Note on the $k$-Dimensional Jensen Inequality

Schaefer, Martin

doi:10.1214/aop/1176996102

Cited by 16 publications

(4 citation statements)

References 3 publications

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“…As the harmonic mean function is concave, [11] the Jensen's inequality applied to a Tdimensional random vector (Schaefer, 1976) gives us:…”

Section: Asymptotic Results Forbmentioning

confidence: 99%

A weighted Fama-MacBeth two-step panel regression procedure: asymptotic properties, finite-sample adjustment, and performance

Lee

2020

SEF

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Purpose In a recent paper, Yoon and Lee (2019) (YL hereafter) propose a weighted Fama and MacBeth (FMB hereafter) two-step panel regression procedure and provide evidence that their weighted FMB procedure produces more efficient coefficient estimators than the usual unweighted FMB procedure. The purpose of this study is to supplement and improve their weighted FMB procedure, as they provide neither asymptotic results (i.e. consistency and asymptotic distribution) nor evidence on how close their standard error estimator is to the true standard error. Design/methodology/approach First, asymptotic results for the weighted FMB coefficient estimator are provided. Second, a finite-sample-adjusted standard error estimator is provided. Finally, the performance of the adjusted standard error estimator compared to the true standard error is assessed. Findings It is found that the standard error estimator proposed by Yoon and Lee (2019) is asymptotically consistent, although the finite-sample-adjusted standard error estimator proposed in this study works better and helps to reduce bias. The findings of Yoon and Lee (2019) are confirmed even when the average R2 over time is very small with about 1% or 0.1%. Originality/value The findings of this study strongly suggest that the weighted FMB regression procedure, in particular the finite-sample-adjusted procedure proposed here, is a computationally simple but more powerful alternative to the usual unweighted FMB procedure. In addition, to the best of the authors’ knowledge, this is the first study that presents a formal proof of the asymptotic distribution for the FMB coefficient estimator.

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“…As the harmonic mean function is concave, [11] the Jensen's inequality applied to a Tdimensional random vector (Schaefer, 1976) gives us:…”

Section: Asymptotic Results Forbmentioning

confidence: 99%

A weighted Fama-MacBeth two-step panel regression procedure: asymptotic properties, finite-sample adjustment, and performance

Lee

2020

SEF

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“…( 3) cannot exceed the minimum of the expectations in Eq. ( 11); see also Schaefer (1976). In particular, the absolute deviation between the output objective U and the envelope objective Ū,…”

Section: Lemma 14 (Envelope Bound) For Anymentioning

confidence: 99%

Optimal Matching of Random Parts

Weber

2021

SSRN Journal

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“…The relevance of Jensen's inequality to operational risk capital estimation appears to be the joint fact that the only severities used (and permitted) in operational risk capital estimation are medium-to heavy-tailed, and the severity quantile being estimated is so extremely large: under these conditions, VaR appears to always be a convex function, like g( ), of the parameters of the severity distribution, which here is the vector β (we can visualize β as a single parameter without loss of generality as the multivariate case for Jensen's inequality is well established (see Schaefer, 1976). Consequently, the capital estimation, V = g( β ), will be biased upwards.…”

Section: Jensen's Inequalitymentioning

confidence: 99%

Estimating operational risk capital with greater accuracy, precision and robustness

Opdyke¹

2014

JOP

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The largest US banks and Systemically Important Financial Institutions are required by regulatory mandate to estimate the operational risk capital they must hold using an Advanced Measurement Approach (AMA) as defined by the Basel II/III Accords. Most of these institutions use the Loss Distribution Approach (LDA) which defines the aggregate loss distribution as the convolution of a frequency distribution and a severity distribution representing the number and magnitude of losses, respectively. Capital is a Value-at-Risk estimate of this annual loss distribution (i.e. the quantile corresponding to the 99.9%tile, representing a one-in-a-thousand-year loss, on average). In practice, the severity distribution drives the capital estimate, which is essentially a very large quantile of the estimated severity distribution. Unfortunately, when using LDA with any of the widely used severity distributions (i.e. heavy-tailed, skewed distributions), all unbiased estimators of severity distribution parameters appear to generate biased capital estimates due to Jensen's Inequality: VaR always appears to be a convex function of these severities' parameter estimates because the (severity) quantile being estimated is so large and the severities are heavy-tailed. The resulting bias means that capital requirements always will be overstated, and this inflation is sometimes enormous (sometimes even billions of dollars at the unit-of-measure level). Herein I present an estimator of capital that essentially eliminates this upward bias when used with any commonly used severity parameter estimator. The Reduced-bias Capital Estimator (RCE), consequently, is more consistent with regulatory intent regarding the responsible implementation of the LDA framework than other implementations that fail to mitigate, if not eliminate this bias. RCE also notably increases the precision of the capital estimate and consistently increases its robustness to violations of the i.i.d. data presumption (which are endemic to operational risk loss event data). So with greater capital accuracy, precision, and robustness, RCE lowers capital requirements at both the unit-of-measure and enterprise levels, increases capital stability from quarter to quarter, ceteris paribus, and does both while more accurately and precisely reflecting regulatory intent. RCE is straightforward to explain, understand, and implement using any major statistical software package.

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Note on the $k$-Dimensional Jensen Inequality

Cited by 16 publications

References 3 publications

A weighted Fama-MacBeth two-step panel regression procedure: asymptotic properties, finite-sample adjustment, and performance

A weighted Fama-MacBeth two-step panel regression procedure: asymptotic properties, finite-sample adjustment, and performance

Optimal Matching of Random Parts

Estimating operational risk capital with greater accuracy, precision and robustness

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