The vibration and acoustic signals collected from rotating machinery are often non-stationary and aperiodic, and it is a challenge to post-process and extract the defect sensitive health indicators. In this paper, we demonstrate how the estimated Hurst exponent of such measured data can be advantageous for analyzing non-stationary and aperiodic data due to its self-similarity and scale-invariance properties. To illustrate this, the paper demonstrates detection of fault diagnostics of a multi-stage gearbox operating under fluctuating speeds through estimated Hurst exponent of the raw vibration and acoustic signals as a health indicator. Thirteen health states of the gearbox are considered, and the raw vibration and acoustic signals are collected. The Hurst exponents are calculated using three different approaches: generalized Hurst exponent (q = 1, 2, 3, and 4), rescale range statistical (R/S) analysis, and dispersion analysis from the vibration and acoustic signals. Three different health indicator datasets are formulated and subjected to feature learning through conventional machine-learning (decision tree and support vector machine) and advanced machine-learning (deep-learning) classifiers. The effectiveness of these datasets while discriminating between the health states of the gearbox is investigated, yielding classification accuracies of 96.4% when compared with the individual health indicator datasets. The ability of the fault diagnosis and defect severity analysis with reduced reliance on the signal post-processing algorithms is demonstrated.