In Bangladesh, drought has negative consequences on agriculture, environments, livelihood, and food security. However, a comprehensive statistical assessment of drought indices has rarely been found in the existing literature. To address this issue, firstly, this paper used daily temperature and precipitation data from the Bangladesh Meteorological Department (BMD) to calculate Standardized Precipitation Index (SPI) and Standardized Precipitation Evapotranspiration Index (SPEI). After that, we performed a statistical assessment, for instance, Pearson correlation coefficient, cross-correlation, cross-wavelet transform, and root mean square error, to identify the strengths of SPI and SPEI. Our findings showed that though both indices had a strong correlation with each other, SPEI performed better than SPI because evaporative demand has a positive impact on defining drought conditions in Bangladesh. Temperature and rainfall indices, for example, warm spell duration indicator (WSDI) and maximum amount of rain (Rx5day) that falls in five consecutive days, have been applied to find out the relationship between climate extremes and drought indices using cross-wavelet transform. Our results suggested that climate extremes such as WSDI and Rx5day have an influence on droughts in Bangladesh. Our results recommended that multi indices of drought assessment should be required in order to draw a robust conclusion.