Only five moderate and large earthquakes (M_w ≥5.7) in India—three in the Indian shield region and two in the Himalayan arc region—have given rise to multiple strong ground-motion recordings. Near-source data are available for only two of these events. The Bhuj earthquake (M_w 7.6), which occurred in the shield region, gave rise to useful recordings at distances exceeding 550 km. Because of the scarcity of the data, we use the stochastic method to estimate ground motions. We assume that (1) S waves dominate at R < 100 km and Lg waves at R ≥ 100 km, (2) Q = 508f^0.48 is valid for the Indian shield as well as the Himalayan arc region, (3) the effective duration is given by fc^-1 + 0.05R, where fc is the corner frequency, and R is the hypocentral distance in kilometer, and (4) the acceleration spectra are sharply cut off beyond 35 Hz. We use two finite-source stochastic models. One is an approximate model that reduces to the ω²-source model at distances greater that about twice the source dimension. This model has the advantage that the ground motion is controlled by the familiar stress parameter, Δσ. In the other finite-source model, which is more reliable for near-source ground-motion estimation, the high-frequency radiation is controlled by the strength factor, sfact, a quantity that is physically related to the maximum slip rate on the fault. We estimate Δσ needed to fit the observed Amax and Vmax data of each earthquake (which are mostly in the far field). The corresponding sfact is obtained by requiring that the predicted curves from the two models match each other in the far field up to a distance of about 500 km. The results show: (1) The Δσ that explains Amax data for shield events may be a function of depth, increasing from ∼50 bars at 10 km to ∼400 bars at 36 km. The corresponding sfact values range from 1.0-2.0. The Δσ values for the two Himalayan arc events are 75 and 150 bars (sfact = 1.0 and 1.4). (2) The Δσ required to explain Vmax data is, roughly, half the corresponding value for Amax, while the same sfact explains both sets of data. (3) The available far-field Amax and Vmax data for the Bhuj mainshock are well explained by Δσ = 200 and 100 bars, respectively, or, equivalently, by sfact = 1.4. The predicted Amax and Vmax in the epicentral region of this earthquake are 0.80 to 0.95 g and 40 to 55 cm/sec, respectively.