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In practice, managers often face a trade‐off of choosing between committed and contingent pricing under cost uncertainty because contingent pricing may increase the firm's profit but may decrease consumer surplus and market share. This article compares the two pricing strategies under cost uncertainty in terms of the firm's profit, consumer surplus, and market share. We find conditions under which committed pricing would lead to higher consumer surplus and market share than contingent pricing and conditions under vice versa. The results highlight the properties of consumers' valuation distribution in the trade‐offs of choosing between committed and contingent pricing, specifically the properties of the so‐called Mills' ratio which is defined as the reciprocal of hazard rate. According to the curvature of Mills' ratio, we define three types of distributions. We then categorize some commonly used continuous distributions into the three types which lead to different preferences in the trade‐offs. Finally, we fit two sets of household income data from China and the United States into income distributions as the proxies of consumers' valuation distributions in an economy to illustrate those trade‐offs embedded in the two pricing strategies. [ABSTRACT FROM AUTHOR]

Abstract We establish some new inequalities for the modified Bessel-type function λν,σ(β)(x) $\lambda _{\nu ,\sigma }^{(\beta )} (x )$ studied by Glaeske et al. [in J. Comput. Appl. Math. 118(1–2):151–168, 2000] as the kernel of an integral transformation that modifies Krätzel’s integral transformation. The inequalities obtained are closely related to the generalized Hurwitz–Lerch zeta function and complementary incomplete gamma function. We also deduce some useful inequalities for the modified Bessel function of the second kind Kν(x) $K_{\nu } (x )$ and Mills’ ratio M(x) $\mathsf{M} (x )$ as worthwhile applications of our main results.

Abstract We introduce logarithmic generalized Maxwell distribution motivated by Vodă (Math. Rep. 11:171-179, 2009), which is an extension of the generalized Maxwell distribution. Some interesting properties of this distribution are studied and the asymptotic distribution of the partial maximum of an i.i.d. sequence from the logarithmic generalized Maxwell distribution is gained. The expansion of the limit distribution from the normalized maxima is established under the optimal norming constants, which shows the rate of convergence of the distribution for normalized maximum to the extreme limit.

We introduce logarithmic generalized Maxwell distribution motivated by Vodă (Math. Rep. 11:171-179, 2009), which is an extension of the generalized Maxwell distribution. Some interesting properties of this distribution are studied and the asymptotic distribution of the partial maximum of an i.i.d. sequence from the logarithmic generalized Maxwell distribution is gained. The expansion of the limit distribution from the normalized maxima is established under the optimal norming constants, which shows the rate of convergence of the distribution for normalized maximum to the extreme limit. [ABSTRACT FROM AUTHOR]

Some lower and upper bounds of multivariate Gaussian probability are given based on the univariate Mills’ ratio. These bounds are sharper than known ones on the multivariate Mills’ ratio in many case. [ABSTRACT FROM PUBLISHER]

Recently, Parsa and Klugman (2011) proposed a generalization of ordinary least squares regression, which they called copula regression. Though theoretically appealing, implementation, especially calibration, of copula regression is generally more involved than for generalized linear models. In this paper a linear approximation to copula regression, for which implementation is similar to that for least squares regression, will be introduced. We proceed by investigating the connection between the proposed approximation to copula regression, and copula regression itself. In particular, we develop a set of criteria which ensure a predictable bias in the estimates from the linear approximation to copula regression. [ABSTRACT FROM AUTHOR]

Logarithmic general error distribution, an extension of the log-normal distribution, is proposed. Some interesting properties of the log GED are derived. These properties are applied to establish the asymptotic behavior of the ratio of probability densities and the ratio of the tails of the logarithmic general error and log-normal distributions, and to derive the asymptotic distribution of the partial maximum of an independent and identically distributed sequence obeying the log GED. [ABSTRACT FROM PUBLISHER]

