# Truncation and Variance in Scale Mixtures

## Abstract

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* = ∞.

## Keywords

Hazard Rate Random Environment Lifetime Distribution Graph Covering Scale Mixture## Preview

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