We present J-band (1.15-1.35 mu m) spectroscopy of a sample of nine galaxie
s showing some degree of LINER activity (classical LINERs, weak-[O I] LINER
s, and transition objects), together with H-band spectroscopy for some of t
hem. A careful subtraction of the stellar continuum allows us to obtain rel
iable [Fe II] 1.2567 mu m/Pa beta line ratios. We conclude that different t
ypes of LINERs (i.e., photoionized by a stellar continuum or by an active g
alactic nucleus) cannot be easily distinguished based solely on the [Fe II]
1.2567 mu m/Pa beta line ratio. The emission-line properties of many LINER
s can be explained in terms of an aging starburst. The optical line ratios
of these LINERs are reproduced by a model with a metal-rich H II region com
ponent photoionized with a single stellar temperature T-* = 38,000 K, plus
a supernova remnant (SNR) component. The [Fe nl line is predominantly excit
ed by shocks produced by SNRs in starbursts and starburst-dominated LINERs,
while Pa beta tracks H II regions ionized by massive young stars. The cont
ribution from SNRs to the overall emission-line spectrum is constrained by
the [Fe II] 1.2567 mu m/Pa beta line ratio. Although our models for aging s
tarbursts are constrained only by these infrared lines, they consistently e
xplain the optical spectra of the galaxies also. The LINER-starburst connec
tion is tested by predicting the time dependence of the ratio of the ionizi
ng luminosity (L-ion) to the supernova rate (SNr), L-ion/(SNr). We predict
the relative number of starbursts to starburst-dominated LINERs (aging star
bursts) and show that it is in approximate agreement with survey findings f
or nearby galaxies.