On Kedlaya type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlaya-type inequality $$\mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big(x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big)$$ holds for an arbitrary $(x_n)$ ($\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_n)$ is a vector with corresponding (non-normalized) weights $(\lambda_n)$ and $\mathscr{M}_{i=1}^n(x_i,\lambda_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$\mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big)$$ holds for every $(x_n)$ and $(\lambda_n)$ such that the sequence $(\frac{\lambda_k}{\lambda_1+\cdots+\lambda_k})$ is decreasing.Comment: J. Inequal. Appl. (2018

European economic sentiment indicator: an empirical reappraisal

In the last five decades the European Economic Sentiment Indicator (ESI) has positioned itself as a high-quality leading indicator of overall economic activity. Relying on data from five distinct business and consumer survey sectors (industry, retail trade, services, construction and the consumer sector), ESI is conceptualized as a weighted average of the chosen 15 response balances. However, the official methodology of calculating ESI is quite flawed because of the arbitrarily chosen balance response weights. This paper proposes two alternative methods for obtaining novel weights aimed at enhancing ESI\u27s forecasting power. Specifically, the weights are determined by minimizing the root mean square error in simple GDP forecasting regression equations; and by maximizing the correlation coefficient between ESI and GDP growth for various lead lengths (up to 12 months). Both employed methods seem to considerably increase ESI\u27s forecasting accuracy in 26 individual European Union countries. The obtained results are quite robust across specifications