The Halfin-Whitt regime, or the quality-and-efficiency-driven (QED) regime, for multiserver systems refers to a situation with many servers, a critical load, and yet favorable system performance. We apply this regime to the classical multiserver loss system with slow retrials. We derive nondegenerate limiting expressions for the main steady-state performance measures, including the retrial rate and the blocking probability. It is shown that the economies of scale associated with the QED regime persist for systems with retrials, although in situations when the load becomes extremely critical the retrials cause deteriorated performance. Most of our results are obtained by a detailed analysis of Cohen's equation that defines the retrial rate in an implicit way. The limiting expressions are established by studying prelimit behavior and exploiting the connection between Cohen's equation and Mills' ratio for the Gaussian and Poisson distributions. [ABSTRACT FROM AUTHOR]

Motivated by Tiku and Vaughan, we consider a type of distribution which is a generalization of the short-tailed symmetric distribution. In particular, we consider the asymptotic behavior of Mills’ ratios for the distribution family. Meanwhile, we obtain the asymptotic distribution of the partial maximum of an independent and identically distributed sequence from the distribution. [ABSTRACT FROM AUTHOR]

Laplace gave a divergent expansion for Mills’ ratio. Truncated to k terms, a bound is known for the remainder. We extend this result to repeated integrals of the normal density. Taylor series are also given for these. [ABSTRACT FROM AUTHOR]

Expressions are given for repeated upper tail integrals of the univariate normal density (and so also for the general Hermite function) for both positive and negative arguments. The expressions involve moments of the form E( x + i N) and E1 / ( x + N), where N is a unit normal random variable. Laplace transforms are provided for the Hermite functions and the moments. The practical use of these expressions is illustrated. [ABSTRACT FROM AUTHOR]

An inequality on the tail behavior of the general error distribution and an asymptotic Mills-type ratio are established. Two applications are provided. The first application considers the asymptotic behavior of the ratio of probability densities and the ratio of the tails of the general error and normal distributions. The second application establishes the asymptotic distribution of the partial maximum of an independent and identically distributed sequence from the general error distribution. [ABSTRACT FROM AUTHOR]

Motivated by Finner et al. (Commun. Stat., Theory Methods 37: 175–179, 2008), we investigate the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of the Laplace and logistic distributions. In particular, we consider the asymptotic behavior of the ratio of the pdfs (cdfs) of the Laplace and Student's t-distributions (likewise for the logistic and Student's t-distributions) as the degrees of freedom parameter approaches infinity in an appropriate way. As by products, we obtain Mills' ratios for the Laplace and logistic distributions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the monotonicity properties of some functions involving the Mills'' ratio of the standard normal law. From these we deduce some new functional inequalities involving the Mills'' ratio, and we show that the Mills'' ratio is strictly completely monotonic. At the end of this paper we present some Turán-type inequalities for Mills'' ratio. [Copyright &y& Elsevier]

We investigate the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of Student's t-distribution with ν > 0 degrees of freedom (tν for short) for ν tending to infinity when the argument x = xν of the pdf (cdf) depends on ν and tends to ± ∞ (-∞). To this end, we consider the ratio of the pdf's (cdf's) of the tν- and the standard normal distribution. Depending on the choice of the argument xν, the pdf-ratio (cdf-ratio) tends to 1, a fixed value greater than 1, or to ∞. As a byproduct, we obtain a result for Mill' ratio when xν → -∞. [ABSTRACT FROM AUTHOR]

For general linear models with normally distributed random errors, the probability of a Type II error decreases exponentially as a function of sample size. This potentially rapid decline reemphasizes the importance of performing power calculations. [ABSTRACT FROM AUTHOR]

We establish some new inequalities for the modified Bessel-type function λν,σ(β)(x) studied by Glaeske et al. [in J. Comput. Appl. Math. 118(1-2):151-168, 2000] as the kernel of an integral transformation that modifies Krätzel's integral transformation. The inequalities obtained are closely related to the generalized Hurwitz-Lerch zeta function and complementary incomplete gamma function. We also deduce some useful inequalities for the modified Bessel function of the second kind Kν(x) and Mills' ratio M(x) as worthwhile applications of our main results. [ABSTRACT FROM AUTHOR]