Density of the Ratio of Two Normal Random Variables and Applications

Pham‐Gia, T.; Turkkan, N.; Marchand, Éric

doi:10.1080/03610920600683689

Cited by 109 publications

(96 citation statements)

References 15 publications

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“…A non-trivial problem arises in case both variables are random, and often correlated, as their ratio has an unknown, and possibly bi-modal distribution (Pham-Gia et al, 2006). There is an ongoing research in this field and several solutions to this problem have been proposed (e.g.…”

Section: Welfare M E a S U R E Smentioning

confidence: 99%

Measuring network effects in mobile telecommunications markets with stated-preference valuation methods

Czajkowski

Sobolewski

2011

IJMNE

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in ABSTRACTThis paper demonstrates how stated-preference methods can be applied to modeling consumers' preferences in the field of mobile telecommunications, and to measuring and the valuation of network effects. We illustrate this with a case study of mobile phone operators in Poland. We utilize the Choice Experiment method and present the respondents with hypothetical choices of mobile phone operators, while explicitly controlling for network effects in the form of other users in the same network. Based on the hypothetical choices consumers make we construct a conditional random parameters multinomial logit model to analyze their preferences. This approach allows us to calculate welfare effects associated with alternatives, as well as marginal rates of substitution (and hence implicit prices) of the attributes used to describe the choices, such as operator brand and distribution of family and friends between available mobile networks. The latter constitutes a network effect as consumer's utility is influenced by the number (or ratio) of members of his or her family, friends and other users subscribed to the same operator. Our results confirm the existence of a strong network effect, which is related to the size of the social network group a particular subscriber belongs to, rather than the absolute size of the mobile operator's customer base. We observe that there are two sources of this 'gross' network effect -pecuniary (arising from possible price discounts for on-net calls) and non-pecuniary, and demonstrate a way to disaggregate them. In addition, we find that brand perception and brand loyalty are important determinants of operator choice. Finally, through the application of a non-market valuation method we are able to calculate monetary values of the network effect and brand loyalty, and both turn out to be relatively high. The results might be of a particular interest to mobile phone operators and regulatory authorities -we find that the capacity for vigorous price competition between mobile operators is limited due to significant non-price barriers which mitigate subscribers' mobility in the market. We demonstrate a way to measure these effects in monetary terms based on modeling of consumer preferences.

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Section: Welfare M E a S U R E Smentioning

confidence: 99%

Measuring network effects in mobile telecommunications markets with stated-preference valuation methods

Czajkowski

Sobolewski

2011

IJMNE

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“…In this example, we show how mathStatica's new TransformProduct function solves a celebrated problem (Tooze, Kosmelj, and Blejec 2004;Fieller 1932;Geary 1930;Hayya, Armstrong, and Gressis 1975;Hinkley 1969;Kamerud 1978;Marsaglia 1965Marsaglia , 2006Pham-Gia, Turkkan, and Marchand 2006;Rose and Smith 2002): What is the PDF of the ratio of two general normal distributions, N (µ 1 , σ 1 ) and N (µ 2 , σ 2 )? To see this, note that density g(y) is the PDF of Y = 1/Z where Z ∼N (µ 2 , σ 2 ).…”

Section: Core Functionsmentioning

confidence: 99%

mathStatica2.5

Stokes¹

2012

J. Stat. Soft., S. Rev.

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“…Both Marsaglia ([11], [12]) and Pham-Gia et. al [13], give closed form expressions of the resulting density and distribution functions. The most applicable form results from the fact that the denominator ofβ CS , i.e.…”

Section: Asymptotic Distribution Ofβ Csmentioning

confidence: 99%

On the asymptotic distribution of an alternative measure of kurtosis

Suaray

2015

IJASP

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Pearson defined the fourth standardized moment of a symmetric distribution as its kurtosis. There has been much discussion in the literature concerning both the meaning of kurtosis, as well as the effectiveness of the classical sample kurtosis as a reliable measure of peakedness and tail weight. In this paper, we consider an alternative measure, developed by Crow and Siddiqui [1], used to describe kurtosis. Its value is calculated for a number of common distributions, and a derivation of its asymptotic distribution is given. Simulations follow, which reveal an interesting connection to the literature on the ratio of normal random variables.

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Density of the Ratio of Two Normal Random Variables and Applications

Cited by 109 publications

References 15 publications

Measuring network effects in mobile telecommunications markets with stated-preference valuation methods

Measuring network effects in mobile telecommunications markets with stated-preference valuation methods

mathStatica2.5

On the asymptotic distribution of an alternative measure of kurtosis

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