Truncation and Variance in Scale Mixtures
William C. Bauldry, Jaimie L. Hebert · Birkhäuser Boston eBooks · 1993
Let X be a continuous nonnegative random variable with density f, mean µ, and finite variance σ 2. Mullooly (1988), hereafter simply Mullooly, has shown that if f is positive on the interior of its support, $$\mathop {\lim }\limits_{x \to 0} f(x) >0\frac{\sigma }{\mu } > 1$$ , and $$\frac{\sigma }{\mu } >1$$ , then σ 2 may be increased by truncation. Denote by σ 2(t), the variance of the truncated random variable X t ≡ I(t, ∞)(X), where I A is an indicator on the set A. Specifically, Mullooly demonstrates that for densities satisfying these conditions, there exists a real number T > 0 such that σ 2(t) > σ 2 for all t∈ (0,T). We shall call T the variance inflation boundary for X t . When σ 2(t) > σ 2 for all t ∈ (0, ∞), we say that T = ∞.