.We also provide a multidimensional extension of our inequality and discuss future applications.